@testset "4.3.2.1 (a+b tan)^m (c+d tan)^n" begin
    (a, b, c, d, e, f, m, n, p, x, ) = @variables a b c d e f m n p x

    #= ::Package:: =#

    #= ::Title::Closed:: =#
    #=Integrands*of*the*form*(a+b*tan(e+f*x))^m*(d*tan(e+f*x))^n=#


    #= ::Section::Closed:: =#
    #=Integrands*of*the*form*(a+im*a*tan(e+f*x))^m*(d*tan(e+f*x))^n=#


    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*(a+im*a*tan(e+f*x))^m*(d*tan(e+f*x))^n=#


    #= ::Subsubsection::Closed:: =#
    #=m>0=#


    @test_int [tan(c + d*x)^5*(a + im*a*tan(c + d*x)), x, 6, (-im)*a*x - (a*log(cos(c + d*x)))/d + (im*a*tan(c + d*x))/d - (a*tan(c + d*x)^2)/(2*d) - ((im/3)*a*tan(c + d*x)^3)/d + (a*tan(c + d*x)^4)/(4*d) + ((im/5)*a*tan(c + d*x)^5)/d]
    @test_int [tan(c + d*x)^4*(a + im*a*tan(c + d*x)), x, 5, a*x - (im*a*log(cos(c + d*x)))/d - (a*tan(c + d*x))/d - ((im/2)*a*tan(c + d*x)^2)/d + (a*tan(c + d*x)^3)/(3*d) + ((im/4)*a*tan(c + d*x)^4)/d]
    @test_int [tan(c + d*x)^3*(a + im*a*tan(c + d*x)), x, 4, im*a*x + (a*log(cos(c + d*x)))/d - (im*a*tan(c + d*x))/d + (a*tan(c + d*x)^2)/(2*d) + ((im/3)*a*tan(c + d*x)^3)/d]
    @test_int [tan(c + d*x)^2*(a + im*a*tan(c + d*x)), x, 3, -(a*x) + (im*a*log(cos(c + d*x)))/d + (a*tan(c + d*x))/d + ((im/2)*a*tan(c + d*x)^2)/d]
    @test_int [tan(c + d*x)*(a + im*a*tan(c + d*x)), x, 2, (-im)*a*x - (a*log(cos(c + d*x)))/d + (im*a*tan(c + d*x))/d]
    @test_int [a + im*a*tan(c + d*x), x, 2, a*x - (im*a*log(cos(c + d*x)))/d]
    @test_int [cot(c + d*x)*(a + im*a*tan(c + d*x)), x, 2, im*a*x + (a*log(sin(c + d*x)))/d]
    @test_int [cot(c + d*x)^2*(a + im*a*tan(c + d*x)), x, 3, -(a*x) - (a*cot(c + d*x))/d + (im*a*log(sin(c + d*x)))/d]
    @test_int [cot(c + d*x)^3*(a + im*a*tan(c + d*x)), x, 4, (-im)*a*x - (im*a*cot(c + d*x))/d - (a*cot(c + d*x)^2)/(2*d) - (a*log(sin(c + d*x)))/d]
    @test_int [cot(c + d*x)^4*(a + im*a*tan(c + d*x)), x, 5, a*x + (a*cot(c + d*x))/d - ((im/2)*a*cot(c + d*x)^2)/d - (a*cot(c + d*x)^3)/(3*d) - (im*a*log(sin(c + d*x)))/d]
    @test_int [cot(c + d*x)^5*(a + im*a*tan(c + d*x)), x, 6, im*a*x + (im*a*cot(c + d*x))/d + (a*cot(c + d*x)^2)/(2*d) - ((im/3)*a*cot(c + d*x)^3)/d - (a*cot(c + d*x)^4)/(4*d) + (a*log(sin(c + d*x)))/d]
    @test_int [cot(c + d*x)^6*(a + im*a*tan(c + d*x)), x, 7, -(a*x) - (a*cot(c + d*x))/d + ((im/2)*a*cot(c + d*x)^2)/d + (a*cot(c + d*x)^3)/(3*d) - ((im/4)*a*cot(c + d*x)^4)/d - (a*cot(c + d*x)^5)/(5*d) + (im*a*log(sin(c + d*x)))/d]


    @test_int [tan(c + d*x)^4*(a + im*a*tan(c + d*x))^2, x, 6, 2*a^2*x - ((2*im)*a^2*log(cos(c + d*x)))/d - (2*a^2*tan(c + d*x))/d - (im*a^2*tan(c + d*x)^2)/d + (2*a^2*tan(c + d*x)^3)/(3*d) + ((im/2)*a^2*tan(c + d*x)^4)/d - (a^2*tan(c + d*x)^5)/(5*d)]
    @test_int [tan(c + d*x)^3*(a + im*a*tan(c + d*x))^2, x, 5, (2*im)*a^2*x + (2*a^2*log(cos(c + d*x)))/d - ((2*im)*a^2*tan(c + d*x))/d + (a^2*tan(c + d*x)^2)/d + (((2*im)/3)*a^2*tan(c + d*x)^3)/d - (a^2*tan(c + d*x)^4)/(4*d)]
    @test_int [tan(c + d*x)^2*(a + im*a*tan(c + d*x))^2, x, 3, -2*a^2*x + (2*im*a^2*log(cos(c + d*x)))/d + (a^2*tan(c + d*x))/d - (im*(a + im*a*tan(c + d*x))^3)/(3*a*d)]
    @test_int [tan(c + d*x)*(a + im*a*tan(c + d*x))^2, x, 3, -2*im*a^2*x - (2*a^2*log(cos(c + d*x)))/d + (im*a^2*tan(c + d*x))/d + (a + im*a*tan(c + d*x))^2/(2*d)]
    @test_int [(a + im*a*tan(c + d*x))^2, x, 2, 2*a^2*x - ((2*im)*a^2*log(cos(c + d*x)))/d - (a^2*tan(c + d*x))/d]
    @test_int [cot(c + d*x)*(a + im*a*tan(c + d*x))^2, x, 3, (2*im)*a^2*x + (a^2*log(cos(c + d*x)))/d + (a^2*log(sin(c + d*x)))/d]
    @test_int [cot(c + d*x)^2*(a + im*a*tan(c + d*x))^2, x, 3, -2*a^2*x - (a^2*cot(c + d*x))/d + ((2*im)*a^2*log(sin(c + d*x)))/d]
    @test_int [cot(c + d*x)^3*(a + im*a*tan(c + d*x))^2, x, 4, (-2*im)*a^2*x - ((2*im)*a^2*cot(c + d*x))/d - (a^2*cot(c + d*x)^2)/(2*d) - (2*a^2*log(sin(c + d*x)))/d]
    @test_int [cot(c + d*x)^4*(a + im*a*tan(c + d*x))^2, x, 5, 2*a^2*x + (2*a^2*cot(c + d*x))/d - (im*a^2*cot(c + d*x)^2)/d - (a^2*cot(c + d*x)^3)/(3*d) - ((2*im)*a^2*log(sin(c + d*x)))/d]
    @test_int [cot(c + d*x)^5*(a + im*a*tan(c + d*x))^2, x, 6, (2*im)*a^2*x + ((2*im)*a^2*cot(c + d*x))/d + (a^2*cot(c + d*x)^2)/d - (((2*im)/3)*a^2*cot(c + d*x)^3)/d - (a^2*cot(c + d*x)^4)/(4*d) + (2*a^2*log(sin(c + d*x)))/d]
    @test_int [cot(c + d*x)^6*(a + im*a*tan(c + d*x))^2, x, 7, -2*a^2*x - (2*a^2*cot(c + d*x))/d + (im*a^2*cot(c + d*x)^2)/d + (2*a^2*cot(c + d*x)^3)/(3*d) - ((im/2)*a^2*cot(c + d*x)^4)/d - (a^2*cot(c + d*x)^5)/(5*d) + ((2*im)*a^2*log(sin(c + d*x)))/d]


    @test_int [tan(c + d*x)^3*(a + im*a*tan(c + d*x))^3, x, 6, (4*im)*a^3*x + (4*a^3*log(cos(c + d*x)))/d - ((4*im)*a^3*tan(c + d*x))/d + (2*a^3*tan(c + d*x)^2)/d + (((4*im)/3)*a^3*tan(c + d*x)^3)/d - (11*a^3*tan(c + d*x)^4)/(20*d) - (tan(c + d*x)^4*(a^3 + im*a^3*tan(c + d*x)))/(5*d)]
    @test_int [tan(c + d*x)^2*(a + im*a*tan(c + d*x))^3, x, 4, -4*a^3*x + ((4*im)*a^3*log(cos(c + d*x)))/d + (2*a^3*tan(c + d*x))/d - ((im/2)*a*(a + im*a*tan(c + d*x))^2)/d - ((im/4)*(a + im*a*tan(c + d*x))^4)/(a*d)]
    @test_int [tan(c + d*x)*(a + im*a*tan(c + d*x))^3, x, 4, (-4*im)*a^3*x - (4*a^3*log(cos(c + d*x)))/d + ((2*im)*a^3*tan(c + d*x))/d + (a*(a + im*a*tan(c + d*x))^2)/(2*d) + (a + im*a*tan(c + d*x))^3/(3*d)]
    @test_int [(a + im*a*tan(c + d*x))^3, x, 3, 4*a^3*x - ((4*im)*a^3*log(cos(c + d*x)))/d - (2*a^3*tan(c + d*x))/d + ((im/2)*a*(a + im*a*tan(c + d*x))^2)/d]
    @test_int [cot(c + d*x)*(a + im*a*tan(c + d*x))^3, x, 5, (4*im)*a^3*x + (3*a^3*log(cos(c + d*x)))/d + (a^3*log(sin(c + d*x)))/d - (a^3 + im*a^3*tan(c + d*x))/d]
    @test_int [cot(c + d*x)^2*(a + im*a*tan(c + d*x))^3, x, 5, -4*a^3*x + (im*a^3*log(cos(c + d*x)))/d + ((3*im)*a^3*log(sin(c + d*x)))/d - (cot(c + d*x)*(a^3 + im*a^3*tan(c + d*x)))/d]
    @test_int [cot(c + d*x)^3*(a + im*a*tan(c + d*x))^3, x, 4, (-4*im)*a^3*x - ((2*im)*a^3*cot(c + d*x))/d - (4*a^3*log(sin(c + d*x)))/d - (a*cot(c + d*x)^2*(a + im*a*tan(c + d*x))^2)/(2*d)]
    @test_int [cot(c + d*x)^4*(a + im*a*tan(c + d*x))^3, x, 5, 4*a^3*x + (2*a^3*cot(c + d*x))/d - ((4*im)*a^3*log(sin(c + d*x)))/d - ((im/2)*a*cot(c + d*x)^2*(a + im*a*tan(c + d*x))^2)/d - (cot(c + d*x)^3*(a + im*a*tan(c + d*x))^3)/(3*d)]
    @test_int [cot(c + d*x)^5*(a + im*a*tan(c + d*x))^3, x, 6, (4*im)*a^3*x + ((4*im)*a^3*cot(c + d*x))/d + (2*a^3*cot(c + d*x)^2)/d - (((3*im)/4)*a^3*cot(c + d*x)^3)/d + (4*a^3*log(sin(c + d*x)))/d - (cot(c + d*x)^4*(a^3 + im*a^3*tan(c + d*x)))/(4*d)]
    @test_int [cot(c + d*x)^6*(a + im*a*tan(c + d*x))^3, x, 7, -4*a^3*x - (4*a^3*cot(c + d*x))/d + ((2*im)*a^3*cot(c + d*x)^2)/d + (4*a^3*cot(c + d*x)^3)/(3*d) - (((11*im)/20)*a^3*cot(c + d*x)^4)/d + ((4*im)*a^3*log(sin(c + d*x)))/d - (cot(c + d*x)^5*(a^3 + im*a^3*tan(c + d*x)))/(5*d)]


    @test_int [tan(c + d*x)^3*(a + im*a*tan(c + d*x))^4, x, 7, (8*im)*a^4*x + (8*a^4*log(cos(c + d*x)))/d - ((8*im)*a^4*tan(c + d*x))/d + (4*a^4*tan(c + d*x)^2)/d + (((8*im)/3)*a^4*tan(c + d*x)^3)/d - (67*a^4*tan(c + d*x)^4)/(60*d) - (tan(c + d*x)^4*(a^2 + im*a^2*tan(c + d*x))^2)/(6*d) - (7*tan(c + d*x)^4*(a^4 + im*a^4*tan(c + d*x)))/(15*d)]
    @test_int [tan(c + d*x)^2*(a + im*a*tan(c + d*x))^4, x, 5, -8*a^4*x + ((8*im)*a^4*log(cos(c + d*x)))/d + (4*a^4*tan(c + d*x))/d - ((im/3)*a*(a + im*a*tan(c + d*x))^3)/d - ((im/5)*(a + im*a*tan(c + d*x))^5)/(a*d) - (im*(a^2 + im*a^2*tan(c + d*x))^2)/d]
    @test_int [tan(c + d*x)*(a + im*a*tan(c + d*x))^4, x, 5, (-8*im)*a^4*x - (8*a^4*log(cos(c + d*x)))/d + ((4*im)*a^4*tan(c + d*x))/d + (a*(a + im*a*tan(c + d*x))^3)/(3*d) + (a + im*a*tan(c + d*x))^4/(4*d) + (a^2 + im*a^2*tan(c + d*x))^2/d]
    @test_int [(a + im*a*tan(c + d*x))^4, x, 4, 8*a^4*x - ((8*im)*a^4*log(cos(c + d*x)))/d - (4*a^4*tan(c + d*x))/d + ((im/3)*a*(a + im*a*tan(c + d*x))^3)/d + (im*(a^2 + im*a^2*tan(c + d*x))^2)/d]
    @test_int [cot(c + d*x)*(a + im*a*tan(c + d*x))^4, x, 6, (8*im)*a^4*x + (7*a^4*log(cos(c + d*x)))/d + (a^4*log(sin(c + d*x)))/d - (a^2 + im*a^2*tan(c + d*x))^2/(2*d) - (3*(a^4 + im*a^4*tan(c + d*x)))/d]
    @test_int [cot(c + d*x)^2*(a + im*a*tan(c + d*x))^4, x, 5, -8*a^4*x + ((4*im)*a^4*log(cos(c + d*x)))/d + ((4*im)*a^4*log(sin(c + d*x)))/d - (cot(c + d*x)*(a^2 + im*a^2*tan(c + d*x))^2)/d]
    @test_int [cot(c + d*x)^3*(a + im*a*tan(c + d*x))^4, x, 6, (-8*im)*a^4*x - (a^4*log(cos(c + d*x)))/d - (7*a^4*log(sin(c + d*x)))/d - (cot(c + d*x)^2*(a^2 + im*a^2*tan(c + d*x))^2)/(2*d) - ((3*im)*cot(c + d*x)*(a^4 + im*a^4*tan(c + d*x)))/d]
    @test_int [cot(c + d*x)^4*(a + im*a*tan(c + d*x))^4, x, 5, 8*a^4*x + (4*a^4*cot(c + d*x))/d - ((8*im)*a^4*log(sin(c + d*x)))/d - (a*cot(c + d*x)^3*(a + im*a*tan(c + d*x))^3)/(3*d) - (im*cot(c + d*x)^2*(a^2 + im*a^2*tan(c + d*x))^2)/d]
    @test_int [cot(c + d*x)^5*(a + im*a*tan(c + d*x))^4, x, 6, (8*im)*a^4*x + ((4*im)*a^4*cot(c + d*x))/d + (8*a^4*log(sin(c + d*x)))/d - ((im/3)*a*cot(c + d*x)^3*(a + im*a*tan(c + d*x))^3)/d - (cot(c + d*x)^4*(a + im*a*tan(c + d*x))^4)/(4*d) + (cot(c + d*x)^2*(a^2 + im*a^2*tan(c + d*x))^2)/d]
    @test_int [cot(c + d*x)^6*(a + im*a*tan(c + d*x))^4, x, 7, -8*a^4*x - (8*a^4*cot(c + d*x))/d + ((4*im)*a^4*cot(c + d*x)^2)/d + (23*a^4*cot(c + d*x)^3)/(15*d) + ((8*im)*a^4*log(sin(c + d*x)))/d - (cot(c + d*x)^5*(a^2 + im*a^2*tan(c + d*x))^2)/(5*d) - (((3*im)/5)*cot(c + d*x)^4*(a^4 + im*a^4*tan(c + d*x)))/d]
    @test_int [cot(c + d*x)^7*(a + im*a*tan(c + d*x))^4, x, 8, (-8*im)*a^4*x - ((8*im)*a^4*cot(c + d*x))/d - (4*a^4*cot(c + d*x)^2)/d + (((8*im)/3)*a^4*cot(c + d*x)^3)/d + (67*a^4*cot(c + d*x)^4)/(60*d) - (8*a^4*log(sin(c + d*x)))/d - (cot(c + d*x)^6*(a^2 + im*a^2*tan(c + d*x))^2)/(6*d) - (((7*im)/15)*cot(c + d*x)^5*(a^4 + im*a^4*tan(c + d*x)))/d]


    #= ::Subsubsection::Closed:: =#
    #=m<0=#


    @test_int [tan(c + d*x)^6/(a + im*a*tan(c + d*x)), x, 6, (5*x)/(2*a) + ((3*im)*log(cos(c + d*x)))/(a*d) - (5*tan(c + d*x))/(2*a*d) + (((3*im)/2)*tan(c + d*x)^2)/(a*d) + (5*tan(c + d*x)^3)/(6*a*d) - (((3*im)/4)*tan(c + d*x)^4)/(a*d) - tan(c + d*x)^5/(2*d*(a + im*a*tan(c + d*x)))]
    @test_int [tan(c + d*x)^5/(a + im*a*tan(c + d*x)), x, 5, (((-5*im)/2)*x)/a + (2*log(cos(c + d*x)))/(a*d) + (((5*im)/2)*tan(c + d*x))/(a*d) + tan(c + d*x)^2/(a*d) - (((5*im)/6)*tan(c + d*x)^3)/(a*d) - tan(c + d*x)^4/(2*d*(a + im*a*tan(c + d*x)))]
    @test_int [tan(c + d*x)^4/(a + im*a*tan(c + d*x)), x, 4, (-3*x)/(2*a) - ((2*im)*log(cos(c + d*x)))/(a*d) + (3*tan(c + d*x))/(2*a*d) - (im*tan(c + d*x)^2)/(a*d) - tan(c + d*x)^3/(2*d*(a + im*a*tan(c + d*x)))]
    @test_int [tan(c + d*x)^3/(a + im*a*tan(c + d*x)), x, 3, (((3*im)/2)*x)/a - log(cos(c + d*x))/(a*d) - (((3*im)/2)*tan(c + d*x))/(a*d) - tan(c + d*x)^2/(2*d*(a + im*a*tan(c + d*x)))]
    @test_int [tan(c + d*x)^2/(a + im*a*tan(c + d*x)), x, 3, x/(2*a) + (im*log(cos(c + d*x)))/(a*d) - im/(2*d*(a + im*a*tan(c + d*x)))]
    @test_int [tan(c + d*x)^1/(a + im*a*tan(c + d*x)), x, 2, -((im*x)/(2*a)) - 1/(2*d*(a + im*a*tan(c + d*x)))]
    @test_int [(a + im*a*tan(c + d*x))^(-1), x, 2, x/(2*a) + (im/2)/(d*(a + im*a*tan(c + d*x)))]
    @test_int [cot(c + d*x)/(a + im*a*tan(c + d*x)), x, 4, ((-im/2)*x)/a + log(sin(c + d*x))/(a*d) + 1/(2*d*(a + im*a*tan(c + d*x)))]
    @test_int [cot(c + d*x)^2/(a + im*a*tan(c + d*x)), x, 4, (-3*x)/(2*a) - (3*cot(c + d*x))/(2*a*d) - (im*log(sin(c + d*x)))/(a*d) + cot(c + d*x)/(2*d*(a + im*a*tan(c + d*x)))]
    @test_int [cot(c + d*x)^3/(a + im*a*tan(c + d*x)), x, 5, (((3*im)/2)*x)/a + (((3*im)/2)*cot(c + d*x))/(a*d) - cot(c + d*x)^2/(a*d) - (2*log(sin(c + d*x)))/(a*d) + cot(c + d*x)^2/(2*d*(a + im*a*tan(c + d*x)))]
    @test_int [cot(c + d*x)^4/(a + im*a*tan(c + d*x)), x, 6, (5*x)/(2*a) + (5*cot(c + d*x))/(2*a*d) + (im*cot(c + d*x)^2)/(a*d) - (5*cot(c + d*x)^3)/(6*a*d) + ((2*im)*log(sin(c + d*x)))/(a*d) + cot(c + d*x)^3/(2*d*(a + im*a*tan(c + d*x)))]


    @test_int [tan(c + d*x)^6/(a + im*a*tan(c + d*x))^2, x, 6, (-25*x)/(4*a^2) - ((6*im)*log(cos(c + d*x)))/(a^2*d) + (25*tan(c + d*x))/(4*a^2*d) - ((3*im)*tan(c + d*x)^2)/(a^2*d) - (25*tan(c + d*x)^3)/(12*a^2*d) + (((3*im)/2)*tan(c + d*x)^4)/(a^2*d*(1 + im*tan(c + d*x))) - tan(c + d*x)^5/(4*d*(a + im*a*tan(c + d*x))^2)]
    @test_int [tan(c + d*x)^5/(a + im*a*tan(c + d*x))^2, x, 5, (((15*im)/4)*x)/a^2 - (4*log(cos(c + d*x)))/(a^2*d) - (((15*im)/4)*tan(c + d*x))/(a^2*d) - (2*tan(c + d*x)^2)/(a^2*d) + (((5*im)/4)*tan(c + d*x)^3)/(a^2*d*(1 + im*tan(c + d*x))) - tan(c + d*x)^4/(4*d*(a + im*a*tan(c + d*x))^2)]
    @test_int [tan(c + d*x)^4/(a + im*a*tan(c + d*x))^2, x, 4, (9*x)/(4*a^2) + ((2*im)*log(cos(c + d*x)))/(a^2*d) - (9*tan(c + d*x))/(4*a^2*d) + (im*tan(c + d*x)^2)/(a^2*d*(1 + im*tan(c + d*x))) - tan(c + d*x)^3/(4*d*(a + im*a*tan(c + d*x))^2)]
    @test_int [tan(c + d*x)^3/(a + im*a*tan(c + d*x))^2, x, 6, -((3*im*x)/(4*a^2)) + log(cos(c + d*x))/(a^2*d) - 3/(4*a^2*d*(1 + im*tan(c + d*x))) - tan(c + d*x)^2/(4*d*(a + im*a*tan(c + d*x))^2)]
    @test_int [tan(c + d*x)^2/(a + im*a*tan(c + d*x))^2, x, 3, -(x/(4*a^2)) + (3*im)/(4*a^2*d*(1 + im*tan(c + d*x))) - im/(4*d*(a + im*a*tan(c + d*x))^2)]
    @test_int [tan(c + d*x)^1/(a + im*a*tan(c + d*x))^2, x, 3, ((-im/4)*x)/a^2 - 1/(4*d*(a + im*a*tan(c + d*x))^2) + 1/(4*d*(a^2 + im*a^2*tan(c + d*x)))]
    @test_int [tan(c + d*x)^0/(a + im*a*tan(c + d*x))^2, x, 3, x/(4*a^2) + (im/4)/(d*(a + im*a*tan(c + d*x))^2) + (im/4)/(d*(a^2 + im*a^2*tan(c + d*x)))]
    @test_int [cot(c + d*x)^1/(a + im*a*tan(c + d*x))^2, x, 4, (((-3*im)/4)*x)/a^2 + log(sin(c + d*x))/(a^2*d) + 3/(4*a^2*d*(1 + im*tan(c + d*x))) + 1/(4*d*(a + im*a*tan(c + d*x))^2)]
    @test_int [cot(c + d*x)^2/(a + im*a*tan(c + d*x))^2, x, 5, (-9*x)/(4*a^2) - (9*cot(c + d*x))/(4*a^2*d) - ((2*im)*log(sin(c + d*x)))/(a^2*d) + cot(c + d*x)/(a^2*d*(1 + im*tan(c + d*x))) + cot(c + d*x)/(4*d*(a + im*a*tan(c + d*x))^2)]
    @test_int [cot(c + d*x)^3/(a + im*a*tan(c + d*x))^2, x, 6, (((15*im)/4)*x)/a^2 + (((15*im)/4)*cot(c + d*x))/(a^2*d) - (2*cot(c + d*x)^2)/(a^2*d) - (4*log(sin(c + d*x)))/(a^2*d) + (5*cot(c + d*x)^2)/(4*a^2*d*(1 + im*tan(c + d*x))) + cot(c + d*x)^2/(4*d*(a + im*a*tan(c + d*x))^2)]


    @test_int [tan(c + d*x)^6/(a + im*a*tan(c + d*x))^3, x, 6, (55*x)/(8*a^3) + ((7*im)*log(cos(c + d*x)))/(a^3*d) - (55*tan(c + d*x))/(8*a^3*d) + (((7*im)/2)*tan(c + d*x)^2)/(a^3*d) - tan(c + d*x)^5/(6*d*(a + im*a*tan(c + d*x))^3) + (((13*im)/24)*tan(c + d*x)^4)/(a*d*(a + im*a*tan(c + d*x))^2) + (55*tan(c + d*x)^3)/(24*d*(a^3 + im*a^3*tan(c + d*x)))]
    @test_int [tan(c + d*x)^5/(a + im*a*tan(c + d*x))^3, x, 5, (((-25*im)/8)*x)/a^3 + (3*log(cos(c + d*x)))/(a^3*d) + (((25*im)/8)*tan(c + d*x))/(a^3*d) - tan(c + d*x)^4/(6*d*(a + im*a*tan(c + d*x))^3) + (((11*im)/24)*tan(c + d*x)^3)/(a*d*(a + im*a*tan(c + d*x))^2) + (3*tan(c + d*x)^2)/(2*d*(a^3 + im*a^3*tan(c + d*x)))]
    @test_int [tan(c + d*x)^4/(a + im*a*tan(c + d*x))^3, x, 7, -((7*x)/(8*a^3)) - (im*log(cos(c + d*x)))/(a^3*d) - tan(c + d*x)^3/(6*d*(a + im*a*tan(c + d*x))^3) + (3*im*tan(c + d*x)^2)/(8*a*d*(a + im*a*tan(c + d*x))^2) + (7*im)/(8*d*(a^3 + im*a^3*tan(c + d*x)))]
    @test_int [tan(c + d*x)^3/(a + im*a*tan(c + d*x))^3, x, 4, (im*x)/(8*a^3) + 3/(8*a^3*d*(1 + im*tan(c + d*x))) + (im*tan(c + d*x)^3)/(6*d*(a + im*a*tan(c + d*x))^3) - 1/(8*a*d*(a + im*a*tan(c + d*x))^2)]
    @test_int [tan(c + d*x)^2/(a + im*a*tan(c + d*x))^3, x, 4, -(x/(8*a^3)) - im/(6*d*(a + im*a*tan(c + d*x))^3) + (3*im)/(8*a*d*(a + im*a*tan(c + d*x))^2) - im/(8*d*(a^3 + im*a^3*tan(c + d*x)))]
    @test_int [tan(c + d*x)/(a + im*a*tan(c + d*x))^3, x, 4, ((-im/8)*x)/a^3 - 1/(6*d*(a + im*a*tan(c + d*x))^3) + 1/(8*a*d*(a + im*a*tan(c + d*x))^2) + 1/(8*d*(a^3 + im*a^3*tan(c + d*x)))]
    @test_int [(a + im*a*tan(c + d*x))^(-3), x, 4, x/(8*a^3) + (im/6)/(d*(a + im*a*tan(c + d*x))^3) + (im/8)/(a*d*(a + im*a*tan(c + d*x))^2) + (im/8)/(d*(a^3 + im*a^3*tan(c + d*x)))]
    @test_int [cot(c + d*x)/(a + im*a*tan(c + d*x))^3, x, 5, (((-7*im)/8)*x)/a^3 + log(sin(c + d*x))/(a^3*d) + 1/(6*d*(a + im*a*tan(c + d*x))^3) + 3/(8*a*d*(a + im*a*tan(c + d*x))^2) + 7/(8*d*(a^3 + im*a^3*tan(c + d*x)))]
    @test_int [cot(c + d*x)^2/(a + im*a*tan(c + d*x))^3, x, 6, (-25*x)/(8*a^3) - (25*cot(c + d*x))/(8*a^3*d) - ((3*im)*log(sin(c + d*x)))/(a^3*d) + cot(c + d*x)/(6*d*(a + im*a*tan(c + d*x))^3) + (11*cot(c + d*x))/(24*a*d*(a + im*a*tan(c + d*x))^2) + (3*cot(c + d*x))/(2*d*(a^3 + im*a^3*tan(c + d*x)))]


    @test_int [tan(c + d*x)^6/(a + im*a*tan(c + d*x))^4, x, 6, (-65*x)/(16*a^4) - ((4*im)*log(cos(c + d*x)))/(a^4*d) + (65*tan(c + d*x))/(16*a^4*d) - ((2*im)*tan(c + d*x)^2)/(a^4*d*(1 + im*tan(c + d*x))) + (31*tan(c + d*x)^3)/(48*a^4*d*(1 + im*tan(c + d*x))^2) - tan(c + d*x)^5/(8*d*(a + im*a*tan(c + d*x))^4) + (((7*im)/24)*tan(c + d*x)^4)/(a*d*(a + im*a*tan(c + d*x))^3)]
    @test_int [tan(c + d*x)^5/(a + im*a*tan(c + d*x))^4, x, 8, (15*im*x)/(16*a^4) - log(cos(c + d*x))/(a^4*d) + 15/(16*a^4*d*(1 + im*tan(c + d*x))) + (7*tan(c + d*x)^2)/(16*a^4*d*(1 + im*tan(c + d*x))^2) - tan(c + d*x)^4/(8*d*(a + im*a*tan(c + d*x))^4) + (im*tan(c + d*x)^3)/(4*a*d*(a + im*a*tan(c + d*x))^3)]
    @test_int [tan(c + d*x)^4/(a + im*a*tan(c + d*x))^4, x, 5, x/(16*a^4) - (3*im)/(16*a^4*d*(1 + im*tan(c + d*x))) + (im*tan(c + d*x)^4)/(8*d*(a + im*a*tan(c + d*x))^4) + tan(c + d*x)^3/(12*a*d*(a + im*a*tan(c + d*x))^3) + im/(16*d*(a^2 + im*a^2*tan(c + d*x))^2)]
    @test_int [tan(c + d*x)^3/(a + im*a*tan(c + d*x))^4, x, 5, (im*x)/(16*a^4) + 3/(16*a^4*d*(1 + im*tan(c + d*x))) + tan(c + d*x)^4/(8*d*(a + im*a*tan(c + d*x))^4) + (im*tan(c + d*x)^3)/(12*a*d*(a + im*a*tan(c + d*x))^3) - 1/(16*d*(a^2 + im*a^2*tan(c + d*x))^2)]
    @test_int [tan(c + d*x)^2/(a + im*a*tan(c + d*x))^4, x, 5, -(x/(16*a^4)) - im/(8*d*(a + im*a*tan(c + d*x))^4) + im/(4*a*d*(a + im*a*tan(c + d*x))^3) - im/(16*d*(a^2 + im*a^2*tan(c + d*x))^2) - im/(16*d*(a^4 + im*a^4*tan(c + d*x)))]
    @test_int [tan(c + d*x)/(a + im*a*tan(c + d*x))^4, x, 5, ((-im/16)*x)/a^4 - 1/(8*d*(a + im*a*tan(c + d*x))^4) + 1/(12*a*d*(a + im*a*tan(c + d*x))^3) + 1/(16*d*(a^2 + im*a^2*tan(c + d*x))^2) + 1/(16*d*(a^4 + im*a^4*tan(c + d*x)))]
    @test_int [(a + im*a*tan(c + d*x))^(-4), x, 5, x/(16*a^4) + (im/8)/(d*(a + im*a*tan(c + d*x))^4) + (im/12)/(a*d*(a + im*a*tan(c + d*x))^3) + (im/16)/(d*(a^2 + im*a^2*tan(c + d*x))^2) + (im/16)/(d*(a^4 + im*a^4*tan(c + d*x)))]
    @test_int [cot(c + d*x)/(a + im*a*tan(c + d*x))^4, x, 6, (((-15*im)/16)*x)/a^4 + log(sin(c + d*x))/(a^4*d) + 7/(16*a^4*d*(1 + im*tan(c + d*x))^2) + 15/(16*a^4*d*(1 + im*tan(c + d*x))) + 1/(8*d*(a + im*a*tan(c + d*x))^4) + 1/(4*a*d*(a + im*a*tan(c + d*x))^3)]
    @test_int [cot(c + d*x)^2/(a + im*a*tan(c + d*x))^4, x, 7, (-65*x)/(16*a^4) - (65*cot(c + d*x))/(16*a^4*d) - ((4*im)*log(sin(c + d*x)))/(a^4*d) + (31*cot(c + d*x))/(48*a^4*d*(1 + im*tan(c + d*x))^2) + (2*cot(c + d*x))/(a^4*d*(1 + im*tan(c + d*x))) + cot(c + d*x)/(8*d*(a + im*a*tan(c + d*x))^4) + (7*cot(c + d*x))/(24*a*d*(a + im*a*tan(c + d*x))^3)]


    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*(a+im*a*tan(e+f*x))^(m/2)*(d*tan(e+f*x))^n=#


    #= ::Subsubsection::Closed:: =#
    #=m>0=#


    @test_int [tan(c + d*x)^4*sqrt(a + im*a*tan(c + d*x)), x, 6, -((im*sqrt(2)*sqrt(a)*atanh(sqrt(a + im*a*tan(c + d*x))/(sqrt(2)*sqrt(a))))/d) + (8*im*sqrt(a + im*a*tan(c + d*x)))/(35*d) - (2*im*tan(c + d*x)^2*sqrt(a + im*a*tan(c + d*x)))/(35*d) + (2*tan(c + d*x)^3*sqrt(a + im*a*tan(c + d*x)))/(7*d) + (62*im*(a + im*a*tan(c + d*x))^(3/2))/(105*a*d)]
    @test_int [tan(c + d*x)^3*sqrt(a + im*a*tan(c + d*x)), x, 5, (sqrt(2)*sqrt(a)*atanh(sqrt(a + im*a*tan(c + d*x))/(sqrt(2)*sqrt(a))))/d - (8*sqrt(a + im*a*tan(c + d*x)))/(5*d) + (2*tan(c + d*x)^2*sqrt(a + im*a*tan(c + d*x)))/(5*d) - (2*(a + im*a*tan(c + d*x))^(3/2))/(15*a*d)]
    @test_int [tan(c + d*x)^2*sqrt(a + im*a*tan(c + d*x)), x, 3, (im*sqrt(2)*sqrt(a)*atanh(sqrt(a + im*a*tan(c + d*x))/(sqrt(2)*sqrt(a))))/d - (((2*im)/3)*(a + im*a*tan(c + d*x))^(3/2))/(a*d)]
    @test_int [tan(c + d*x)*sqrt(a + im*a*tan(c + d*x)), x, 3, -((sqrt(2)*sqrt(a)*atanh(sqrt(a + im*a*tan(c + d*x))/(sqrt(2)*sqrt(a))))/d) + (2*sqrt(a + im*a*tan(c + d*x)))/d]
    @test_int [sqrt(a + im*a*tan(c + d*x)), x, 2, ((-im)*sqrt(2)*sqrt(a)*atanh(sqrt(a + im*a*tan(c + d*x))/(sqrt(2)*sqrt(a))))/d]
    @test_int [cot(c + d*x)*sqrt(a + im*a*tan(c + d*x)), x, 6, (-2*sqrt(a)*atanh(sqrt(a + im*a*tan(c + d*x))/sqrt(a)))/d + (sqrt(2)*sqrt(a)*atanh(sqrt(a + im*a*tan(c + d*x))/(sqrt(2)*sqrt(a))))/d]
    @test_int [cot(c + d*x)^2*sqrt(a + im*a*tan(c + d*x)), x, 8, ((-im)*sqrt(a)*atanh(sqrt(a + im*a*tan(c + d*x))/sqrt(a)))/d + (im*sqrt(2)*sqrt(a)*atanh(sqrt(a + im*a*tan(c + d*x))/(sqrt(2)*sqrt(a))))/d - (cot(c + d*x)*sqrt(a + im*a*tan(c + d*x)))/d]
    @test_int [cot(c + d*x)^3*sqrt(a + im*a*tan(c + d*x)), x, 8, (7*sqrt(a)*atanh(sqrt(a + im*a*tan(c + d*x))/sqrt(a)))/(4*d) - (sqrt(2)*sqrt(a)*atanh(sqrt(a + im*a*tan(c + d*x))/(sqrt(2)*sqrt(a))))/d - ((im/4)*cot(c + d*x)*sqrt(a + im*a*tan(c + d*x)))/d - (cot(c + d*x)^2*sqrt(a + im*a*tan(c + d*x)))/(2*d)]


    @test_int [tan(c + d*x)^3*(a + im*a*tan(c + d*x))^(3/2), x, 7, (2*sqrt(2)*a^(3/2)*atanh(sqrt(a + im*a*tan(c + d*x))/(sqrt(2)*sqrt(a))))/d + (((2*im)/7)*a^2*tan(c + d*x)^3)/(d*sqrt(a + im*a*tan(c + d*x))) - (2*a^2*tan(c + d*x)^4)/(7*d*sqrt(a + im*a*tan(c + d*x))) - (64*a*sqrt(a + im*a*tan(c + d*x)))/(35*d) + (16*a*tan(c + d*x)^2*sqrt(a + im*a*tan(c + d*x)))/(35*d) - (76*(a + im*a*tan(c + d*x))^(3/2))/(105*d)]
    @test_int [tan(c + d*x)^2*(a + im*a*tan(c + d*x))^(3/2), x, 4, ((2*im)*sqrt(2)*a^(3/2)*atanh(sqrt(a + im*a*tan(c + d*x))/(sqrt(2)*sqrt(a))))/d - ((2*im)*a*sqrt(a + im*a*tan(c + d*x)))/d - (((2*im)/5)*(a + im*a*tan(c + d*x))^(5/2))/(a*d)]
    @test_int [tan(c + d*x)*(a + im*a*tan(c + d*x))^(3/2), x, 4, (-2*sqrt(2)*a^(3/2)*atanh(sqrt(a + im*a*tan(c + d*x))/(sqrt(2)*sqrt(a))))/d + (2*a*sqrt(a + im*a*tan(c + d*x)))/d + (2*(a + im*a*tan(c + d*x))^(3/2))/(3*d)]
    @test_int [(a + im*a*tan(c + d*x))^(3/2), x, 3, ((-2*im)*sqrt(2)*a^(3/2)*atanh(sqrt(a + im*a*tan(c + d*x))/(sqrt(2)*sqrt(a))))/d + ((2*im)*a*sqrt(a + im*a*tan(c + d*x)))/d]
    @test_int [cot(c + d*x)*(a + im*a*tan(c + d*x))^(3/2), x, 6, (-2*a^(3/2)*atanh(sqrt(a + im*a*tan(c + d*x))/sqrt(a)))/d + (2*sqrt(2)*a^(3/2)*atanh(sqrt(a + im*a*tan(c + d*x))/(sqrt(2)*sqrt(a))))/d]
    @test_int [cot(c + d*x)^2*(a + im*a*tan(c + d*x))^(3/2), x, 8, ((-3*im)*a^(3/2)*atanh(sqrt(a + im*a*tan(c + d*x))/sqrt(a)))/d + ((2*im)*sqrt(2)*a^(3/2)*atanh(sqrt(a + im*a*tan(c + d*x))/(sqrt(2)*sqrt(a))))/d - (im*a^2)/(d*sqrt(a + im*a*tan(c + d*x))) - (a^2*cot(c + d*x))/(d*sqrt(a + im*a*tan(c + d*x)))]
    @test_int [cot(c + d*x)^3*(a + im*a*tan(c + d*x))^(3/2), x, 9, (11*a^(3/2)*atanh(sqrt(a + im*a*tan(c + d*x))/sqrt(a)))/(4*d) - (2*sqrt(2)*a^(3/2)*atanh(sqrt(a + im*a*tan(c + d*x))/(sqrt(2)*sqrt(a))))/d - ((im/2)*a^2*cot(c + d*x))/(d*sqrt(a + im*a*tan(c + d*x))) - (a^2*cot(c + d*x)^2)/(2*d*sqrt(a + im*a*tan(c + d*x))) - (((5*im)/4)*a*cot(c + d*x)*sqrt(a + im*a*tan(c + d*x)))/d]


    @test_int [tan(c + d*x)^3*(a + im*a*tan(c + d*x))^(5/2), x, 7, (4*sqrt(2)*a^(5/2)*atanh(sqrt(a + im*a*tan(c + d*x))/(sqrt(2)*sqrt(a))))/d - (368*a^2*sqrt(a + im*a*tan(c + d*x)))/(105*d) + (92*a^2*tan(c + d*x)^2*sqrt(a + im*a*tan(c + d*x)))/(105*d) + (((38*im)/63)*a^2*tan(c + d*x)^3*sqrt(a + im*a*tan(c + d*x)))/d - (2*a^2*tan(c + d*x)^4*sqrt(a + im*a*tan(c + d*x)))/(9*d) - (472*a*(a + im*a*tan(c + d*x))^(3/2))/(315*d)]
    @test_int [tan(c + d*x)^2*(a + im*a*tan(c + d*x))^(5/2), x, 5, ((4*im)*sqrt(2)*a^(5/2)*atanh(sqrt(a + im*a*tan(c + d*x))/(sqrt(2)*sqrt(a))))/d - ((4*im)*a^2*sqrt(a + im*a*tan(c + d*x)))/d - (((2*im)/3)*a*(a + im*a*tan(c + d*x))^(3/2))/d - (((2*im)/7)*(a + im*a*tan(c + d*x))^(7/2))/(a*d)]
    @test_int [tan(c + d*x)*(a + im*a*tan(c + d*x))^(5/2), x, 5, (-4*sqrt(2)*a^(5/2)*atanh(sqrt(a + im*a*tan(c + d*x))/(sqrt(2)*sqrt(a))))/d + (4*a^2*sqrt(a + im*a*tan(c + d*x)))/d + (2*a*(a + im*a*tan(c + d*x))^(3/2))/(3*d) + (2*(a + im*a*tan(c + d*x))^(5/2))/(5*d)]
    @test_int [(a + im*a*tan(c + d*x))^(5/2), x, 4, ((-4*im)*sqrt(2)*a^(5/2)*atanh(sqrt(a + im*a*tan(c + d*x))/(sqrt(2)*sqrt(a))))/d + ((4*im)*a^2*sqrt(a + im*a*tan(c + d*x)))/d + (((2*im)/3)*a*(a + im*a*tan(c + d*x))^(3/2))/d]
    @test_int [cot(c + d*x)*(a + im*a*tan(c + d*x))^(5/2), x, 7, (-2*a^(5/2)*atanh(sqrt(a + im*a*tan(c + d*x))/sqrt(a)))/d + (4*sqrt(2)*a^(5/2)*atanh(sqrt(a + im*a*tan(c + d*x))/(sqrt(2)*sqrt(a))))/d - (2*a^2*sqrt(a + im*a*tan(c + d*x)))/d]
    @test_int [cot(c + d*x)^2*(a + im*a*tan(c + d*x))^(5/2), x, 7, ((-5*im)*a^(5/2)*atanh(sqrt(a + im*a*tan(c + d*x))/sqrt(a)))/d + ((4*im)*sqrt(2)*a^(5/2)*atanh(sqrt(a + im*a*tan(c + d*x))/(sqrt(2)*sqrt(a))))/d - (a^2*cot(c + d*x)*sqrt(a + im*a*tan(c + d*x)))/d]
    @test_int [cot(c + d*x)^3*(a + im*a*tan(c + d*x))^(5/2), x, 8, (23*a^(5/2)*atanh(sqrt(a + im*a*tan(c + d*x))/sqrt(a)))/(4*d) - (4*sqrt(2)*a^(5/2)*atanh(sqrt(a + im*a*tan(c + d*x))/(sqrt(2)*sqrt(a))))/d - (((9*im)/4)*a^2*cot(c + d*x)*sqrt(a + im*a*tan(c + d*x)))/d - (a^2*cot(c + d*x)^2*sqrt(a + im*a*tan(c + d*x)))/(2*d)]
    @test_int [cot(c + d*x)^4*(a + im*a*tan(c + d*x))^(5/2), x, 9, (((45*im)/8)*a^(5/2)*atanh(sqrt(a + im*a*tan(c + d*x))/sqrt(a)))/d - ((4*im)*sqrt(2)*a^(5/2)*atanh(sqrt(a + im*a*tan(c + d*x))/(sqrt(2)*sqrt(a))))/d + (19*a^2*cot(c + d*x)*sqrt(a + im*a*tan(c + d*x)))/(8*d) - (((13*im)/12)*a^2*cot(c + d*x)^2*sqrt(a + im*a*tan(c + d*x)))/d - (a^2*cot(c + d*x)^3*sqrt(a + im*a*tan(c + d*x)))/(3*d)]


    @test_int [(a + im*a*tan(c + d*x))^(7/2), x, 5, ((-8*im)*sqrt(2)*a^(7/2)*atanh(sqrt(a + im*a*tan(c + d*x))/(sqrt(2)*sqrt(a))))/d + ((8*im)*a^3*sqrt(a + im*a*tan(c + d*x)))/d + (((4*im)/3)*a^2*(a + im*a*tan(c + d*x))^(3/2))/d + (((2*im)/5)*a*(a + im*a*tan(c + d*x))^(5/2))/d]


    #= ::Subsubsection::Closed:: =#
    #=m<0=#


    @test_int [tan(c + d*x)^5/sqrt(a + im*a*tan(c + d*x)), x, 7, -(atanh(sqrt(a + im*a*tan(c + d*x))/(sqrt(2)*sqrt(a)))/(sqrt(2)*sqrt(a)*d)) - tan(c + d*x)^4/(d*sqrt(a + im*a*tan(c + d*x))) - (188*sqrt(a + im*a*tan(c + d*x)))/(35*a*d) + (47*tan(c + d*x)^2*sqrt(a + im*a*tan(c + d*x)))/(35*a*d) - (9*im*tan(c + d*x)^3*sqrt(a + im*a*tan(c + d*x)))/(7*a*d) + (223*(a + im*a*tan(c + d*x))^(3/2))/(105*a^2*d)]
    @test_int [tan(c + d*x)^4/sqrt(a + im*a*tan(c + d*x)), x, 6, -((im*atanh(sqrt(a + im*a*tan(c + d*x))/(sqrt(2)*sqrt(a))))/(sqrt(2)*sqrt(a)*d)) - tan(c + d*x)^3/(d*sqrt(a + im*a*tan(c + d*x))) + (28*im*sqrt(a + im*a*tan(c + d*x)))/(5*a*d) - (7*im*tan(c + d*x)^2*sqrt(a + im*a*tan(c + d*x)))/(5*a*d) - (23*im*(a + im*a*tan(c + d*x))^(3/2))/(15*a^2*d)]
    @test_int [tan(c + d*x)^3/sqrt(a + im*a*tan(c + d*x)), x, 5, atanh(sqrt(a + im*a*tan(c + d*x))/(sqrt(2)*sqrt(a)))/(sqrt(2)*sqrt(a)*d) - tan(c + d*x)^2/(d*sqrt(a + im*a*tan(c + d*x))) + (4*sqrt(a + im*a*tan(c + d*x)))/(a*d) - (5*(a + im*a*tan(c + d*x))^(3/2))/(3*a^2*d)]
    @test_int [tan(c + d*x)^2/sqrt(a + im*a*tan(c + d*x)), x, 4, (im*atanh(sqrt(a + im*a*tan(c + d*x))/(sqrt(2)*sqrt(a))))/(sqrt(2)*sqrt(a)*d) - im/(d*sqrt(a + im*a*tan(c + d*x))) - ((2*im)*sqrt(a + im*a*tan(c + d*x)))/(a*d)]
    @test_int [tan(c + d*x)^1/sqrt(a + im*a*tan(c + d*x)), x, 3, -(atanh(sqrt(a + im*a*tan(c + d*x))/(sqrt(2)*sqrt(a)))/(sqrt(2)*sqrt(a)*d)) - 1/(d*sqrt(a + im*a*tan(c + d*x)))]
    @test_int [tan(c + d*x)^0/sqrt(a + im*a*tan(c + d*x)), x, 3, ((-im)*atanh(sqrt(a + im*a*tan(c + d*x))/(sqrt(2)*sqrt(a))))/(sqrt(2)*sqrt(a)*d) + im/(d*sqrt(a + im*a*tan(c + d*x)))]
    @test_int [cot(c + d*x)^1/sqrt(a + im*a*tan(c + d*x)), x, 7, (-2*atanh(sqrt(a + im*a*tan(c + d*x))/sqrt(a)))/(sqrt(a)*d) + atanh(sqrt(a + im*a*tan(c + d*x))/(sqrt(2)*sqrt(a)))/(sqrt(2)*sqrt(a)*d) + 1/(d*sqrt(a + im*a*tan(c + d*x)))]
    @test_int [cot(c + d*x)^2/sqrt(a + im*a*tan(c + d*x)), x, 8, (im*atanh(sqrt(a + im*a*tan(c + d*x))/sqrt(a)))/(sqrt(a)*d) + (im*atanh(sqrt(a + im*a*tan(c + d*x))/(sqrt(2)*sqrt(a))))/(sqrt(2)*sqrt(a)*d) + cot(c + d*x)/(d*sqrt(a + im*a*tan(c + d*x))) - (2*cot(c + d*x)*sqrt(a + im*a*tan(c + d*x)))/(a*d)]
    @test_int [cot(c + d*x)^3/sqrt(a + im*a*tan(c + d*x)), x, 9, (11*atanh(sqrt(a + im*a*tan(c + d*x))/sqrt(a)))/(4*sqrt(a)*d) - atanh(sqrt(a + im*a*tan(c + d*x))/(sqrt(2)*sqrt(a)))/(sqrt(2)*sqrt(a)*d) + cot(c + d*x)^2/(d*sqrt(a + im*a*tan(c + d*x))) + (7*im*cot(c + d*x)*sqrt(a + im*a*tan(c + d*x)))/(4*a*d) - (3*cot(c + d*x)^2*sqrt(a + im*a*tan(c + d*x)))/(2*a*d)]


    @test_int [tan(c + d*x)^5/(a + im*a*tan(c + d*x))^(3/2), x, 7, -atanh(sqrt(a + im*a*tan(c + d*x))/(sqrt(2)*sqrt(a)))/(2*sqrt(2)*a^(3/2)*d) - tan(c + d*x)^4/(3*d*(a + im*a*tan(c + d*x))^(3/2)) + (((19*im)/6)*tan(c + d*x)^3)/(a*d*sqrt(a + im*a*tan(c + d*x))) + (78*sqrt(a + im*a*tan(c + d*x)))/(5*a^2*d) - (39*tan(c + d*x)^2*sqrt(a + im*a*tan(c + d*x)))/(10*a^2*d) - (151*(a + im*a*tan(c + d*x))^(3/2))/(30*a^3*d)]
    @test_int [tan(c + d*x)^4/(a + im*a*tan(c + d*x))^(3/2), x, 6, ((-im/2)*atanh(sqrt(a + im*a*tan(c + d*x))/(sqrt(2)*sqrt(a))))/(sqrt(2)*a^(3/2)*d) - tan(c + d*x)^3/(3*d*(a + im*a*tan(c + d*x))^(3/2)) + (((5*im)/2)*tan(c + d*x)^2)/(a*d*sqrt(a + im*a*tan(c + d*x))) - ((10*im)*sqrt(a + im*a*tan(c + d*x)))/(a^2*d) + (((7*im)/2)*(a + im*a*tan(c + d*x))^(3/2))/(a^3*d)]
    @test_int [tan(c + d*x)^3/(a + im*a*tan(c + d*x))^(3/2), x, 5, atanh(sqrt(a + im*a*tan(c + d*x))/(sqrt(2)*sqrt(a)))/(2*sqrt(2)*a^(3/2)*d) - tan(c + d*x)^2/(3*d*(a + im*a*tan(c + d*x))^(3/2)) - 11/(6*a*d*sqrt(a + im*a*tan(c + d*x))) - (7*sqrt(a + im*a*tan(c + d*x)))/(3*a^2*d)]
    @test_int [tan(c + d*x)^2/(a + im*a*tan(c + d*x))^(3/2), x, 4, (im*atanh(sqrt(a + im*a*tan(c + d*x))/(sqrt(2)*sqrt(a))))/(2*sqrt(2)*a^(3/2)*d) - im/(3*d*(a + im*a*tan(c + d*x))^(3/2)) + (3*im)/(2*a*d*sqrt(a + im*a*tan(c + d*x)))]
    @test_int [tan(c + d*x)^1/(a + im*a*tan(c + d*x))^(3/2), x, 4, -atanh(sqrt(a + im*a*tan(c + d*x))/(sqrt(2)*sqrt(a)))/(2*sqrt(2)*a^(3/2)*d) - 1/(3*d*(a + im*a*tan(c + d*x))^(3/2)) + 1/(2*a*d*sqrt(a + im*a*tan(c + d*x)))]
    @test_int [tan(c + d*x)^0/(a + im*a*tan(c + d*x))^(3/2), x, 4, ((-im/2)*atanh(sqrt(a + im*a*tan(c + d*x))/(sqrt(2)*sqrt(a))))/(sqrt(2)*a^(3/2)*d) + (im/3)/(d*(a + im*a*tan(c + d*x))^(3/2)) + (im/2)/(a*d*sqrt(a + im*a*tan(c + d*x)))]
    @test_int [cot(c + d*x)^1/(a + im*a*tan(c + d*x))^(3/2), x, 8, (-2*atanh(sqrt(a + im*a*tan(c + d*x))/sqrt(a)))/(a^(3/2)*d) + atanh(sqrt(a + im*a*tan(c + d*x))/(sqrt(2)*sqrt(a)))/(2*sqrt(2)*a^(3/2)*d) + 1/(3*d*(a + im*a*tan(c + d*x))^(3/2)) + 3/(2*a*d*sqrt(a + im*a*tan(c + d*x)))]
    @test_int [cot(c + d*x)^2/(a + im*a*tan(c + d*x))^(3/2), x, 9, ((3*im)*atanh(sqrt(a + im*a*tan(c + d*x))/sqrt(a)))/(a^(3/2)*d) + ((im/2)*atanh(sqrt(a + im*a*tan(c + d*x))/(sqrt(2)*sqrt(a))))/(sqrt(2)*a^(3/2)*d) + cot(c + d*x)/(3*d*(a + im*a*tan(c + d*x))^(3/2)) + (13*cot(c + d*x))/(6*a*d*sqrt(a + im*a*tan(c + d*x))) - (7*cot(c + d*x)*sqrt(a + im*a*tan(c + d*x)))/(2*a^2*d)]
    @test_int [cot(c + d*x)^3/(a + im*a*tan(c + d*x))^(3/2), x, 10, (23*atanh(sqrt(a + im*a*tan(c + d*x))/sqrt(a)))/(4*a^(3/2)*d) - atanh(sqrt(a + im*a*tan(c + d*x))/(sqrt(2)*sqrt(a)))/(2*sqrt(2)*a^(3/2)*d) + cot(c + d*x)^2/(3*d*(a + im*a*tan(c + d*x))^(3/2)) + (17*cot(c + d*x)^2)/(6*a*d*sqrt(a + im*a*tan(c + d*x))) + (((21*im)/4)*cot(c + d*x)*sqrt(a + im*a*tan(c + d*x)))/(a^2*d) - (11*cot(c + d*x)^2*sqrt(a + im*a*tan(c + d*x)))/(3*a^2*d)]


    @test_int [tan(c + d*x)^5/(a + im*a*tan(c + d*x))^(5/2), x, 7, -atanh(sqrt(a + im*a*tan(c + d*x))/(sqrt(2)*sqrt(a)))/(4*sqrt(2)*a^(5/2)*d) - tan(c + d*x)^4/(5*d*(a + im*a*tan(c + d*x))^(5/2)) + (((7*im)/10)*tan(c + d*x)^3)/(a*d*(a + im*a*tan(c + d*x))^(3/2)) + (89*tan(c + d*x)^2)/(20*a^2*d*sqrt(a + im*a*tan(c + d*x))) - (89*sqrt(a + im*a*tan(c + d*x)))/(5*a^3*d) + (361*(a + im*a*tan(c + d*x))^(3/2))/(60*a^4*d)]
    @test_int [tan(c + d*x)^4/(a + im*a*tan(c + d*x))^(5/2), x, 6, ((-im/4)*atanh(sqrt(a + im*a*tan(c + d*x))/(sqrt(2)*sqrt(a))))/(sqrt(2)*a^(5/2)*d) - tan(c + d*x)^3/(5*d*(a + im*a*tan(c + d*x))^(5/2)) + (((17*im)/30)*tan(c + d*x)^2)/(a*d*(a + im*a*tan(c + d*x))^(3/2)) + ((151*im)/60)/(a^2*d*sqrt(a + im*a*tan(c + d*x))) + (((83*im)/30)*sqrt(a + im*a*tan(c + d*x)))/(a^3*d)]
    @test_int [tan(c + d*x)^3/(a + im*a*tan(c + d*x))^(5/2), x, 5, atanh(sqrt(a + im*a*tan(c + d*x))/(sqrt(2)*sqrt(a)))/(4*sqrt(2)*a^(5/2)*d) - tan(c + d*x)^2/(5*d*(a + im*a*tan(c + d*x))^(5/2)) - 13/(30*a*d*(a + im*a*tan(c + d*x))^(3/2)) + 31/(20*a^2*d*sqrt(a + im*a*tan(c + d*x)))]
    @test_int [tan(c + d*x)^2/(a + im*a*tan(c + d*x))^(5/2), x, 5, (im*atanh(sqrt(a + im*a*tan(c + d*x))/(sqrt(2)*sqrt(a))))/(4*sqrt(2)*a^(5/2)*d) - im/(5*d*(a + im*a*tan(c + d*x))^(5/2)) + im/(2*a*d*(a + im*a*tan(c + d*x))^(3/2)) - im/(4*a^2*d*sqrt(a + im*a*tan(c + d*x)))]
    @test_int [tan(c + d*x)^1/(a + im*a*tan(c + d*x))^(5/2), x, 5, -atanh(sqrt(a + im*a*tan(c + d*x))/(sqrt(2)*sqrt(a)))/(4*sqrt(2)*a^(5/2)*d) - 1/(5*d*(a + im*a*tan(c + d*x))^(5/2)) + 1/(6*a*d*(a + im*a*tan(c + d*x))^(3/2)) + 1/(4*a^2*d*sqrt(a + im*a*tan(c + d*x)))]
    @test_int [tan(c + d*x)^0/(a + im*a*tan(c + d*x))^(5/2), x, 5, ((-im/4)*atanh(sqrt(a + im*a*tan(c + d*x))/(sqrt(2)*sqrt(a))))/(sqrt(2)*a^(5/2)*d) + (im/5)/(d*(a + im*a*tan(c + d*x))^(5/2)) + (im/6)/(a*d*(a + im*a*tan(c + d*x))^(3/2)) + (im/4)/(a^2*d*sqrt(a + im*a*tan(c + d*x)))]
    @test_int [cot(c + d*x)^1/(a + im*a*tan(c + d*x))^(5/2), x, 9, (-2*atanh(sqrt(a + im*a*tan(c + d*x))/sqrt(a)))/(a^(5/2)*d) + atanh(sqrt(a + im*a*tan(c + d*x))/(sqrt(2)*sqrt(a)))/(4*sqrt(2)*a^(5/2)*d) + 1/(5*d*(a + im*a*tan(c + d*x))^(5/2)) + 1/(2*a*d*(a + im*a*tan(c + d*x))^(3/2)) + 7/(4*a^2*d*sqrt(a + im*a*tan(c + d*x)))]
    @test_int [cot(c + d*x)^2/(a + im*a*tan(c + d*x))^(5/2), x, 10, ((5*im)*atanh(sqrt(a + im*a*tan(c + d*x))/sqrt(a)))/(a^(5/2)*d) + ((im/4)*atanh(sqrt(a + im*a*tan(c + d*x))/(sqrt(2)*sqrt(a))))/(sqrt(2)*a^(5/2)*d) + cot(c + d*x)/(5*d*(a + im*a*tan(c + d*x))^(5/2)) + (19*cot(c + d*x))/(30*a*d*(a + im*a*tan(c + d*x))^(3/2)) + (41*cot(c + d*x))/(12*a^2*d*sqrt(a + im*a*tan(c + d*x))) - (21*cot(c + d*x)*sqrt(a + im*a*tan(c + d*x)))/(4*a^3*d)]


    @test_int [(a + im*a*tan(c + d*x))^(-7/2), x, 6, ((-im/8)*atanh(sqrt(a + im*a*tan(c + d*x))/(sqrt(2)*sqrt(a))))/(sqrt(2)*a^(7/2)*d) + (im/7)/(d*(a + im*a*tan(c + d*x))^(7/2)) + (im/10)/(a*d*(a + im*a*tan(c + d*x))^(5/2)) + (im/12)/(a^2*d*(a + im*a*tan(c + d*x))^(3/2)) + (im/8)/(a^3*d*sqrt(a + im*a*tan(c + d*x)))]


    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*(a+im*a*tan(e+f*x))^m*(d*tan(e+f*x))^(n/2)=#


    #= ::Subsubsection::Closed:: =#
    #=m>0=#


    @test_int [(d*tan(e + f*x))^(5/2)*(a + im*a*tan(e + f*x)), x, 5, (-2*(-1)^(3/4)*a*d^(5/2)*atan(((-1)^(3/4)*sqrt(d*tan(e + f*x)))/sqrt(d)))/f - ((2*im)*a*d^2*sqrt(d*tan(e + f*x)))/f + (2*a*d*(d*tan(e + f*x))^(3/2))/(3*f) + (((2*im)/5)*a*(d*tan(e + f*x))^(5/2))/f]
    @test_int [(d*tan(e + f*x))^(3/2)*(a + im*a*tan(e + f*x)), x, 4, (2*(-1)^(1/4)*a*d^(3/2)*atan(((-1)^(3/4)*sqrt(d*tan(e + f*x)))/sqrt(d)))/f + (2*a*d*sqrt(d*tan(e + f*x)))/f + (((2*im)/3)*a*(d*tan(e + f*x))^(3/2))/f]
    @test_int [sqrt(d*tan(e + f*x))*(a + im*a*tan(e + f*x)), x, 3, (2*(-1)^(3/4)*a*sqrt(d)*atan(((-1)^(3/4)*sqrt(d*tan(e + f*x)))/sqrt(d)))/f + ((2*im)*a*sqrt(d*tan(e + f*x)))/f]
    @test_int [(a + im*a*tan(e + f*x))/sqrt(d*tan(e + f*x)), x, 2, (-2*(-1)^(1/4)*a*atan(((-1)^(3/4)*sqrt(d*tan(e + f*x)))/sqrt(d)))/(sqrt(d)*f)]
    @test_int [(a + im*a*tan(e + f*x))/(d*tan(e + f*x))^(3/2), x, 3, (-2*(-1)^(3/4)*a*atan(((-1)^(3/4)*sqrt(d*tan(e + f*x)))/sqrt(d)))/(d^(3/2)*f) - (2*a)/(d*f*sqrt(d*tan(e + f*x)))]
    @test_int [(a + im*a*tan(e + f*x))/(d*tan(e + f*x))^(5/2), x, 4, (2*(-1)^(1/4)*a*atan(((-1)^(3/4)*sqrt(d*tan(e + f*x)))/sqrt(d)))/(d^(5/2)*f) - (2*a)/(3*d*f*(d*tan(e + f*x))^(3/2)) - ((2*im)*a)/(d^2*f*sqrt(d*tan(e + f*x)))]
    @test_int [(a + im*a*tan(e + f*x))/(d*tan(e + f*x))^(7/2), x, 5, (2*(-1)^(3/4)*a*atan(((-1)^(3/4)*sqrt(d*tan(e + f*x)))/sqrt(d)))/(d^(7/2)*f) - (2*a)/(5*d*f*(d*tan(e + f*x))^(5/2)) - (((2*im)/3)*a)/(d^2*f*(d*tan(e + f*x))^(3/2)) + (2*a)/(d^3*f*sqrt(d*tan(e + f*x)))]

    @test_int [(d*tan(e + f*x))^(5/2)*(a - im*a*tan(e + f*x)), x, 5, (2*(-1)^(3/4)*a*d^(5/2)*atanh(((-1)^(3/4)*sqrt(d*tan(e + f*x)))/sqrt(d)))/f + (2*im*a*d^2*sqrt(d*tan(e + f*x)))/f + (2*a*d*(d*tan(e + f*x))^(3/2))/(3*f) - (2*im*a*(d*tan(e + f*x))^(5/2))/(5*f)]
    @test_int [(d*tan(e + f*x))^(3/2)*(a - im*a*tan(e + f*x)), x, 4, (2*(-1)^(1/4)*a*d^(3/2)*atanh(((-1)^(3/4)*sqrt(d*tan(e + f*x)))/sqrt(d)))/f + (2*a*d*sqrt(d*tan(e + f*x)))/f - (2*im*a*(d*tan(e + f*x))^(3/2))/(3*f)]
    @test_int [sqrt(d*tan(e + f*x))*(a - im*a*tan(e + f*x)), x, 3, -((2*(-1)^(3/4)*a*sqrt(d)*atanh(((-1)^(3/4)*sqrt(d*tan(e + f*x)))/sqrt(d)))/f) - (2*im*a*sqrt(d*tan(e + f*x)))/f]
    @test_int [(a - im*a*tan(e + f*x))/sqrt(d*tan(e + f*x)), x, 2, -((2*(-1)^(1/4)*a*atanh(((-1)^(3/4)*sqrt(d*tan(e + f*x)))/sqrt(d)))/(sqrt(d)*f))]
    @test_int [(a - im*a*tan(e + f*x))/(d*tan(e + f*x))^(3/2), x, 3, (2*(-1)^(3/4)*a*atanh(((-1)^(3/4)*sqrt(d*tan(e + f*x)))/sqrt(d)))/(d^(3/2)*f) - (2*a)/(d*f*sqrt(d*tan(e + f*x)))]
    @test_int [(a - im*a*tan(e + f*x))/(d*tan(e + f*x))^(5/2), x, 4, (2*(-1)^(1/4)*a*atanh(((-1)^(3/4)*sqrt(d*tan(e + f*x)))/sqrt(d)))/(d^(5/2)*f) - (2*a)/(3*d*f*(d*tan(e + f*x))^(3/2)) + (2*im*a)/(d^2*f*sqrt(d*tan(e + f*x)))]
    @test_int [(a - im*a*tan(e + f*x))/(d*tan(e + f*x))^(7/2), x, 5, -((2*(-1)^(3/4)*a*atanh(((-1)^(3/4)*sqrt(d*tan(e + f*x)))/sqrt(d)))/(d^(7/2)*f)) - (2*a)/(5*d*f*(d*tan(e + f*x))^(5/2)) + (2*im*a)/(3*d^2*f*(d*tan(e + f*x))^(3/2)) + (2*a)/(d^3*f*sqrt(d*tan(e + f*x)))]


    @test_int [(d*tan(e + f*x))^(5/2)*(a + im*a*tan(e + f*x))^2, x, 6, (-4*(-1)^(3/4)*a^2*d^(5/2)*atan(((-1)^(3/4)*sqrt(d*tan(e + f*x)))/sqrt(d)))/f - ((4*im)*a^2*d^2*sqrt(d*tan(e + f*x)))/f + (4*a^2*d*(d*tan(e + f*x))^(3/2))/(3*f) + (((4*im)/5)*a^2*(d*tan(e + f*x))^(5/2))/f - (2*a^2*(d*tan(e + f*x))^(7/2))/(7*d*f)]
    @test_int [(d*tan(e + f*x))^(3/2)*(a + im*a*tan(e + f*x))^2, x, 5, (4*(-1)^(1/4)*a^2*d^(3/2)*atan(((-1)^(3/4)*sqrt(d*tan(e + f*x)))/sqrt(d)))/f + (4*a^2*d*sqrt(d*tan(e + f*x)))/f + (((4*im)/3)*a^2*(d*tan(e + f*x))^(3/2))/f - (2*a^2*(d*tan(e + f*x))^(5/2))/(5*d*f)]
    @test_int [sqrt(d*tan(e + f*x))*(a + im*a*tan(e + f*x))^2, x, 4, (4*(-1)^(3/4)*a^2*sqrt(d)*atan(((-1)^(3/4)*sqrt(d*tan(e + f*x)))/sqrt(d)))/f + ((4*im)*a^2*sqrt(d*tan(e + f*x)))/f - (2*a^2*(d*tan(e + f*x))^(3/2))/(3*d*f)]
    @test_int [(a + im*a*tan(e + f*x))^2/sqrt(d*tan(e + f*x)), x, 3, (-4*(-1)^(1/4)*a^2*atan(((-1)^(3/4)*sqrt(d*tan(e + f*x)))/sqrt(d)))/(sqrt(d)*f) - (2*a^2*sqrt(d*tan(e + f*x)))/(d*f)]
    @test_int [(a + im*a*tan(e + f*x))^2/(d*tan(e + f*x))^(3/2), x, 3, (-4*(-1)^(3/4)*a^2*atan(((-1)^(3/4)*sqrt(d*tan(e + f*x)))/sqrt(d)))/(d^(3/2)*f) - (2*a^2)/(d*f*sqrt(d*tan(e + f*x)))]
    @test_int [(a + im*a*tan(e + f*x))^2/(d*tan(e + f*x))^(5/2), x, 4, (4*(-1)^(1/4)*a^2*atan(((-1)^(3/4)*sqrt(d*tan(e + f*x)))/sqrt(d)))/(d^(5/2)*f) - (2*a^2)/(3*d*f*(d*tan(e + f*x))^(3/2)) - ((4*im)*a^2)/(d^2*f*sqrt(d*tan(e + f*x)))]
    @test_int [(a + im*a*tan(e + f*x))^2/(d*tan(e + f*x))^(7/2), x, 5, (4*(-1)^(3/4)*a^2*atan(((-1)^(3/4)*sqrt(d*tan(e + f*x)))/sqrt(d)))/(d^(7/2)*f) - (2*a^2)/(5*d*f*(d*tan(e + f*x))^(5/2)) - (((4*im)/3)*a^2)/(d^2*f*(d*tan(e + f*x))^(3/2)) + (4*a^2)/(d^3*f*sqrt(d*tan(e + f*x)))]


    @test_int [(d*tan(e + f*x))^(5/2)*(a + im*a*tan(e + f*x))^3, x, 7, (-8*(-1)^(3/4)*a^3*d^(5/2)*atan(((-1)^(3/4)*sqrt(d*tan(e + f*x)))/sqrt(d)))/f - ((8*im)*a^3*d^2*sqrt(d*tan(e + f*x)))/f + (8*a^3*d*(d*tan(e + f*x))^(3/2))/(3*f) + (((8*im)/5)*a^3*(d*tan(e + f*x))^(5/2))/f - (40*a^3*(d*tan(e + f*x))^(7/2))/(63*d*f) - (2*(d*tan(e + f*x))^(7/2)*(a^3 + im*a^3*tan(e + f*x)))/(9*d*f)]
    @test_int [(d*tan(e + f*x))^(3/2)*(a + im*a*tan(e + f*x))^3, x, 6, (8*(-1)^(1/4)*a^3*d^(3/2)*atan(((-1)^(3/4)*sqrt(d*tan(e + f*x)))/sqrt(d)))/f + (8*a^3*d*sqrt(d*tan(e + f*x)))/f + (((8*im)/3)*a^3*(d*tan(e + f*x))^(3/2))/f - (32*a^3*(d*tan(e + f*x))^(5/2))/(35*d*f) - (2*(d*tan(e + f*x))^(5/2)*(a^3 + im*a^3*tan(e + f*x)))/(7*d*f)]
    @test_int [sqrt(d*tan(e + f*x))*(a + im*a*tan(e + f*x))^3, x, 5, (8*(-1)^(3/4)*a^3*sqrt(d)*atan(((-1)^(3/4)*sqrt(d*tan(e + f*x)))/sqrt(d)))/f + ((8*im)*a^3*sqrt(d*tan(e + f*x)))/f - (8*a^3*(d*tan(e + f*x))^(3/2))/(5*d*f) - (2*(d*tan(e + f*x))^(3/2)*(a^3 + im*a^3*tan(e + f*x)))/(5*d*f)]
    @test_int [(a + im*a*tan(e + f*x))^3/sqrt(d*tan(e + f*x)), x, 4, (-8*(-1)^(1/4)*a^3*atan(((-1)^(3/4)*sqrt(d*tan(e + f*x)))/sqrt(d)))/(sqrt(d)*f) - (16*a^3*sqrt(d*tan(e + f*x)))/(3*d*f) - (2*sqrt(d*tan(e + f*x))*(a^3 + im*a^3*tan(e + f*x)))/(3*d*f)]
    @test_int [(a + im*a*tan(e + f*x))^3/(d*tan(e + f*x))^(3/2), x, 4, (-8*(-1)^(3/4)*a^3*atan(((-1)^(3/4)*sqrt(d*tan(e + f*x)))/sqrt(d)))/(d^(3/2)*f) - (2*(a^3 + im*a^3*tan(e + f*x)))/(d*f*sqrt(d*tan(e + f*x)))]
    @test_int [(a + im*a*tan(e + f*x))^3/(d*tan(e + f*x))^(5/2), x, 4, (8*(-1)^(1/4)*a^3*atan(((-1)^(3/4)*sqrt(d*tan(e + f*x)))/sqrt(d)))/(d^(5/2)*f) - (((16*im)/3)*a^3)/(d^2*f*sqrt(d*tan(e + f*x))) - (2*(a^3 + im*a^3*tan(e + f*x)))/(3*d*f*(d*tan(e + f*x))^(3/2))]
    @test_int [(a + im*a*tan(e + f*x))^3/(d*tan(e + f*x))^(7/2), x, 5, (8*(-1)^(3/4)*a^3*atan(((-1)^(3/4)*sqrt(d*tan(e + f*x)))/sqrt(d)))/(d^(7/2)*f) - (((8*im)/5)*a^3)/(d^2*f*(d*tan(e + f*x))^(3/2)) + (8*a^3)/(d^3*f*sqrt(d*tan(e + f*x))) - (2*(a^3 + im*a^3*tan(e + f*x)))/(5*d*f*(d*tan(e + f*x))^(5/2))]
    @test_int [(a + im*a*tan(e + f*x))^3/(d*tan(e + f*x))^(9/2), x, 6, (-8*(-1)^(1/4)*a^3*atan(((-1)^(3/4)*sqrt(d*tan(e + f*x)))/sqrt(d)))/(d^(9/2)*f) - (((32*im)/35)*a^3)/(d^2*f*(d*tan(e + f*x))^(5/2)) + (8*a^3)/(3*d^3*f*(d*tan(e + f*x))^(3/2)) + ((8*im)*a^3)/(d^4*f*sqrt(d*tan(e + f*x))) - (2*(a^3 + im*a^3*tan(e + f*x)))/(7*d*f*(d*tan(e + f*x))^(7/2))]


    #= ::Subsubsection::Closed:: =#
    #=m<0=#


    @test_int [(d*tan(e + f*x))^(7/2)/(a + im*a*tan(e + f*x)), x, 13, ((5/4 - (7*im)/4)*d^(7/2)*atan(1 - (sqrt(2)*sqrt(d*tan(e + f*x)))/sqrt(d)))/(sqrt(2)*a*f) - ((5/4 - (7*im)/4)*d^(7/2)*atan(1 + (sqrt(2)*sqrt(d*tan(e + f*x)))/sqrt(d)))/(sqrt(2)*a*f) + ((5/8 + (7*im)/8)*d^(7/2)*log(sqrt(d) + sqrt(d)*tan(e + f*x) - sqrt(2)*sqrt(d*tan(e + f*x))))/(sqrt(2)*a*f) - ((5/8 + (7*im)/8)*d^(7/2)*log(sqrt(d) + sqrt(d)*tan(e + f*x) + sqrt(2)*sqrt(d*tan(e + f*x))))/(sqrt(2)*a*f) + (5*d^3*sqrt(d*tan(e + f*x)))/(2*a*f) - (7*im*d^2*(d*tan(e + f*x))^(3/2))/(6*a*f) - (d*(d*tan(e + f*x))^(5/2))/(2*f*(a + im*a*tan(e + f*x)))]
    @test_int [(d*tan(e + f*x))^(5/2)/(a + im*a*tan(e + f*x)), x, 12, -(((3/4 + (5*im)/4)*d^(5/2)*atan(1 - (sqrt(2)*sqrt(d*tan(e + f*x)))/sqrt(d)))/(sqrt(2)*a*f)) + ((3/4 + (5*im)/4)*d^(5/2)*atan(1 + (sqrt(2)*sqrt(d*tan(e + f*x)))/sqrt(d)))/(sqrt(2)*a*f) + ((3/8 - (5*im)/8)*d^(5/2)*log(sqrt(d) + sqrt(d)*tan(e + f*x) - sqrt(2)*sqrt(d*tan(e + f*x))))/(sqrt(2)*a*f) - ((3/8 - (5*im)/8)*d^(5/2)*log(sqrt(d) + sqrt(d)*tan(e + f*x) + sqrt(2)*sqrt(d*tan(e + f*x))))/(sqrt(2)*a*f) - (5*im*d^2*sqrt(d*tan(e + f*x)))/(2*a*f) - (d*(d*tan(e + f*x))^(3/2))/(2*f*(a + im*a*tan(e + f*x)))]
    @test_int [(d*tan(e + f*x))^(3/2)/(a + im*a*tan(e + f*x)), x, 11, -(((1/4 - (3*im)/4)*d^(3/2)*atan(1 - (sqrt(2)*sqrt(d*tan(e + f*x)))/sqrt(d)))/(sqrt(2)*a*f)) + ((1/4 - (3*im)/4)*d^(3/2)*atan(1 + (sqrt(2)*sqrt(d*tan(e + f*x)))/sqrt(d)))/(sqrt(2)*a*f) - ((1/8 + (3*im)/8)*d^(3/2)*log(sqrt(d) + sqrt(d)*tan(e + f*x) - sqrt(2)*sqrt(d*tan(e + f*x))))/(sqrt(2)*a*f) + ((1/8 + (3*im)/8)*d^(3/2)*log(sqrt(d) + sqrt(d)*tan(e + f*x) + sqrt(2)*sqrt(d*tan(e + f*x))))/(sqrt(2)*a*f) - (d*sqrt(d*tan(e + f*x)))/(2*f*(a + im*a*tan(e + f*x)))]
    @test_int [sqrt(d*tan(e + f*x))/(a + im*a*tan(e + f*x)), x, 3, ((-1)^(3/4)*sqrt(d)*atan(((-1)^(3/4)*sqrt(d*tan(e + f*x)))/sqrt(d)))/(2*a*f) + ((im/2)*sqrt(d*tan(e + f*x)))/(f*(a + im*a*tan(e + f*x)))]
    @test_int [1/(sqrt(d*tan(e + f*x))*(a + im*a*tan(e + f*x))), x, 11, -(((3/4 - im/4)*atan(1 - (sqrt(2)*sqrt(d*tan(e + f*x)))/sqrt(d)))/(sqrt(2)*a*sqrt(d)*f)) + ((3/4 - im/4)*atan(1 + (sqrt(2)*sqrt(d*tan(e + f*x)))/sqrt(d)))/(sqrt(2)*a*sqrt(d)*f) - ((3/8 + im/8)*log(sqrt(d) + sqrt(d)*tan(e + f*x) - sqrt(2)*sqrt(d*tan(e + f*x))))/(sqrt(2)*a*sqrt(d)*f) + ((3/8 + im/8)*log(sqrt(d) + sqrt(d)*tan(e + f*x) + sqrt(2)*sqrt(d*tan(e + f*x))))/(sqrt(2)*a*sqrt(d)*f) + sqrt(d*tan(e + f*x))/(2*d*f*(a + im*a*tan(e + f*x)))]
    @test_int [1/((d*tan(e + f*x))^(3/2)*(a + im*a*tan(e + f*x))), x, 12, ((5/4 + (3*im)/4)*atan(1 - (sqrt(2)*sqrt(d*tan(e + f*x)))/sqrt(d)))/(sqrt(2)*a*d^(3/2)*f) - ((5/4 + (3*im)/4)*atan(1 + (sqrt(2)*sqrt(d*tan(e + f*x)))/sqrt(d)))/(sqrt(2)*a*d^(3/2)*f) - ((5/8 - (3*im)/8)*log(sqrt(d) + sqrt(d)*tan(e + f*x) - sqrt(2)*sqrt(d*tan(e + f*x))))/(sqrt(2)*a*d^(3/2)*f) + ((5/8 - (3*im)/8)*log(sqrt(d) + sqrt(d)*tan(e + f*x) + sqrt(2)*sqrt(d*tan(e + f*x))))/(sqrt(2)*a*d^(3/2)*f) - 5/(2*a*d*f*sqrt(d*tan(e + f*x))) + 1/(2*d*f*sqrt(d*tan(e + f*x))*(a + im*a*tan(e + f*x)))]
    @test_int [1/((d*tan(e + f*x))^(5/2)*(a + im*a*tan(e + f*x))), x, 13, ((7/4 - (5*im)/4)*atan(1 - (sqrt(2)*sqrt(d*tan(e + f*x)))/sqrt(d)))/(sqrt(2)*a*d^(5/2)*f) - ((7/4 - (5*im)/4)*atan(1 + (sqrt(2)*sqrt(d*tan(e + f*x)))/sqrt(d)))/(sqrt(2)*a*d^(5/2)*f) + ((7/8 + (5*im)/8)*log(sqrt(d) + sqrt(d)*tan(e + f*x) - sqrt(2)*sqrt(d*tan(e + f*x))))/(sqrt(2)*a*d^(5/2)*f) - ((7/8 + (5*im)/8)*log(sqrt(d) + sqrt(d)*tan(e + f*x) + sqrt(2)*sqrt(d*tan(e + f*x))))/(sqrt(2)*a*d^(5/2)*f) - 7/(6*a*d*f*(d*tan(e + f*x))^(3/2)) + (5*im)/(2*a*d^2*f*sqrt(d*tan(e + f*x))) + 1/(2*d*f*(d*tan(e + f*x))^(3/2)*(a + im*a*tan(e + f*x)))]


    @test_int [(d*tan(e + f*x))^(9/2)/(a + im*a*tan(e + f*x))^2, x, 14, -(((49/16 + (45*im)/16)*d^(9/2)*atan(1 - (sqrt(2)*sqrt(d*tan(e + f*x)))/sqrt(d)))/(sqrt(2)*a^2*f)) + ((49/16 + (45*im)/16)*d^(9/2)*atan(1 + (sqrt(2)*sqrt(d*tan(e + f*x)))/sqrt(d)))/(sqrt(2)*a^2*f) + ((49/32 - (45*im)/32)*d^(9/2)*log(sqrt(d) + sqrt(d)*tan(e + f*x) - sqrt(2)*sqrt(d*tan(e + f*x))))/(sqrt(2)*a^2*f) - ((49/32 - (45*im)/32)*d^(9/2)*log(sqrt(d) + sqrt(d)*tan(e + f*x) + sqrt(2)*sqrt(d*tan(e + f*x))))/(sqrt(2)*a^2*f) - (45*im*d^4*sqrt(d*tan(e + f*x)))/(8*a^2*f) - (49*d^3*(d*tan(e + f*x))^(3/2))/(24*a^2*f) + (9*im*d^2*(d*tan(e + f*x))^(5/2))/(8*a^2*f*(1 + im*tan(e + f*x))) - (d*(d*tan(e + f*x))^(7/2))/(4*f*(a + im*a*tan(e + f*x))^2)]
    @test_int [(d*tan(e + f*x))^(7/2)/(a + im*a*tan(e + f*x))^2, x, 13, -(((25/16 - (21*im)/16)*d^(7/2)*atan(1 - (sqrt(2)*sqrt(d*tan(e + f*x)))/sqrt(d)))/(sqrt(2)*a^2*f)) + ((25/16 - (21*im)/16)*d^(7/2)*atan(1 + (sqrt(2)*sqrt(d*tan(e + f*x)))/sqrt(d)))/(sqrt(2)*a^2*f) - ((25/32 + (21*im)/32)*d^(7/2)*log(sqrt(d) + sqrt(d)*tan(e + f*x) - sqrt(2)*sqrt(d*tan(e + f*x))))/(sqrt(2)*a^2*f) + ((25/32 + (21*im)/32)*d^(7/2)*log(sqrt(d) + sqrt(d)*tan(e + f*x) + sqrt(2)*sqrt(d*tan(e + f*x))))/(sqrt(2)*a^2*f) - (25*d^3*sqrt(d*tan(e + f*x)))/(8*a^2*f) + (7*im*d^2*(d*tan(e + f*x))^(3/2))/(8*a^2*f*(1 + im*tan(e + f*x))) - (d*(d*tan(e + f*x))^(5/2))/(4*f*(a + im*a*tan(e + f*x))^2)]
    @test_int [(d*tan(e + f*x))^(5/2)/(a + im*a*tan(e + f*x))^2, x, 12, ((9/16 + (5*im)/16)*d^(5/2)*atan(1 - (sqrt(2)*sqrt(d*tan(e + f*x)))/sqrt(d)))/(sqrt(2)*a^2*f) - ((9/16 + (5*im)/16)*d^(5/2)*atan(1 + (sqrt(2)*sqrt(d*tan(e + f*x)))/sqrt(d)))/(sqrt(2)*a^2*f) - ((9/32 - (5*im)/32)*d^(5/2)*log(sqrt(d) + sqrt(d)*tan(e + f*x) - sqrt(2)*sqrt(d*tan(e + f*x))))/(sqrt(2)*a^2*f) + ((9/32 - (5*im)/32)*d^(5/2)*log(sqrt(d) + sqrt(d)*tan(e + f*x) + sqrt(2)*sqrt(d*tan(e + f*x))))/(sqrt(2)*a^2*f) + (5*im*d^2*sqrt(d*tan(e + f*x)))/(8*a^2*f*(1 + im*tan(e + f*x))) - (d*(d*tan(e + f*x))^(3/2))/(4*f*(a + im*a*tan(e + f*x))^2)]
    @test_int [(d*tan(e + f*x))^(3/2)/(a + im*a*tan(e + f*x))^2, x, 12, ((1/16 + (3*im)/16)*d^(3/2)*atan(1 - (sqrt(2)*sqrt(d*tan(e + f*x)))/sqrt(d)))/(sqrt(2)*a^2*f) - ((1/16 + (3*im)/16)*d^(3/2)*atan(1 + (sqrt(2)*sqrt(d*tan(e + f*x)))/sqrt(d)))/(sqrt(2)*a^2*f) + ((1/32 - (3*im)/32)*d^(3/2)*log(sqrt(d) + sqrt(d)*tan(e + f*x) - sqrt(2)*sqrt(d*tan(e + f*x))))/(sqrt(2)*a^2*f) - ((1/32 - (3*im)/32)*d^(3/2)*log(sqrt(d) + sqrt(d)*tan(e + f*x) + sqrt(2)*sqrt(d*tan(e + f*x))))/(sqrt(2)*a^2*f) + (3*d*sqrt(d*tan(e + f*x)))/(8*a^2*f*(1 + im*tan(e + f*x))) - (d*sqrt(d*tan(e + f*x)))/(4*f*(a + im*a*tan(e + f*x))^2)]
    @test_int [sqrt(d*tan(e + f*x))/(a + im*a*tan(e + f*x))^2, x, 12, -(((1/16 - (3*im)/16)*sqrt(d)*atan(1 - (sqrt(2)*sqrt(d*tan(e + f*x)))/sqrt(d)))/(sqrt(2)*a^2*f)) + ((1/16 - (3*im)/16)*sqrt(d)*atan(1 + (sqrt(2)*sqrt(d*tan(e + f*x)))/sqrt(d)))/(sqrt(2)*a^2*f) + ((1/32 + (3*im)/32)*sqrt(d)*log(sqrt(d) + sqrt(d)*tan(e + f*x) - sqrt(2)*sqrt(d*tan(e + f*x))))/(sqrt(2)*a^2*f) - ((1/32 + (3*im)/32)*sqrt(d)*log(sqrt(d) + sqrt(d)*tan(e + f*x) + sqrt(2)*sqrt(d*tan(e + f*x))))/(sqrt(2)*a^2*f) + (im*sqrt(d*tan(e + f*x)))/(8*a^2*f*(1 + im*tan(e + f*x))) + (im*sqrt(d*tan(e + f*x)))/(4*f*(a + im*a*tan(e + f*x))^2)]
    @test_int [1/(sqrt(d*tan(e + f*x))*(a + im*a*tan(e + f*x))^2), x, 12, -(((9/16 - (5*im)/16)*atan(1 - (sqrt(2)*sqrt(d*tan(e + f*x)))/sqrt(d)))/(sqrt(2)*a^2*sqrt(d)*f)) + ((9/16 - (5*im)/16)*atan(1 + (sqrt(2)*sqrt(d*tan(e + f*x)))/sqrt(d)))/(sqrt(2)*a^2*sqrt(d)*f) - ((9/32 + (5*im)/32)*log(sqrt(d) + sqrt(d)*tan(e + f*x) - sqrt(2)*sqrt(d*tan(e + f*x))))/(sqrt(2)*a^2*sqrt(d)*f) + ((9/32 + (5*im)/32)*log(sqrt(d) + sqrt(d)*tan(e + f*x) + sqrt(2)*sqrt(d*tan(e + f*x))))/(sqrt(2)*a^2*sqrt(d)*f) + (5*sqrt(d*tan(e + f*x)))/(8*a^2*d*f*(1 + im*tan(e + f*x))) + sqrt(d*tan(e + f*x))/(4*d*f*(a + im*a*tan(e + f*x))^2)]
    @test_int [1/((d*tan(e + f*x))^(3/2)*(a + im*a*tan(e + f*x))^2), x, 13, ((25/16 + (21*im)/16)*atan(1 - (sqrt(2)*sqrt(d*tan(e + f*x)))/sqrt(d)))/(sqrt(2)*a^2*d^(3/2)*f) - ((25/16 + (21*im)/16)*atan(1 + (sqrt(2)*sqrt(d*tan(e + f*x)))/sqrt(d)))/(sqrt(2)*a^2*d^(3/2)*f) - ((25/32 - (21*im)/32)*log(sqrt(d) + sqrt(d)*tan(e + f*x) - sqrt(2)*sqrt(d*tan(e + f*x))))/(sqrt(2)*a^2*d^(3/2)*f) + ((25/32 - (21*im)/32)*log(sqrt(d) + sqrt(d)*tan(e + f*x) + sqrt(2)*sqrt(d*tan(e + f*x))))/(sqrt(2)*a^2*d^(3/2)*f) - 25/(8*a^2*d*f*sqrt(d*tan(e + f*x))) + 7/(8*a^2*d*f*(1 + im*tan(e + f*x))*sqrt(d*tan(e + f*x))) + 1/(4*d*f*sqrt(d*tan(e + f*x))*(a + im*a*tan(e + f*x))^2)]
    @test_int [1/((d*tan(e + f*x))^(5/2)*(a + im*a*tan(e + f*x))^2), x, 14, ((49/16 - (45*im)/16)*atan(1 - (sqrt(2)*sqrt(d*tan(e + f*x)))/sqrt(d)))/(sqrt(2)*a^2*d^(5/2)*f) - ((49/16 - (45*im)/16)*atan(1 + (sqrt(2)*sqrt(d*tan(e + f*x)))/sqrt(d)))/(sqrt(2)*a^2*d^(5/2)*f) + ((49/32 + (45*im)/32)*log(sqrt(d) + sqrt(d)*tan(e + f*x) - sqrt(2)*sqrt(d*tan(e + f*x))))/(sqrt(2)*a^2*d^(5/2)*f) - ((49/32 + (45*im)/32)*log(sqrt(d) + sqrt(d)*tan(e + f*x) + sqrt(2)*sqrt(d*tan(e + f*x))))/(sqrt(2)*a^2*d^(5/2)*f) - 49/(24*a^2*d*f*(d*tan(e + f*x))^(3/2)) + 9/(8*a^2*d*f*(1 + im*tan(e + f*x))*(d*tan(e + f*x))^(3/2)) + (45*im)/(8*a^2*d^2*f*sqrt(d*tan(e + f*x))) + 1/(4*d*f*(d*tan(e + f*x))^(3/2)*(a + im*a*tan(e + f*x))^2)]


    @test_int [(d*tan(e + f*x))^(9/2)/(a + im*a*tan(e + f*x))^3, x, 14, ((7/4 + (15*im)/8)*d^(9/2)*atan(1 - (sqrt(2)*sqrt(d*tan(e + f*x)))/sqrt(d)))/(sqrt(2)*a^3*f) - ((7/4 + (15*im)/8)*d^(9/2)*atan(1 + (sqrt(2)*sqrt(d*tan(e + f*x)))/sqrt(d)))/(sqrt(2)*a^3*f) - ((7/8 - (15*im)/16)*d^(9/2)*log(sqrt(d) + sqrt(d)*tan(e + f*x) - sqrt(2)*sqrt(d*tan(e + f*x))))/(sqrt(2)*a^3*f) + ((7/8 - (15*im)/16)*d^(9/2)*log(sqrt(d) + sqrt(d)*tan(e + f*x) + sqrt(2)*sqrt(d*tan(e + f*x))))/(sqrt(2)*a^3*f) + (15*im*d^4*sqrt(d*tan(e + f*x)))/(4*a^3*f) - (d*(d*tan(e + f*x))^(7/2))/(6*f*(a + im*a*tan(e + f*x))^3) + (5*im*d^2*(d*tan(e + f*x))^(5/2))/(12*a*f*(a + im*a*tan(e + f*x))^2) + (7*d^3*(d*tan(e + f*x))^(3/2))/(6*f*(a^3 + im*a^3*tan(e + f*x)))]
    @test_int [(d*tan(e + f*x))^(7/2)/(a + im*a*tan(e + f*x))^3, x, 13, ((5/16 - (7*im)/16)*d^(7/2)*atan(1 - (sqrt(2)*sqrt(d*tan(e + f*x)))/sqrt(d)))/(sqrt(2)*a^3*f) - ((5/16 - (7*im)/16)*d^(7/2)*atan(1 + (sqrt(2)*sqrt(d*tan(e + f*x)))/sqrt(d)))/(sqrt(2)*a^3*f) + ((5/32 + (7*im)/32)*d^(7/2)*log(sqrt(d) + sqrt(d)*tan(e + f*x) - sqrt(2)*sqrt(d*tan(e + f*x))))/(sqrt(2)*a^3*f) - ((5/32 + (7*im)/32)*d^(7/2)*log(sqrt(d) + sqrt(d)*tan(e + f*x) + sqrt(2)*sqrt(d*tan(e + f*x))))/(sqrt(2)*a^3*f) - (d*(d*tan(e + f*x))^(5/2))/(6*f*(a + im*a*tan(e + f*x))^3) + (im*d^2*(d*tan(e + f*x))^(3/2))/(3*a*f*(a + im*a*tan(e + f*x))^2) + (5*d^3*sqrt(d*tan(e + f*x)))/(8*f*(a^3 + im*a^3*tan(e + f*x)))]
    @test_int [(d*tan(e + f*x))^(5/2)/(a + im*a*tan(e + f*x))^3, x, 16, (d^(5/2)*atan(1 - (sqrt(2)*sqrt(d*tan(e + f*x)))/sqrt(d)))/(8*sqrt(2)*a^3*f) - (d^(5/2)*atan(1 + (sqrt(2)*sqrt(d*tan(e + f*x)))/sqrt(d)))/(8*sqrt(2)*a^3*f) - (d^(5/2)*log(sqrt(d) + sqrt(d)*tan(e + f*x) - sqrt(2)*sqrt(d*tan(e + f*x))))/(16*sqrt(2)*a^3*f) + (d^(5/2)*log(sqrt(d) + sqrt(d)*tan(e + f*x) + sqrt(2)*sqrt(d*tan(e + f*x))))/(16*sqrt(2)*a^3*f) - (d*(d*tan(e + f*x))^(3/2))/(6*f*(a + im*a*tan(e + f*x))^3) + ((im/4)*d^2*sqrt(d*tan(e + f*x)))/(a*f*(a + im*a*tan(e + f*x))^2) - ((im/4)*d^2*sqrt(d*tan(e + f*x)))/(f*(a^3 + im*a^3*tan(e + f*x)))]
    @test_int [(d*tan(e + f*x))^(3/2)/(a + im*a*tan(e + f*x))^3, x, 7, ((-1)^(1/4)*d^(3/2)*atan(((-1)^(3/4)*sqrt(d*tan(e + f*x)))/sqrt(d)))/(8*a^3*f) - (d*sqrt(d*tan(e + f*x)))/(6*f*(a + im*a*tan(e + f*x))^3) + (d*sqrt(d*tan(e + f*x)))/(6*a*f*(a + im*a*tan(e + f*x))^2) + (d*sqrt(d*tan(e + f*x)))/(8*f*(a^3 + im*a^3*tan(e + f*x)))]
    @test_int [sqrt(d*tan(e + f*x))/(a + im*a*tan(e + f*x))^3, x, 14, ((im/8)*sqrt(d)*atan(1 - (sqrt(2)*sqrt(d*tan(e + f*x)))/sqrt(d)))/(sqrt(2)*a^3*f) - ((im/8)*sqrt(d)*atan(1 + (sqrt(2)*sqrt(d*tan(e + f*x)))/sqrt(d)))/(sqrt(2)*a^3*f) + ((im/16)*sqrt(d)*log(sqrt(d) + sqrt(d)*tan(e + f*x) - sqrt(2)*sqrt(d*tan(e + f*x))))/(sqrt(2)*a^3*f) - ((im/16)*sqrt(d)*log(sqrt(d) + sqrt(d)*tan(e + f*x) + sqrt(2)*sqrt(d*tan(e + f*x))))/(sqrt(2)*a^3*f) + ((im/6)*sqrt(d*tan(e + f*x)))/(f*(a + im*a*tan(e + f*x))^3) + ((im/12)*sqrt(d*tan(e + f*x)))/(a*f*(a + im*a*tan(e + f*x))^2)]
    @test_int [1/(sqrt(d*tan(e + f*x))*(a + im*a*tan(e + f*x))^3), x, 13, -(((7/16 - (5*im)/16)*atan(1 - (sqrt(2)*sqrt(d*tan(e + f*x)))/sqrt(d)))/(sqrt(2)*a^3*sqrt(d)*f)) + ((7/16 - (5*im)/16)*atan(1 + (sqrt(2)*sqrt(d*tan(e + f*x)))/sqrt(d)))/(sqrt(2)*a^3*sqrt(d)*f) - ((7/32 + (5*im)/32)*log(sqrt(d) + sqrt(d)*tan(e + f*x) - sqrt(2)*sqrt(d*tan(e + f*x))))/(sqrt(2)*a^3*sqrt(d)*f) + ((7/32 + (5*im)/32)*log(sqrt(d) + sqrt(d)*tan(e + f*x) + sqrt(2)*sqrt(d*tan(e + f*x))))/(sqrt(2)*a^3*sqrt(d)*f) + sqrt(d*tan(e + f*x))/(6*d*f*(a + im*a*tan(e + f*x))^3) + sqrt(d*tan(e + f*x))/(3*a*d*f*(a + im*a*tan(e + f*x))^2) + (5*sqrt(d*tan(e + f*x)))/(8*d*f*(a^3 + im*a^3*tan(e + f*x)))]
    @test_int [1/((d*tan(e + f*x))^(3/2)*(a + im*a*tan(e + f*x))^3), x, 14, ((15/8 + (7*im)/4)*atan(1 - (sqrt(2)*sqrt(d*tan(e + f*x)))/sqrt(d)))/(sqrt(2)*a^3*d^(3/2)*f) - ((15/8 + (7*im)/4)*atan(1 + (sqrt(2)*sqrt(d*tan(e + f*x)))/sqrt(d)))/(sqrt(2)*a^3*d^(3/2)*f) - ((15/16 - (7*im)/8)*log(sqrt(d) + sqrt(d)*tan(e + f*x) - sqrt(2)*sqrt(d*tan(e + f*x))))/(sqrt(2)*a^3*d^(3/2)*f) + ((15/16 - (7*im)/8)*log(sqrt(d) + sqrt(d)*tan(e + f*x) + sqrt(2)*sqrt(d*tan(e + f*x))))/(sqrt(2)*a^3*d^(3/2)*f) - 15/(4*a^3*d*f*sqrt(d*tan(e + f*x))) + 1/(6*d*f*sqrt(d*tan(e + f*x))*(a + im*a*tan(e + f*x))^3) + 5/(12*a*d*f*sqrt(d*tan(e + f*x))*(a + im*a*tan(e + f*x))^2) + 7/(6*d*f*sqrt(d*tan(e + f*x))*(a^3 + im*a^3*tan(e + f*x)))]


    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*(a+im*a*tan(e+f*x))^(m/2)*(d*tan(e+f*x))^(n/2)=#


    #= ::Subsubsection::Closed:: =#
    #=m>0=#


    @test_int [tan(c + d*x)^(5/2)*sqrt(a + im*a*tan(c + d*x)), x, 9, (7*(-1)^(3/4)*sqrt(a)*atan(((-1)^(3/4)*sqrt(a)*sqrt(tan(c + d*x)))/sqrt(a + im*a*tan(c + d*x))))/(4*d) + ((1 + im)*sqrt(a)*atanh(((1 + im)*sqrt(a)*sqrt(tan(c + d*x)))/sqrt(a + im*a*tan(c + d*x))))/d - (im*sqrt(tan(c + d*x))*sqrt(a + im*a*tan(c + d*x)))/(4*d) + (tan(c + d*x)^(3/2)*sqrt(a + im*a*tan(c + d*x)))/(2*d)]
    @test_int [tan(c + d*x)^(3/2)*sqrt(a + im*a*tan(c + d*x)), x, 9, -(((-1)^(1/4)*sqrt(a)*atan(((-1)^(3/4)*sqrt(a)*sqrt(tan(c + d*x)))/sqrt(a + im*a*tan(c + d*x))))/d) - ((1 - im)*sqrt(a)*atanh(((1 + im)*sqrt(a)*sqrt(tan(c + d*x)))/sqrt(a + im*a*tan(c + d*x))))/d + (sqrt(tan(c + d*x))*sqrt(a + im*a*tan(c + d*x)))/d]
    @test_int [sqrt(tan(c + d*x))*sqrt(a + im*a*tan(c + d*x)), x, 7, -((2*(-1)^(3/4)*sqrt(a)*atan(((-1)^(3/4)*sqrt(a)*sqrt(tan(c + d*x)))/sqrt(a + im*a*tan(c + d*x))))/d) - ((1 + im)*sqrt(a)*atanh(((1 + im)*sqrt(a)*sqrt(tan(c + d*x)))/sqrt(a + im*a*tan(c + d*x))))/d]
    @test_int [sqrt(a + im*a*tan(c + d*x))/sqrt(tan(c + d*x)), x, 2, ((1 - im)*sqrt(a)*atanh(((1 + im)*sqrt(a)*sqrt(tan(c + d*x)))/sqrt(a + im*a*tan(c + d*x))))/d]
    @test_int [sqrt(a + im*a*tan(c + d*x))/tan(c + d*x)^(3/2), x, 3, ((1 + im)*sqrt(a)*atanh(((1 + im)*sqrt(a)*sqrt(tan(c + d*x)))/sqrt(a + im*a*tan(c + d*x))))/d - (2*sqrt(a + im*a*tan(c + d*x)))/(d*sqrt(tan(c + d*x)))]
    @test_int [sqrt(a + im*a*tan(c + d*x))/tan(c + d*x)^(5/2), x, 5, ((-1 + im)*sqrt(a)*atanh(((1 + im)*sqrt(a)*sqrt(tan(c + d*x)))/sqrt(a + im*a*tan(c + d*x))))/d - (2*sqrt(a + im*a*tan(c + d*x)))/(3*d*tan(c + d*x)^(3/2)) - (((2*im)/3)*sqrt(a + im*a*tan(c + d*x)))/(d*sqrt(tan(c + d*x)))]
    @test_int [sqrt(a + im*a*tan(c + d*x))/tan(c + d*x)^(7/2), x, 6, ((-1 - im)*sqrt(a)*atanh(((1 + im)*sqrt(a)*sqrt(tan(c + d*x)))/sqrt(a + im*a*tan(c + d*x))))/d - (2*sqrt(a + im*a*tan(c + d*x)))/(5*d*tan(c + d*x)^(5/2)) - (((2*im)/15)*sqrt(a + im*a*tan(c + d*x)))/(d*tan(c + d*x)^(3/2)) + (26*sqrt(a + im*a*tan(c + d*x)))/(15*d*sqrt(tan(c + d*x)))]


    @test_int [tan(c + d*x)^(5/2)*(a + im*a*tan(c + d*x))^(3/2), x, 11, (23*(-1)^(3/4)*a^(3/2)*atan(((-1)^(3/4)*sqrt(a)*sqrt(tan(c + d*x)))/sqrt(a + im*a*tan(c + d*x))))/(8*d) + ((2 + 2*im)*a^(3/2)*atanh(((1 + im)*sqrt(a)*sqrt(tan(c + d*x)))/sqrt(a + im*a*tan(c + d*x))))/d + (im*a^2*tan(c + d*x)^(5/2))/(3*d*sqrt(a + im*a*tan(c + d*x))) - (a^2*tan(c + d*x)^(7/2))/(3*d*sqrt(a + im*a*tan(c + d*x))) - (9*im*a*sqrt(tan(c + d*x))*sqrt(a + im*a*tan(c + d*x)))/(8*d) + (7*a*tan(c + d*x)^(3/2)*sqrt(a + im*a*tan(c + d*x)))/(12*d)]
    @test_int [tan(c + d*x)^(3/2)*(a + im*a*tan(c + d*x))^(3/2), x, 10, -((11*(-1)^(1/4)*a^(3/2)*atan(((-1)^(3/4)*sqrt(a)*sqrt(tan(c + d*x)))/sqrt(a + im*a*tan(c + d*x))))/(4*d)) - ((2 - 2*im)*a^(3/2)*atanh(((1 + im)*sqrt(a)*sqrt(tan(c + d*x)))/sqrt(a + im*a*tan(c + d*x))))/d + (im*a^2*tan(c + d*x)^(3/2))/(2*d*sqrt(a + im*a*tan(c + d*x))) - (a^2*tan(c + d*x)^(5/2))/(2*d*sqrt(a + im*a*tan(c + d*x))) + (5*a*sqrt(tan(c + d*x))*sqrt(a + im*a*tan(c + d*x)))/(4*d)]
    @test_int [sqrt(tan(c + d*x))*(a + im*a*tan(c + d*x))^(3/2), x, 9, -((3*(-1)^(3/4)*a^(3/2)*atan(((-1)^(3/4)*sqrt(a)*sqrt(tan(c + d*x)))/sqrt(a + im*a*tan(c + d*x))))/d) - ((2 + 2*im)*a^(3/2)*atanh(((1 + im)*sqrt(a)*sqrt(tan(c + d*x)))/sqrt(a + im*a*tan(c + d*x))))/d + (im*a^2*sqrt(tan(c + d*x)))/(d*sqrt(a + im*a*tan(c + d*x))) - (a^2*tan(c + d*x)^(3/2))/(d*sqrt(a + im*a*tan(c + d*x)))]
    @test_int [(a + im*a*tan(c + d*x))^(3/2)/sqrt(tan(c + d*x)), x, 7, (2*(-1)^(1/4)*a^(3/2)*atan(((-1)^(3/4)*sqrt(a)*sqrt(tan(c + d*x)))/sqrt(a + im*a*tan(c + d*x))))/d + ((2 - 2*im)*a^(3/2)*atanh(((1 + im)*sqrt(a)*sqrt(tan(c + d*x)))/sqrt(a + im*a*tan(c + d*x))))/d]
    @test_int [(a + im*a*tan(c + d*x))^(3/2)/tan(c + d*x)^(3/2), x, 3, ((2 + 2*im)*a^(3/2)*atanh(((1 + im)*sqrt(a)*sqrt(tan(c + d*x)))/sqrt(a + im*a*tan(c + d*x))))/d - (2*a*sqrt(a + im*a*tan(c + d*x)))/(d*sqrt(tan(c + d*x)))]
    @test_int [(a + im*a*tan(c + d*x))^(3/2)/tan(c + d*x)^(5/2), x, 4, ((-2 + 2*im)*a^(3/2)*atanh(((1 + im)*sqrt(a)*sqrt(tan(c + d*x)))/sqrt(a + im*a*tan(c + d*x))))/d - ((2*im)*a*sqrt(a + im*a*tan(c + d*x)))/(d*sqrt(tan(c + d*x))) - (2*(a + im*a*tan(c + d*x))^(3/2))/(3*d*tan(c + d*x)^(3/2))]
    @test_int [(a + im*a*tan(c + d*x))^(3/2)/tan(c + d*x)^(7/2), x, 7, ((-2 - 2*im)*a^(3/2)*atanh(((1 + im)*sqrt(a)*sqrt(tan(c + d*x)))/sqrt(a + im*a*tan(c + d*x))))/d - (2*a^2)/(5*d*tan(c + d*x)^(5/2)*sqrt(a + im*a*tan(c + d*x))) - (((2*im)/5)*a^2)/(d*tan(c + d*x)^(3/2)*sqrt(a + im*a*tan(c + d*x))) - (((4*im)/5)*a*sqrt(a + im*a*tan(c + d*x)))/(d*tan(c + d*x)^(3/2)) + (12*a*sqrt(a + im*a*tan(c + d*x)))/(5*d*sqrt(tan(c + d*x)))]
    @test_int [(a + im*a*tan(c + d*x))^(3/2)/tan(c + d*x)^(9/2), x, 8, ((2 - 2*im)*a^(3/2)*atanh(((1 + im)*sqrt(a)*sqrt(tan(c + d*x)))/sqrt(a + im*a*tan(c + d*x))))/d - (2*a^2)/(7*d*tan(c + d*x)^(7/2)*sqrt(a + im*a*tan(c + d*x))) - (((2*im)/7)*a^2)/(d*tan(c + d*x)^(5/2)*sqrt(a + im*a*tan(c + d*x))) - (((16*im)/35)*a*sqrt(a + im*a*tan(c + d*x)))/(d*tan(c + d*x)^(5/2)) + (76*a*sqrt(a + im*a*tan(c + d*x)))/(105*d*tan(c + d*x)^(3/2)) + (((268*im)/105)*a*sqrt(a + im*a*tan(c + d*x)))/(d*sqrt(tan(c + d*x)))]


    @test_int [tan(c + d*x)^(5/2)*(a + im*a*tan(c + d*x))^(5/2), x, 11, (363*(-1)^(3/4)*a^(5/2)*atan(((-1)^(3/4)*sqrt(a)*sqrt(tan(c + d*x)))/sqrt(a + im*a*tan(c + d*x))))/(64*d) + ((4 + 4*im)*a^(5/2)*atanh(((1 + im)*sqrt(a)*sqrt(tan(c + d*x)))/sqrt(a + im*a*tan(c + d*x))))/d - (149*im*a^2*sqrt(tan(c + d*x))*sqrt(a + im*a*tan(c + d*x)))/(64*d) + (107*a^2*tan(c + d*x)^(3/2)*sqrt(a + im*a*tan(c + d*x)))/(96*d) + (17*im*a^2*tan(c + d*x)^(5/2)*sqrt(a + im*a*tan(c + d*x)))/(24*d) - (a^2*tan(c + d*x)^(7/2)*sqrt(a + im*a*tan(c + d*x)))/(4*d)]
    @test_int [tan(c + d*x)^(3/2)*(a + im*a*tan(c + d*x))^(5/2), x, 10, -((45*(-1)^(1/4)*a^(5/2)*atan(((-1)^(3/4)*sqrt(a)*sqrt(tan(c + d*x)))/sqrt(a + im*a*tan(c + d*x))))/(8*d)) - ((4 - 4*im)*a^(5/2)*atanh(((1 + im)*sqrt(a)*sqrt(tan(c + d*x)))/sqrt(a + im*a*tan(c + d*x))))/d + (19*a^2*sqrt(tan(c + d*x))*sqrt(a + im*a*tan(c + d*x)))/(8*d) + (13*im*a^2*tan(c + d*x)^(3/2)*sqrt(a + im*a*tan(c + d*x)))/(12*d) - (a^2*tan(c + d*x)^(5/2)*sqrt(a + im*a*tan(c + d*x)))/(3*d)]
    @test_int [sqrt(tan(c + d*x))*(a + im*a*tan(c + d*x))^(5/2), x, 9, -((23*(-1)^(3/4)*a^(5/2)*atan(((-1)^(3/4)*sqrt(a)*sqrt(tan(c + d*x)))/sqrt(a + im*a*tan(c + d*x))))/(4*d)) - ((4 + 4*im)*a^(5/2)*atanh(((1 + im)*sqrt(a)*sqrt(tan(c + d*x)))/sqrt(a + im*a*tan(c + d*x))))/d + (9*im*a^2*sqrt(tan(c + d*x))*sqrt(a + im*a*tan(c + d*x)))/(4*d) - (a^2*tan(c + d*x)^(3/2)*sqrt(a + im*a*tan(c + d*x)))/(2*d)]
    @test_int [(a + im*a*tan(c + d*x))^(5/2)/sqrt(tan(c + d*x)), x, 8, (5*(-1)^(1/4)*a^(5/2)*atan(((-1)^(3/4)*sqrt(a)*sqrt(tan(c + d*x)))/sqrt(a + im*a*tan(c + d*x))))/d + ((4 - 4*im)*a^(5/2)*atanh(((1 + im)*sqrt(a)*sqrt(tan(c + d*x)))/sqrt(a + im*a*tan(c + d*x))))/d - (a^2*sqrt(tan(c + d*x))*sqrt(a + im*a*tan(c + d*x)))/d]
    @test_int [(a + im*a*tan(c + d*x))^(5/2)/tan(c + d*x)^(3/2), x, 8, (2*(-1)^(3/4)*a^(5/2)*atan(((-1)^(3/4)*sqrt(a)*sqrt(tan(c + d*x)))/sqrt(a + im*a*tan(c + d*x))))/d + ((4 + 4*im)*a^(5/2)*atanh(((1 + im)*sqrt(a)*sqrt(tan(c + d*x)))/sqrt(a + im*a*tan(c + d*x))))/d - (2*a^2*sqrt(a + im*a*tan(c + d*x)))/(d*sqrt(tan(c + d*x)))]
    @test_int [(a + im*a*tan(c + d*x))^(5/2)/tan(c + d*x)^(5/2), x, 4, ((-4 + 4*im)*a^(5/2)*atanh(((1 + im)*sqrt(a)*sqrt(tan(c + d*x)))/sqrt(a + im*a*tan(c + d*x))))/d - ((4*im)*a^2*sqrt(a + im*a*tan(c + d*x)))/(d*sqrt(tan(c + d*x))) - (2*a*(a + im*a*tan(c + d*x))^(3/2))/(3*d*tan(c + d*x)^(3/2))]
    @test_int [(a + im*a*tan(c + d*x))^(5/2)/tan(c + d*x)^(7/2), x, 5, ((-4 - 4*im)*a^(5/2)*atanh(((1 + im)*sqrt(a)*sqrt(tan(c + d*x)))/sqrt(a + im*a*tan(c + d*x))))/d + (4*a^2*sqrt(a + im*a*tan(c + d*x)))/(d*sqrt(tan(c + d*x))) - (((2*im)/3)*a*(a + im*a*tan(c + d*x))^(3/2))/(d*tan(c + d*x)^(3/2)) - (2*(a + im*a*tan(c + d*x))^(5/2))/(5*d*tan(c + d*x)^(5/2))]
    @test_int [(a + im*a*tan(c + d*x))^(5/2)/tan(c + d*x)^(9/2), x, 7, ((4 - 4*im)*a^(5/2)*atanh(((1 + im)*sqrt(a)*sqrt(tan(c + d*x)))/sqrt(a + im*a*tan(c + d*x))))/d - (2*a^2*sqrt(a + im*a*tan(c + d*x)))/(7*d*tan(c + d*x)^(7/2)) - (((6*im)/7)*a^2*sqrt(a + im*a*tan(c + d*x)))/(d*tan(c + d*x)^(5/2)) + (32*a^2*sqrt(a + im*a*tan(c + d*x)))/(21*d*tan(c + d*x)^(3/2)) + (((104*im)/21)*a^2*sqrt(a + im*a*tan(c + d*x)))/(d*sqrt(tan(c + d*x)))]
    @test_int [(a + im*a*tan(c + d*x))^(5/2)/tan(c + d*x)^(11/2), x, 8, ((4 + 4*im)*a^(5/2)*atanh(((1 + im)*sqrt(a)*sqrt(tan(c + d*x)))/sqrt(a + im*a*tan(c + d*x))))/d - (2*a^2*sqrt(a + im*a*tan(c + d*x)))/(9*d*tan(c + d*x)^(9/2)) - (((38*im)/63)*a^2*sqrt(a + im*a*tan(c + d*x)))/(d*tan(c + d*x)^(7/2)) + (92*a^2*sqrt(a + im*a*tan(c + d*x)))/(105*d*tan(c + d*x)^(5/2)) + (((472*im)/315)*a^2*sqrt(a + im*a*tan(c + d*x)))/(d*tan(c + d*x)^(3/2)) - (1576*a^2*sqrt(a + im*a*tan(c + d*x)))/(315*d*sqrt(tan(c + d*x)))]


    #= ::Subsubsection::Closed:: =#
    #=m<0=#


    @test_int [tan(c + d*x)^(7/2)/sqrt(a + im*a*tan(c + d*x)), x, 10, (11*(-1)^(1/4)*atan(((-1)^(3/4)*sqrt(a)*sqrt(tan(c + d*x)))/sqrt(a + im*a*tan(c + d*x))))/(4*sqrt(a)*d) + ((1/2 - im/2)*atanh(((1 + im)*sqrt(a)*sqrt(tan(c + d*x)))/sqrt(a + im*a*tan(c + d*x))))/(sqrt(a)*d) - tan(c + d*x)^(5/2)/(d*sqrt(a + im*a*tan(c + d*x))) + (7*sqrt(tan(c + d*x))*sqrt(a + im*a*tan(c + d*x)))/(4*a*d) - (3*im*tan(c + d*x)^(3/2)*sqrt(a + im*a*tan(c + d*x)))/(2*a*d)]
    @test_int [tan(c + d*x)^(5/2)/sqrt(a + im*a*tan(c + d*x)), x, 9, -(((-1)^(3/4)*atan(((-1)^(3/4)*sqrt(a)*sqrt(tan(c + d*x)))/sqrt(a + im*a*tan(c + d*x))))/(sqrt(a)*d)) + ((1/2 + im/2)*atanh(((1 + im)*sqrt(a)*sqrt(tan(c + d*x)))/sqrt(a + im*a*tan(c + d*x))))/(sqrt(a)*d) - tan(c + d*x)^(3/2)/(d*sqrt(a + im*a*tan(c + d*x))) - (2*im*sqrt(tan(c + d*x))*sqrt(a + im*a*tan(c + d*x)))/(a*d)]
    @test_int [tan(c + d*x)^(3/2)/sqrt(a + im*a*tan(c + d*x)), x, 8, -((2*(-1)^(1/4)*atan(((-1)^(3/4)*sqrt(a)*sqrt(tan(c + d*x)))/sqrt(a + im*a*tan(c + d*x))))/(sqrt(a)*d)) - ((1/2 - im/2)*atanh(((1 + im)*sqrt(a)*sqrt(tan(c + d*x)))/sqrt(a + im*a*tan(c + d*x))))/(sqrt(a)*d) - sqrt(tan(c + d*x))/(d*sqrt(a + im*a*tan(c + d*x)))]
    @test_int [tan(c + d*x)^(1/2)/sqrt(a + im*a*tan(c + d*x)), x, 3, ((-1/2 - im/2)*atanh(((1 + im)*sqrt(a)*sqrt(tan(c + d*x)))/sqrt(a + im*a*tan(c + d*x))))/(sqrt(a)*d) + (im*sqrt(tan(c + d*x)))/(d*sqrt(a + im*a*tan(c + d*x)))]
    @test_int [1/(tan(c + d*x)^(1/2)*sqrt(a + im*a*tan(c + d*x))), x, 4, ((1/2 - im/2)*atanh(((1 + im)*sqrt(a)*sqrt(tan(c + d*x)))/sqrt(a + im*a*tan(c + d*x))))/(sqrt(a)*d) + sqrt(tan(c + d*x))/(d*sqrt(a + im*a*tan(c + d*x)))]
    @test_int [1/(tan(c + d*x)^(3/2)*sqrt(a + im*a*tan(c + d*x))), x, 5, ((1/2 + im/2)*atanh(((1 + im)*sqrt(a)*sqrt(tan(c + d*x)))/sqrt(a + im*a*tan(c + d*x))))/(sqrt(a)*d) + 1/(d*sqrt(tan(c + d*x))*sqrt(a + im*a*tan(c + d*x))) - (3*sqrt(a + im*a*tan(c + d*x)))/(a*d*sqrt(tan(c + d*x)))]
    @test_int [1/(tan(c + d*x)^(5/2)*sqrt(a + im*a*tan(c + d*x))), x, 6, -(((1/2 - im/2)*atanh(((1 + im)*sqrt(a)*sqrt(tan(c + d*x)))/sqrt(a + im*a*tan(c + d*x))))/(sqrt(a)*d)) + 1/(d*tan(c + d*x)^(3/2)*sqrt(a + im*a*tan(c + d*x))) - (5*sqrt(a + im*a*tan(c + d*x)))/(3*a*d*tan(c + d*x)^(3/2)) + (7*im*sqrt(a + im*a*tan(c + d*x)))/(3*a*d*sqrt(tan(c + d*x)))]
    @test_int [1/(tan(c + d*x)^(7/2)*sqrt(a + im*a*tan(c + d*x))), x, 7, -(((1/2 + im/2)*atanh(((1 + im)*sqrt(a)*sqrt(tan(c + d*x)))/sqrt(a + im*a*tan(c + d*x))))/(sqrt(a)*d)) + 1/(d*tan(c + d*x)^(5/2)*sqrt(a + im*a*tan(c + d*x))) - (7*sqrt(a + im*a*tan(c + d*x)))/(5*a*d*tan(c + d*x)^(5/2)) + (23*im*sqrt(a + im*a*tan(c + d*x)))/(15*a*d*tan(c + d*x)^(3/2)) + (61*sqrt(a + im*a*tan(c + d*x)))/(15*a*d*sqrt(tan(c + d*x)))]


    @test_int [tan(c + d*x)^(7/2)/(a + im*a*tan(c + d*x))^(3/2), x, 10, -((3*(-1)^(1/4)*atan(((-1)^(3/4)*sqrt(a)*sqrt(tan(c + d*x)))/sqrt(a + im*a*tan(c + d*x))))/(a^(3/2)*d)) + ((1/4 - im/4)*atanh(((1 + im)*sqrt(a)*sqrt(tan(c + d*x)))/sqrt(a + im*a*tan(c + d*x))))/(a^(3/2)*d) - tan(c + d*x)^(5/2)/(3*d*(a + im*a*tan(c + d*x))^(3/2)) + (13*im*tan(c + d*x)^(3/2))/(6*a*d*sqrt(a + im*a*tan(c + d*x))) - (7*sqrt(tan(c + d*x))*sqrt(a + im*a*tan(c + d*x)))/(2*a^2*d)]
    @test_int [tan(c + d*x)^(5/2)/(a + im*a*tan(c + d*x))^(3/2), x, 9, (2*(-1)^(3/4)*atan(((-1)^(3/4)*sqrt(a)*sqrt(tan(c + d*x)))/sqrt(a + im*a*tan(c + d*x))))/(a^(3/2)*d) + ((1/4 + im/4)*atanh(((1 + im)*sqrt(a)*sqrt(tan(c + d*x)))/sqrt(a + im*a*tan(c + d*x))))/(a^(3/2)*d) - tan(c + d*x)^(3/2)/(3*d*(a + im*a*tan(c + d*x))^(3/2)) + (3*im*sqrt(tan(c + d*x)))/(2*a*d*sqrt(a + im*a*tan(c + d*x)))]
    @test_int [tan(c + d*x)^(3/2)/(a + im*a*tan(c + d*x))^(3/2), x, 4, ((-1/4 + im/4)*atanh(((1 + im)*sqrt(a)*sqrt(tan(c + d*x)))/sqrt(a + im*a*tan(c + d*x))))/(a^(3/2)*d) + ((im/3)*tan(c + d*x)^(3/2))/(d*(a + im*a*tan(c + d*x))^(3/2)) + sqrt(tan(c + d*x))/(2*a*d*sqrt(a + im*a*tan(c + d*x)))]
    @test_int [tan(c + d*x)^(1/2)/(a + im*a*tan(c + d*x))^(3/2), x, 4, -(((1/4 + im/4)*atanh(((1 + im)*sqrt(a)*sqrt(tan(c + d*x)))/sqrt(a + im*a*tan(c + d*x))))/(a^(3/2)*d)) + tan(c + d*x)^(3/2)/(3*d*(a + im*a*tan(c + d*x))^(3/2)) + (im*sqrt(tan(c + d*x)))/(2*a*d*sqrt(a + im*a*tan(c + d*x)))]
    @test_int [1/(tan(c + d*x)^(1/2)*(a + im*a*tan(c + d*x))^(3/2)), x, 5, ((1/4 - im/4)*atanh(((1 + im)*sqrt(a)*sqrt(tan(c + d*x)))/sqrt(a + im*a*tan(c + d*x))))/(a^(3/2)*d) + sqrt(tan(c + d*x))/(3*d*(a + im*a*tan(c + d*x))^(3/2)) + (7*sqrt(tan(c + d*x)))/(6*a*d*sqrt(a + im*a*tan(c + d*x)))]
    @test_int [1/(tan(c + d*x)^(3/2)*(a + im*a*tan(c + d*x))^(3/2)), x, 6, ((1/4 + im/4)*atanh(((1 + im)*sqrt(a)*sqrt(tan(c + d*x)))/sqrt(a + im*a*tan(c + d*x))))/(a^(3/2)*d) + 1/(3*d*sqrt(tan(c + d*x))*(a + im*a*tan(c + d*x))^(3/2)) + 11/(6*a*d*sqrt(tan(c + d*x))*sqrt(a + im*a*tan(c + d*x))) - (25*sqrt(a + im*a*tan(c + d*x)))/(6*a^2*d*sqrt(tan(c + d*x)))]
    @test_int [1/(tan(c + d*x)^(5/2)*(a + im*a*tan(c + d*x))^(3/2)), x, 7, ((-1/4 + im/4)*atanh(((1 + im)*sqrt(a)*sqrt(tan(c + d*x)))/sqrt(a + im*a*tan(c + d*x))))/(a^(3/2)*d) + 1/(3*d*tan(c + d*x)^(3/2)*(a + im*a*tan(c + d*x))^(3/2)) + 5/(2*a*d*tan(c + d*x)^(3/2)*sqrt(a + im*a*tan(c + d*x))) - (7*sqrt(a + im*a*tan(c + d*x)))/(2*a^2*d*tan(c + d*x)^(3/2)) + (((13*im)/2)*sqrt(a + im*a*tan(c + d*x)))/(a^2*d*sqrt(tan(c + d*x)))]


    @test_int [tan(c + d*x)^(9/2)/(a + im*a*tan(c + d*x))^(5/2), x, 11, (5*(-1)^(3/4)*atan(((-1)^(3/4)*sqrt(a)*sqrt(tan(c + d*x)))/sqrt(a + im*a*tan(c + d*x))))/(a^(5/2)*d) - ((1/8 + im/8)*atanh(((1 + im)*sqrt(a)*sqrt(tan(c + d*x)))/sqrt(a + im*a*tan(c + d*x))))/(a^(5/2)*d) - tan(c + d*x)^(7/2)/(5*d*(a + im*a*tan(c + d*x))^(5/2)) + (19*im*tan(c + d*x)^(5/2))/(30*a*d*(a + im*a*tan(c + d*x))^(3/2)) + (41*tan(c + d*x)^(3/2))/(12*a^2*d*sqrt(a + im*a*tan(c + d*x))) + (21*im*sqrt(tan(c + d*x))*sqrt(a + im*a*tan(c + d*x)))/(4*a^3*d)]
    @test_int [tan(c + d*x)^(7/2)/(a + im*a*tan(c + d*x))^(5/2), x, 10, (2*(-1)^(1/4)*atan(((-1)^(3/4)*sqrt(a)*sqrt(tan(c + d*x)))/sqrt(a + im*a*tan(c + d*x))))/(a^(5/2)*d) + ((1/8 - im/8)*atanh(((1 + im)*sqrt(a)*sqrt(tan(c + d*x)))/sqrt(a + im*a*tan(c + d*x))))/(a^(5/2)*d) - tan(c + d*x)^(5/2)/(5*d*(a + im*a*tan(c + d*x))^(5/2)) + (im*tan(c + d*x)^(3/2))/(2*a*d*(a + im*a*tan(c + d*x))^(3/2)) + (7*sqrt(tan(c + d*x)))/(4*a^2*d*sqrt(a + im*a*tan(c + d*x)))]
    @test_int [tan(c + d*x)^(5/2)/(a + im*a*tan(c + d*x))^(5/2), x, 5, ((1/8 + im/8)*atanh(((1 + im)*sqrt(a)*sqrt(tan(c + d*x)))/sqrt(a + im*a*tan(c + d*x))))/(a^(5/2)*d) + ((im/5)*tan(c + d*x)^(5/2))/(d*(a + im*a*tan(c + d*x))^(5/2)) + tan(c + d*x)^(3/2)/(6*a*d*(a + im*a*tan(c + d*x))^(3/2)) - ((im/4)*sqrt(tan(c + d*x)))/(a^2*d*sqrt(a + im*a*tan(c + d*x)))]
    @test_int [tan(c + d*x)^(3/2)/(a + im*a*tan(c + d*x))^(5/2), x, 5, ((-1/8 + im/8)*atanh(((1 + im)*sqrt(a)*sqrt(tan(c + d*x)))/sqrt(a + im*a*tan(c + d*x))))/(a^(5/2)*d) + tan(c + d*x)^(5/2)/(5*d*(a + im*a*tan(c + d*x))^(5/2)) + ((im/6)*tan(c + d*x)^(3/2))/(a*d*(a + im*a*tan(c + d*x))^(3/2)) + sqrt(tan(c + d*x))/(4*a^2*d*sqrt(a + im*a*tan(c + d*x)))]
    @test_int [tan(c + d*x)^(1/2)/(a + im*a*tan(c + d*x))^(5/2), x, 6, ((-1/8 - im/8)*atanh(((1 + im)*sqrt(a)*sqrt(tan(c + d*x)))/sqrt(a + im*a*tan(c + d*x))))/(a^(5/2)*d) + ((im/5)*sqrt(tan(c + d*x)))/(d*(a + im*a*tan(c + d*x))^(5/2)) + ((im/10)*sqrt(tan(c + d*x)))/(a*d*(a + im*a*tan(c + d*x))^(3/2)) - ((im/20)*sqrt(tan(c + d*x)))/(a^2*d*sqrt(a + im*a*tan(c + d*x)))]
    @test_int [1/(tan(c + d*x)^(1/2)*(a + im*a*tan(c + d*x))^(5/2)), x, 6, ((1/8 - im/8)*atanh(((1 + im)*sqrt(a)*sqrt(tan(c + d*x)))/sqrt(a + im*a*tan(c + d*x))))/(a^(5/2)*d) + sqrt(tan(c + d*x))/(5*d*(a + im*a*tan(c + d*x))^(5/2)) + (13*sqrt(tan(c + d*x)))/(30*a*d*(a + im*a*tan(c + d*x))^(3/2)) + (67*sqrt(tan(c + d*x)))/(60*a^2*d*sqrt(a + im*a*tan(c + d*x)))]
    @test_int [1/(tan(c + d*x)^(3/2)*(a + im*a*tan(c + d*x))^(5/2)), x, 7, ((1/8 + im/8)*atanh(((1 + im)*sqrt(a)*sqrt(tan(c + d*x)))/sqrt(a + im*a*tan(c + d*x))))/(a^(5/2)*d) + 1/(5*d*sqrt(tan(c + d*x))*(a + im*a*tan(c + d*x))^(5/2)) + 17/(30*a*d*sqrt(tan(c + d*x))*(a + im*a*tan(c + d*x))^(3/2)) + 151/(60*a^2*d*sqrt(tan(c + d*x))*sqrt(a + im*a*tan(c + d*x))) - (317*sqrt(a + im*a*tan(c + d*x)))/(60*a^3*d*sqrt(tan(c + d*x)))]
    @test_int [1/(tan(c + d*x)^(5/2)*(a + im*a*tan(c + d*x))^(5/2)), x, 8, ((-1/8 + im/8)*atanh(((1 + im)*sqrt(a)*sqrt(tan(c + d*x)))/sqrt(a + im*a*tan(c + d*x))))/(a^(5/2)*d) + 1/(5*d*tan(c + d*x)^(3/2)*(a + im*a*tan(c + d*x))^(5/2)) + 7/(10*a*d*tan(c + d*x)^(3/2)*(a + im*a*tan(c + d*x))^(3/2)) + 89/(20*a^2*d*tan(c + d*x)^(3/2)*sqrt(a + im*a*tan(c + d*x))) - (361*sqrt(a + im*a*tan(c + d*x)))/(60*a^3*d*tan(c + d*x)^(3/2)) + (((707*im)/60)*sqrt(a + im*a*tan(c + d*x)))/(a^3*d*sqrt(tan(c + d*x)))]


    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*(a+im*a*tan(e+f*x))^m*(d*tan(e+f*x))^(n/3)=#


    #= ::Subsubsection:: =#
    #=m>0=#


    #= ::Subsubsection::Closed:: =#
    #=m<0=#


    @test_int [tan(c + d*x)^(10/3)/(a + im*a*tan(c + d*x)), x, 25, (7*atan(sqrt(3) - 2*tan(c + d*x)^(1/3)))/(12*a*d) - (7*atan(sqrt(3) + 2*tan(c + d*x)^(1/3)))/(12*a*d) - (5*im*atan((1 - 2*tan(c + d*x)^(2/3))/sqrt(3)))/(2*sqrt(3)*a*d) - (7*atan(tan(c + d*x)^(1/3)))/(6*a*d) - (5*im*log(1 + tan(c + d*x)^(2/3)))/(6*a*d) + (7*log(1 - sqrt(3)*tan(c + d*x)^(1/3) + tan(c + d*x)^(2/3)))/(8*sqrt(3)*a*d) - (7*log(1 + sqrt(3)*tan(c + d*x)^(1/3) + tan(c + d*x)^(2/3)))/(8*sqrt(3)*a*d) + (5*im*log(1 - tan(c + d*x)^(2/3) + tan(c + d*x)^(4/3)))/(12*a*d) + (7*tan(c + d*x)^(1/3))/(2*a*d) - (5*im*tan(c + d*x)^(4/3))/(4*a*d) - tan(c + d*x)^(7/3)/(2*d*(a + im*a*tan(c + d*x)))]
    @test_int [tan(c + d*x)^(8/3)/(a + im*a*tan(c + d*x)), x, 24, -((5*atan(sqrt(3) - 2*tan(c + d*x)^(1/3)))/(12*a*d)) + (5*atan(sqrt(3) + 2*tan(c + d*x)^(1/3)))/(12*a*d) - (2*im*atan((1 - 2*tan(c + d*x)^(2/3))/sqrt(3)))/(sqrt(3)*a*d) + (5*atan(tan(c + d*x)^(1/3)))/(6*a*d) + (2*im*log(1 + tan(c + d*x)^(2/3)))/(3*a*d) + (5*log(1 - sqrt(3)*tan(c + d*x)^(1/3) + tan(c + d*x)^(2/3)))/(8*sqrt(3)*a*d) - (5*log(1 + sqrt(3)*tan(c + d*x)^(1/3) + tan(c + d*x)^(2/3)))/(8*sqrt(3)*a*d) - (im*log(1 - tan(c + d*x)^(2/3) + tan(c + d*x)^(4/3)))/(3*a*d) - (2*im*tan(c + d*x)^(2/3))/(a*d) - tan(c + d*x)^(5/3)/(2*d*(a + im*a*tan(c + d*x)))]
    @test_int [tan(c + d*x)^(4/3)/(a + im*a*tan(c + d*x)), x, 23, -(atan(sqrt(3) - 2*tan(c + d*x)^(1/3))/(12*a*d)) + atan(sqrt(3) + 2*tan(c + d*x)^(1/3))/(12*a*d) + (im*atan((1 - 2*tan(c + d*x)^(2/3))/sqrt(3)))/(sqrt(3)*a*d) + atan(tan(c + d*x)^(1/3))/(6*a*d) + (im*log(1 + tan(c + d*x)^(2/3)))/(3*a*d) - log(1 - sqrt(3)*tan(c + d*x)^(1/3) + tan(c + d*x)^(2/3))/(8*sqrt(3)*a*d) + log(1 + sqrt(3)*tan(c + d*x)^(1/3) + tan(c + d*x)^(2/3))/(8*sqrt(3)*a*d) - (im*log(1 - tan(c + d*x)^(2/3) + tan(c + d*x)^(4/3)))/(6*a*d) - tan(c + d*x)^(1/3)/(2*d*(a + im*a*tan(c + d*x)))]
    @test_int [tan(c + d*x)^(2/3)/(a + im*a*tan(c + d*x)), x, 23, -(atan(sqrt(3) - 2*tan(c + d*x)^(1/3))/(12*a*d)) + atan(sqrt(3) + 2*tan(c + d*x)^(1/3))/(12*a*d) + (im*atan((1 - 2*tan(c + d*x)^(2/3))/sqrt(3)))/(2*sqrt(3)*a*d) + atan(tan(c + d*x)^(1/3))/(6*a*d) - (im*log(1 + tan(c + d*x)^(2/3)))/(6*a*d) + log(1 - sqrt(3)*tan(c + d*x)^(1/3) + tan(c + d*x)^(2/3))/(8*sqrt(3)*a*d) - log(1 + sqrt(3)*tan(c + d*x)^(1/3) + tan(c + d*x)^(2/3))/(8*sqrt(3)*a*d) + (im*log(1 - tan(c + d*x)^(2/3) + tan(c + d*x)^(4/3)))/(12*a*d) + (im*tan(c + d*x)^(2/3))/(2*d*(a + im*a*tan(c + d*x)))]
    @test_int [1/(tan(c + d*x)^(1/3)*(a + im*a*tan(c + d*x))), x, 23, (im*atan(sqrt(3) - 2*tan(c + d*x)^(1/3)))/(12*a*d) - (im*atan(sqrt(3) + 2*tan(c + d*x)^(1/3)))/(12*a*d) - atan((1 - 2*tan(c + d*x)^(2/3))/sqrt(3))/(sqrt(3)*a*d) - (im*atan(tan(c + d*x)^(1/3)))/(6*a*d) + log(1 + tan(c + d*x)^(2/3))/(3*a*d) - (im*log(1 - sqrt(3)*tan(c + d*x)^(1/3) + tan(c + d*x)^(2/3)))/(8*sqrt(3)*a*d) + (im*log(1 + sqrt(3)*tan(c + d*x)^(1/3) + tan(c + d*x)^(2/3)))/(8*sqrt(3)*a*d) - log(1 - tan(c + d*x)^(2/3) + tan(c + d*x)^(4/3))/(6*a*d) + tan(c + d*x)^(2/3)/(2*d*(a + im*a*tan(c + d*x)))]
    @test_int [1/(tan(c + d*x)^(5/3)*(a + im*a*tan(c + d*x))), x, 24, (5*im*atan(sqrt(3) - 2*tan(c + d*x)^(1/3)))/(12*a*d) - (5*im*atan(sqrt(3) + 2*tan(c + d*x)^(1/3)))/(12*a*d) + (2*atan((1 - 2*tan(c + d*x)^(2/3))/sqrt(3)))/(sqrt(3)*a*d) - (5*im*atan(tan(c + d*x)^(1/3)))/(6*a*d) + (2*log(1 + tan(c + d*x)^(2/3)))/(3*a*d) + (5*im*log(1 - sqrt(3)*tan(c + d*x)^(1/3) + tan(c + d*x)^(2/3)))/(8*sqrt(3)*a*d) - (5*im*log(1 + sqrt(3)*tan(c + d*x)^(1/3) + tan(c + d*x)^(2/3)))/(8*sqrt(3)*a*d) - log(1 - tan(c + d*x)^(2/3) + tan(c + d*x)^(4/3))/(3*a*d) - 2/(a*d*tan(c + d*x)^(2/3)) + 1/(2*d*tan(c + d*x)^(2/3)*(a + im*a*tan(c + d*x)))]
    @test_int [1/(tan(c + d*x)^(7/3)*(a + im*a*tan(c + d*x))), x, 25, -((7*im*atan(sqrt(3) - 2*tan(c + d*x)^(1/3)))/(12*a*d)) + (7*im*atan(sqrt(3) + 2*tan(c + d*x)^(1/3)))/(12*a*d) + (5*atan((1 - 2*tan(c + d*x)^(2/3))/sqrt(3)))/(2*sqrt(3)*a*d) + (7*im*atan(tan(c + d*x)^(1/3)))/(6*a*d) - (5*log(1 + tan(c + d*x)^(2/3)))/(6*a*d) + (7*im*log(1 - sqrt(3)*tan(c + d*x)^(1/3) + tan(c + d*x)^(2/3)))/(8*sqrt(3)*a*d) - (7*im*log(1 + sqrt(3)*tan(c + d*x)^(1/3) + tan(c + d*x)^(2/3)))/(8*sqrt(3)*a*d) + (5*log(1 - tan(c + d*x)^(2/3) + tan(c + d*x)^(4/3)))/(12*a*d) - 5/(4*a*d*tan(c + d*x)^(4/3)) + (7*im)/(2*a*d*tan(c + d*x)^(1/3)) + 1/(2*d*tan(c + d*x)^(4/3)*(a + im*a*tan(c + d*x)))]


    @test_int [tan(c + d*x)^(14/3)/(a + im*a*tan(c + d*x))^2, x, 26, -((121*atan(sqrt(3) - 2*tan(c + d*x)^(1/3)))/(72*a^2*d)) + (121*atan(sqrt(3) + 2*tan(c + d*x)^(1/3)))/(72*a^2*d) - (14*im*atan((1 - 2*tan(c + d*x)^(2/3))/sqrt(3)))/(3*sqrt(3)*a^2*d) + (121*atan(tan(c + d*x)^(1/3)))/(36*a^2*d) + (14*im*log(1 + tan(c + d*x)^(2/3)))/(9*a^2*d) + (121*log(1 - sqrt(3)*tan(c + d*x)^(1/3) + tan(c + d*x)^(2/3)))/(48*sqrt(3)*a^2*d) - (121*log(1 + sqrt(3)*tan(c + d*x)^(1/3) + tan(c + d*x)^(2/3)))/(48*sqrt(3)*a^2*d) - (7*im*log(1 - tan(c + d*x)^(2/3) + tan(c + d*x)^(4/3)))/(9*a^2*d) - (14*im*tan(c + d*x)^(2/3))/(3*a^2*d) - (121*tan(c + d*x)^(5/3))/(60*a^2*d) + (7*im*tan(c + d*x)^(8/3))/(6*a^2*d*(1 + im*tan(c + d*x))) - tan(c + d*x)^(11/3)/(4*d*(a + im*a*tan(c + d*x))^2)]
    @test_int [tan(c + d*x)^(10/3)/(a + im*a*tan(c + d*x))^2, x, 25, -((49*atan(sqrt(3) - 2*tan(c + d*x)^(1/3)))/(72*a^2*d)) + (49*atan(sqrt(3) + 2*tan(c + d*x)^(1/3)))/(72*a^2*d) + (5*im*atan((1 - 2*tan(c + d*x)^(2/3))/sqrt(3)))/(3*sqrt(3)*a^2*d) + (49*atan(tan(c + d*x)^(1/3)))/(36*a^2*d) + (5*im*log(1 + tan(c + d*x)^(2/3)))/(9*a^2*d) - (49*log(1 - sqrt(3)*tan(c + d*x)^(1/3) + tan(c + d*x)^(2/3)))/(48*sqrt(3)*a^2*d) + (49*log(1 + sqrt(3)*tan(c + d*x)^(1/3) + tan(c + d*x)^(2/3)))/(48*sqrt(3)*a^2*d) - (5*im*log(1 - tan(c + d*x)^(2/3) + tan(c + d*x)^(4/3)))/(18*a^2*d) - (49*tan(c + d*x)^(1/3))/(12*a^2*d) + (5*im*tan(c + d*x)^(4/3))/(6*a^2*d*(1 + im*tan(c + d*x))) - tan(c + d*x)^(7/3)/(4*d*(a + im*a*tan(c + d*x))^2)]
    @test_int [tan(c + d*x)^(8/3)/(a + im*a*tan(c + d*x))^2, x, 24, (25*atan(sqrt(3) - 2*tan(c + d*x)^(1/3)))/(72*a^2*d) - (25*atan(sqrt(3) + 2*tan(c + d*x)^(1/3)))/(72*a^2*d) + (2*im*atan((1 - 2*tan(c + d*x)^(2/3))/sqrt(3)))/(3*sqrt(3)*a^2*d) - (25*atan(tan(c + d*x)^(1/3)))/(36*a^2*d) - (2*im*log(1 + tan(c + d*x)^(2/3)))/(9*a^2*d) - (25*log(1 - sqrt(3)*tan(c + d*x)^(1/3) + tan(c + d*x)^(2/3)))/(48*sqrt(3)*a^2*d) + (25*log(1 + sqrt(3)*tan(c + d*x)^(1/3) + tan(c + d*x)^(2/3)))/(48*sqrt(3)*a^2*d) + (im*log(1 - tan(c + d*x)^(2/3) + tan(c + d*x)^(4/3)))/(9*a^2*d) + (2*im*tan(c + d*x)^(2/3))/(3*a^2*d*(1 + im*tan(c + d*x))) - tan(c + d*x)^(5/3)/(4*d*(a + im*a*tan(c + d*x))^2)]
    @test_int [tan(c + d*x)^(4/3)/(a + im*a*tan(c + d*x))^2, x, 24, atan(sqrt(3) - 2*tan(c + d*x)^(1/3))/(72*a^2*d) - atan(sqrt(3) + 2*tan(c + d*x)^(1/3))/(72*a^2*d) + (im*atan((1 - 2*tan(c + d*x)^(2/3))/sqrt(3)))/(3*sqrt(3)*a^2*d) - atan(tan(c + d*x)^(1/3))/(36*a^2*d) + (im*log(1 + tan(c + d*x)^(2/3)))/(9*a^2*d) + log(1 - sqrt(3)*tan(c + d*x)^(1/3) + tan(c + d*x)^(2/3))/(48*sqrt(3)*a^2*d) - log(1 + sqrt(3)*tan(c + d*x)^(1/3) + tan(c + d*x)^(2/3))/(48*sqrt(3)*a^2*d) - (im*log(1 - tan(c + d*x)^(2/3) + tan(c + d*x)^(4/3)))/(18*a^2*d) + tan(c + d*x)^(1/3)/(3*a^2*d*(1 + im*tan(c + d*x))) - tan(c + d*x)^(1/3)/(4*d*(a + im*a*tan(c + d*x))^2)]
    @test_int [tan(c + d*x)^(2/3)/(a + im*a*tan(c + d*x))^2, x, 24, -(atan(sqrt(3) - 2*tan(c + d*x)^(1/3))/(72*a^2*d)) + atan(sqrt(3) + 2*tan(c + d*x)^(1/3))/(72*a^2*d) + (im*atan((1 - 2*tan(c + d*x)^(2/3))/sqrt(3)))/(3*sqrt(3)*a^2*d) + atan(tan(c + d*x)^(1/3))/(36*a^2*d) - (im*log(1 + tan(c + d*x)^(2/3)))/(9*a^2*d) + log(1 - sqrt(3)*tan(c + d*x)^(1/3) + tan(c + d*x)^(2/3))/(48*sqrt(3)*a^2*d) - log(1 + sqrt(3)*tan(c + d*x)^(1/3) + tan(c + d*x)^(2/3))/(48*sqrt(3)*a^2*d) + (im*log(1 - tan(c + d*x)^(2/3) + tan(c + d*x)^(4/3)))/(18*a^2*d) + (im*tan(c + d*x)^(2/3))/(3*a^2*d*(1 + im*tan(c + d*x))) + tan(c + d*x)^(5/3)/(4*d*(a + im*a*tan(c + d*x))^2)]
    @test_int [1/(tan(c + d*x)^(1/3)*(a + im*a*tan(c + d*x))^2), x, 24, (7*im*atan(sqrt(3) - 2*tan(c + d*x)^(1/3)))/(72*a^2*d) - (7*im*atan(sqrt(3) + 2*tan(c + d*x)^(1/3)))/(72*a^2*d) - (2*atan((1 - 2*tan(c + d*x)^(2/3))/sqrt(3)))/(3*sqrt(3)*a^2*d) - (7*im*atan(tan(c + d*x)^(1/3)))/(36*a^2*d) + (2*log(1 + tan(c + d*x)^(2/3)))/(9*a^2*d) - (7*im*log(1 - sqrt(3)*tan(c + d*x)^(1/3) + tan(c + d*x)^(2/3)))/(48*sqrt(3)*a^2*d) + (7*im*log(1 + sqrt(3)*tan(c + d*x)^(1/3) + tan(c + d*x)^(2/3)))/(48*sqrt(3)*a^2*d) - log(1 - tan(c + d*x)^(2/3) + tan(c + d*x)^(4/3))/(9*a^2*d) + (7*tan(c + d*x)^(2/3))/(12*a^2*d*(1 + im*tan(c + d*x))) + tan(c + d*x)^(2/3)/(4*d*(a + im*a*tan(c + d*x))^2)]
    @test_int [1/(tan(c + d*x)^(5/3)*(a + im*a*tan(c + d*x))^2), x, 25, (55*im*atan(sqrt(3) - 2*tan(c + d*x)^(1/3)))/(72*a^2*d) - (55*im*atan(sqrt(3) + 2*tan(c + d*x)^(1/3)))/(72*a^2*d) + (8*atan((1 - 2*tan(c + d*x)^(2/3))/sqrt(3)))/(3*sqrt(3)*a^2*d) - (55*im*atan(tan(c + d*x)^(1/3)))/(36*a^2*d) + (8*log(1 + tan(c + d*x)^(2/3)))/(9*a^2*d) + (55*im*log(1 - sqrt(3)*tan(c + d*x)^(1/3) + tan(c + d*x)^(2/3)))/(48*sqrt(3)*a^2*d) - (55*im*log(1 + sqrt(3)*tan(c + d*x)^(1/3) + tan(c + d*x)^(2/3)))/(48*sqrt(3)*a^2*d) - (4*log(1 - tan(c + d*x)^(2/3) + tan(c + d*x)^(4/3)))/(9*a^2*d) - 8/(3*a^2*d*tan(c + d*x)^(2/3)) + 11/(12*a^2*d*(1 + im*tan(c + d*x))*tan(c + d*x)^(2/3)) + 1/(4*d*tan(c + d*x)^(2/3)*(a + im*a*tan(c + d*x))^2)]
    @test_int [1/(tan(c + d*x)^(7/3)*(a + im*a*tan(c + d*x))^2), x, 26, -((91*im*atan(sqrt(3) - 2*tan(c + d*x)^(1/3)))/(72*a^2*d)) + (91*im*atan(sqrt(3) + 2*tan(c + d*x)^(1/3)))/(72*a^2*d) + (25*atan((1 - 2*tan(c + d*x)^(2/3))/sqrt(3)))/(6*sqrt(3)*a^2*d) + (91*im*atan(tan(c + d*x)^(1/3)))/(36*a^2*d) - (25*log(1 + tan(c + d*x)^(2/3)))/(18*a^2*d) + (91*im*log(1 - sqrt(3)*tan(c + d*x)^(1/3) + tan(c + d*x)^(2/3)))/(48*sqrt(3)*a^2*d) - (91*im*log(1 + sqrt(3)*tan(c + d*x)^(1/3) + tan(c + d*x)^(2/3)))/(48*sqrt(3)*a^2*d) + (25*log(1 - tan(c + d*x)^(2/3) + tan(c + d*x)^(4/3)))/(36*a^2*d) - 25/(12*a^2*d*tan(c + d*x)^(4/3)) + 13/(12*a^2*d*(1 + im*tan(c + d*x))*tan(c + d*x)^(4/3)) + (91*im)/(12*a^2*d*tan(c + d*x)^(1/3)) + 1/(4*d*tan(c + d*x)^(4/3)*(a + im*a*tan(c + d*x))^2)]


    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*(a+im*a*tan(e+f*x))^(m/2)*(d*tan(e+f*x))^(n/3)=#


    #= ::Subsubsection::Closed:: =#
    #=m>0=#


    @test_int [(a + im*a*tan(c + d*x))^(1/2)*tan(c + d*x)^(4/3), x, 4, (3*a*AppellF1(7/3, 1/2, 1, 10/3, (-im)*tan(c + d*x), im*tan(c + d*x))*sqrt(1 + im*tan(c + d*x))*tan(c + d*x)^(7/3))/(7*d*sqrt(a + im*a*tan(c + d*x)))]
    @test_int [(a + im*a*tan(c + d*x))^(1/2)*tan(c + d*x)^(2/3), x, 4, (3*a*AppellF1(5/3, 1/2, 1, 8/3, (-im)*tan(c + d*x), im*tan(c + d*x))*sqrt(1 + im*tan(c + d*x))*tan(c + d*x)^(5/3))/(5*d*sqrt(a + im*a*tan(c + d*x)))]
    @test_int [(a + im*a*tan(c + d*x))^(1/2)*tan(c + d*x)^(1/3), x, 4, (3*a*AppellF1(4/3, 1/2, 1, 7/3, (-im)*tan(c + d*x), im*tan(c + d*x))*sqrt(1 + im*tan(c + d*x))*tan(c + d*x)^(4/3))/(4*d*sqrt(a + im*a*tan(c + d*x)))]
    @test_int [(a + im*a*tan(c + d*x))^(1/2)/tan(c + d*x)^(1/3), x, 4, (3*a*AppellF1(2/3, 1/2, 1, 5/3, (-im)*tan(c + d*x), im*tan(c + d*x))*sqrt(1 + im*tan(c + d*x))*tan(c + d*x)^(2/3))/(2*d*sqrt(a + im*a*tan(c + d*x)))]
    @test_int [(a + im*a*tan(c + d*x))^(1/2)/tan(c + d*x)^(2/3), x, 4, (3*a*AppellF1(1/3, 1/2, 1, 4/3, (-im)*tan(c + d*x), im*tan(c + d*x))*sqrt(1 + im*tan(c + d*x))*tan(c + d*x)^(1/3))/(d*sqrt(a + im*a*tan(c + d*x)))]
    @test_int [(a + im*a*tan(c + d*x))^(1/2)/tan(c + d*x)^(4/3), x, 4, -((3*a*AppellF1(-(1/3), 1/2, 1, 2/3, (-im)*tan(c + d*x), im*tan(c + d*x))*sqrt(1 + im*tan(c + d*x)))/(d*tan(c + d*x)^(1/3)*sqrt(a + im*a*tan(c + d*x))))]


    @test_int [(a + im*a*tan(c + d*x))^(3/2)*tan(c + d*x)^(4/3), x, 4, (3*a*AppellF1(7/3, -(1/2), 1, 10/3, (-im)*tan(c + d*x), im*tan(c + d*x))*tan(c + d*x)^(7/3)*sqrt(a + im*a*tan(c + d*x)))/(7*d*sqrt(1 + im*tan(c + d*x)))]
    @test_int [(a + im*a*tan(c + d*x))^(3/2)*tan(c + d*x)^(2/3), x, 4, (3*a*AppellF1(5/3, -(1/2), 1, 8/3, (-im)*tan(c + d*x), im*tan(c + d*x))*tan(c + d*x)^(5/3)*sqrt(a + im*a*tan(c + d*x)))/(5*d*sqrt(1 + im*tan(c + d*x)))]
    @test_int [(a + im*a*tan(c + d*x))^(3/2)*tan(c + d*x)^(1/3), x, 4, (3*a*AppellF1(4/3, -(1/2), 1, 7/3, (-im)*tan(c + d*x), im*tan(c + d*x))*tan(c + d*x)^(4/3)*sqrt(a + im*a*tan(c + d*x)))/(4*d*sqrt(1 + im*tan(c + d*x)))]
    @test_int [(a + im*a*tan(c + d*x))^(3/2)/tan(c + d*x)^(1/3), x, 4, (3*a*AppellF1(2/3, -(1/2), 1, 5/3, (-im)*tan(c + d*x), im*tan(c + d*x))*tan(c + d*x)^(2/3)*sqrt(a + im*a*tan(c + d*x)))/(2*d*sqrt(1 + im*tan(c + d*x)))]
    @test_int [(a + im*a*tan(c + d*x))^(3/2)/tan(c + d*x)^(2/3), x, 4, (3*a*AppellF1(1/3, -(1/2), 1, 4/3, (-im)*tan(c + d*x), im*tan(c + d*x))*tan(c + d*x)^(1/3)*sqrt(a + im*a*tan(c + d*x)))/(d*sqrt(1 + im*tan(c + d*x)))]
    @test_int [(a + im*a*tan(c + d*x))^(3/2)/tan(c + d*x)^(4/3), x, 4, -((3*a*AppellF1(-(1/3), -(1/2), 1, 2/3, (-im)*tan(c + d*x), im*tan(c + d*x))*sqrt(a + im*a*tan(c + d*x)))/(d*sqrt(1 + im*tan(c + d*x))*tan(c + d*x)^(1/3)))]


    #= ::Subsubsection::Closed:: =#
    #=m<0=#


    @test_int [tan(c + d*x)^(4/3)/sqrt(a + im*a*tan(c + d*x)), x, 4, (3*AppellF1(7/3, 3/2, 1, 10/3, (-im)*tan(c + d*x), im*tan(c + d*x))*sqrt(1 + im*tan(c + d*x))*tan(c + d*x)^(7/3))/(7*d*sqrt(a + im*a*tan(c + d*x)))]
    @test_int [tan(c + d*x)^(2/3)/sqrt(a + im*a*tan(c + d*x)), x, 4, (3*AppellF1(5/3, 3/2, 1, 8/3, (-im)*tan(c + d*x), im*tan(c + d*x))*sqrt(1 + im*tan(c + d*x))*tan(c + d*x)^(5/3))/(5*d*sqrt(a + im*a*tan(c + d*x)))]
    @test_int [tan(c + d*x)^(1/3)/sqrt(a + im*a*tan(c + d*x)), x, 4, (3*AppellF1(4/3, 3/2, 1, 7/3, (-im)*tan(c + d*x), im*tan(c + d*x))*sqrt(1 + im*tan(c + d*x))*tan(c + d*x)^(4/3))/(4*d*sqrt(a + im*a*tan(c + d*x)))]
    @test_int [1/(tan(c + d*x)^(1/3)*sqrt(a + im*a*tan(c + d*x))), x, 4, (3*AppellF1(2/3, 3/2, 1, 5/3, (-im)*tan(c + d*x), im*tan(c + d*x))*sqrt(1 + im*tan(c + d*x))*tan(c + d*x)^(2/3))/(2*d*sqrt(a + im*a*tan(c + d*x)))]
    @test_int [1/(tan(c + d*x)^(2/3)*sqrt(a + im*a*tan(c + d*x))), x, 4, (3*AppellF1(1/3, 3/2, 1, 4/3, (-im)*tan(c + d*x), im*tan(c + d*x))*sqrt(1 + im*tan(c + d*x))*tan(c + d*x)^(1/3))/(d*sqrt(a + im*a*tan(c + d*x)))]
    @test_int [1/(tan(c + d*x)^(4/3)*sqrt(a + im*a*tan(c + d*x))), x, 4, -((3*AppellF1(-(1/3), 3/2, 1, 2/3, (-im)*tan(c + d*x), im*tan(c + d*x))*sqrt(1 + im*tan(c + d*x)))/(d*tan(c + d*x)^(1/3)*sqrt(a + im*a*tan(c + d*x))))]


    @test_int [tan(c + d*x)^(4/3)/(a + im*a*tan(c + d*x))^(3/2), x, 4, (3*AppellF1(7/3, 5/2, 1, 10/3, (-im)*tan(c + d*x), im*tan(c + d*x))*sqrt(1 + im*tan(c + d*x))*tan(c + d*x)^(7/3))/(7*a*d*sqrt(a + im*a*tan(c + d*x)))]
    @test_int [tan(c + d*x)^(2/3)/(a + im*a*tan(c + d*x))^(3/2), x, 4, (3*AppellF1(5/3, 5/2, 1, 8/3, (-im)*tan(c + d*x), im*tan(c + d*x))*sqrt(1 + im*tan(c + d*x))*tan(c + d*x)^(5/3))/(5*a*d*sqrt(a + im*a*tan(c + d*x)))]
    @test_int [tan(c + d*x)^(1/3)/(a + im*a*tan(c + d*x))^(3/2), x, 4, (3*AppellF1(4/3, 5/2, 1, 7/3, (-im)*tan(c + d*x), im*tan(c + d*x))*sqrt(1 + im*tan(c + d*x))*tan(c + d*x)^(4/3))/(4*a*d*sqrt(a + im*a*tan(c + d*x)))]
    @test_int [1/(tan(c + d*x)^(1/3)*(a + im*a*tan(c + d*x))^(3/2)), x, 4, (3*AppellF1(2/3, 5/2, 1, 5/3, (-im)*tan(c + d*x), im*tan(c + d*x))*sqrt(1 + im*tan(c + d*x))*tan(c + d*x)^(2/3))/(2*a*d*sqrt(a + im*a*tan(c + d*x)))]
    @test_int [1/(tan(c + d*x)^(2/3)*(a + im*a*tan(c + d*x))^(3/2)), x, 4, (3*AppellF1(1/3, 5/2, 1, 4/3, (-im)*tan(c + d*x), im*tan(c + d*x))*sqrt(1 + im*tan(c + d*x))*tan(c + d*x)^(1/3))/(a*d*sqrt(a + im*a*tan(c + d*x)))]
    @test_int [1/(tan(c + d*x)^(4/3)*(a + im*a*tan(c + d*x))^(3/2)), x, 4, -((3*AppellF1(-(1/3), 5/2, 1, 2/3, (-im)*tan(c + d*x), im*tan(c + d*x))*sqrt(1 + im*tan(c + d*x)))/(a*d*tan(c + d*x)^(1/3)*sqrt(a + im*a*tan(c + d*x))))]


    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*(a+im*a*tan(e+f*x))^(m/3)*(d*tan(e+f*x))^n=#


    #= ::Subsubsection::Closed:: =#
    #=m>0=#


    @test_int [tan(c + d*x)^3*(a + im*a*tan(c + d*x))^(1/3), x, 8, -((im*a^(1/3)*x)/(2*2^(2/3))) + (sqrt(3)*a^(1/3)*atan((a^(1/3) + 2^(2/3)*(a + im*a*tan(c + d*x))^(1/3))/(sqrt(3)*a^(1/3))))/(2^(2/3)*d) - (a^(1/3)*log(cos(c + d*x)))/(2*2^(2/3)*d) - (3*a^(1/3)*log(2^(1/3)*a^(1/3) - (a + im*a*tan(c + d*x))^(1/3)))/(2*2^(2/3)*d) - (18*(a + im*a*tan(c + d*x))^(1/3))/(7*d) + (3*tan(c + d*x)^2*(a + im*a*tan(c + d*x))^(1/3))/(7*d) - (3*(a + im*a*tan(c + d*x))^(4/3))/(28*a*d)]
    @test_int [tan(c + d*x)^2*(a + im*a*tan(c + d*x))^(1/3), x, 6, (a^(1/3)*x)/(2*2^(2/3)) + (im*sqrt(3)*a^(1/3)*atan((a^(1/3) + 2^(2/3)*(a + im*a*tan(c + d*x))^(1/3))/(sqrt(3)*a^(1/3))))/(2^(2/3)*d) - (im*a^(1/3)*log(cos(c + d*x)))/(2*2^(2/3)*d) - (3*im*a^(1/3)*log(2^(1/3)*a^(1/3) - (a + im*a*tan(c + d*x))^(1/3)))/(2*2^(2/3)*d) - (3*im*(a + im*a*tan(c + d*x))^(4/3))/(4*a*d)]
    @test_int [tan(c + d*x)*(a + im*a*tan(c + d*x))^(1/3), x, 6, (im*a^(1/3)*x)/(2*2^(2/3)) - (sqrt(3)*a^(1/3)*atan((a^(1/3) + 2^(2/3)*(a + im*a*tan(c + d*x))^(1/3))/(sqrt(3)*a^(1/3))))/(2^(2/3)*d) + (a^(1/3)*log(cos(c + d*x)))/(2*2^(2/3)*d) + (3*a^(1/3)*log(2^(1/3)*a^(1/3) - (a + im*a*tan(c + d*x))^(1/3)))/(2*2^(2/3)*d) + (3*(a + im*a*tan(c + d*x))^(1/3))/d]
    @test_int [(a + im*a*tan(c + d*x))^(1/3), x, 5, -((a^(1/3)*x)/(2*2^(2/3))) - (im*sqrt(3)*a^(1/3)*atan((a^(1/3) + 2^(2/3)*(a + im*a*tan(c + d*x))^(1/3))/(sqrt(3)*a^(1/3))))/(2^(2/3)*d) + (im*a^(1/3)*log(cos(c + d*x)))/(2*2^(2/3)*d) + (3*im*a^(1/3)*log(2^(1/3)*a^(1/3) - (a + im*a*tan(c + d*x))^(1/3)))/(2*2^(2/3)*d)]
    @test_int [cot(c + d*x)*(a + im*a*tan(c + d*x))^(1/3), x, 11, -((im*a^(1/3)*x)/(2*2^(2/3))) - (sqrt(3)*a^(1/3)*atan((a^(1/3) + 2*(a + im*a*tan(c + d*x))^(1/3))/(sqrt(3)*a^(1/3))))/d + (sqrt(3)*a^(1/3)*atan((a^(1/3) + 2^(2/3)*(a + im*a*tan(c + d*x))^(1/3))/(sqrt(3)*a^(1/3))))/(2^(2/3)*d) - (a^(1/3)*log(cos(c + d*x)))/(2*2^(2/3)*d) - (a^(1/3)*log(tan(c + d*x)))/(2*d) + (3*a^(1/3)*log(a^(1/3) - (a + im*a*tan(c + d*x))^(1/3)))/(2*d) - (3*a^(1/3)*log(2^(1/3)*a^(1/3) - (a + im*a*tan(c + d*x))^(1/3)))/(2*2^(2/3)*d)]
    @test_int [cot(c + d*x)^2*(a + im*a*tan(c + d*x))^(1/3), x, 12, (a^(1/3)*x)/(2*2^(2/3)) - (im*a^(1/3)*atan((a^(1/3) + 2*(a + im*a*tan(c + d*x))^(1/3))/(sqrt(3)*a^(1/3))))/(sqrt(3)*d) + (im*sqrt(3)*a^(1/3)*atan((a^(1/3) + 2^(2/3)*(a + im*a*tan(c + d*x))^(1/3))/(sqrt(3)*a^(1/3))))/(2^(2/3)*d) - (im*a^(1/3)*log(cos(c + d*x)))/(2*2^(2/3)*d) - (im*a^(1/3)*log(tan(c + d*x)))/(6*d) + (im*a^(1/3)*log(a^(1/3) - (a + im*a*tan(c + d*x))^(1/3)))/(2*d) - (3*im*a^(1/3)*log(2^(1/3)*a^(1/3) - (a + im*a*tan(c + d*x))^(1/3)))/(2*2^(2/3)*d) - (cot(c + d*x)*(a + im*a*tan(c + d*x))^(1/3))/d]
    @test_int [cot(c + d*x)^3*(a + im*a*tan(c + d*x))^(1/3), x, 13, (im*a^(1/3)*x)/(2*2^(2/3)) + (8*a^(1/3)*atan((a^(1/3) + 2*(a + im*a*tan(c + d*x))^(1/3))/(sqrt(3)*a^(1/3))))/(3*sqrt(3)*d) - (sqrt(3)*a^(1/3)*atan((a^(1/3) + 2^(2/3)*(a + im*a*tan(c + d*x))^(1/3))/(sqrt(3)*a^(1/3))))/(2^(2/3)*d) + (a^(1/3)*log(cos(c + d*x)))/(2*2^(2/3)*d) + (4*a^(1/3)*log(tan(c + d*x)))/(9*d) - (4*a^(1/3)*log(a^(1/3) - (a + im*a*tan(c + d*x))^(1/3)))/(3*d) + (3*a^(1/3)*log(2^(1/3)*a^(1/3) - (a + im*a*tan(c + d*x))^(1/3)))/(2*2^(2/3)*d) - (im*cot(c + d*x)*(a + im*a*tan(c + d*x))^(1/3))/(6*d) - (cot(c + d*x)^2*(a + im*a*tan(c + d*x))^(1/3))/(2*d)]


    @test_int [(a + im*a*tan(c + d*x))^(2/3), x, 5, -((a^(2/3)*x)/(2*2^(1/3))) + (im*sqrt(3)*a^(2/3)*atan((a^(1/3) + 2^(2/3)*(a + im*a*tan(c + d*x))^(1/3))/(sqrt(3)*a^(1/3))))/(2^(1/3)*d) + (im*a^(2/3)*log(cos(c + d*x)))/(2*2^(1/3)*d) + (3*im*a^(2/3)*log(2^(1/3)*a^(1/3) - (a + im*a*tan(c + d*x))^(1/3)))/(2*2^(1/3)*d)]


    @test_int [tan(c + d*x)^3*(a + im*a*tan(c + d*x))^(4/3), x, 9, -((im*a^(4/3)*x)/2^(2/3)) + (2^(1/3)*sqrt(3)*a^(4/3)*atan((a^(1/3) + 2^(2/3)*(a + im*a*tan(c + d*x))^(1/3))/(sqrt(3)*a^(1/3))))/d - (a^(4/3)*log(cos(c + d*x)))/(2^(2/3)*d) - (3*a^(4/3)*log(2^(1/3)*a^(1/3) - (a + im*a*tan(c + d*x))^(1/3)))/(2^(2/3)*d) - (3*a*(a + im*a*tan(c + d*x))^(1/3))/d - (9*(a + im*a*tan(c + d*x))^(4/3))/(20*d) + (3*tan(c + d*x)^2*(a + im*a*tan(c + d*x))^(4/3))/(10*d) - (6*(a + im*a*tan(c + d*x))^(7/3))/(35*a*d)]
    @test_int [tan(c + d*x)^2*(a + im*a*tan(c + d*x))^(4/3), x, 7, (a^(4/3)*x)/2^(2/3) + (im*2^(1/3)*sqrt(3)*a^(4/3)*atan((a^(1/3) + 2^(2/3)*(a + im*a*tan(c + d*x))^(1/3))/(sqrt(3)*a^(1/3))))/d - (im*a^(4/3)*log(cos(c + d*x)))/(2^(2/3)*d) - (3*im*a^(4/3)*log(2^(1/3)*a^(1/3) - (a + im*a*tan(c + d*x))^(1/3)))/(2^(2/3)*d) - (3*im*a*(a + im*a*tan(c + d*x))^(1/3))/d - (3*im*(a + im*a*tan(c + d*x))^(7/3))/(7*a*d)]
    @test_int [tan(c + d*x)*(a + im*a*tan(c + d*x))^(4/3), x, 7, (im*a^(4/3)*x)/2^(2/3) - (2^(1/3)*sqrt(3)*a^(4/3)*atan((a^(1/3) + 2^(2/3)*(a + im*a*tan(c + d*x))^(1/3))/(sqrt(3)*a^(1/3))))/d + (a^(4/3)*log(cos(c + d*x)))/(2^(2/3)*d) + (3*a^(4/3)*log(2^(1/3)*a^(1/3) - (a + im*a*tan(c + d*x))^(1/3)))/(2^(2/3)*d) + (3*a*(a + im*a*tan(c + d*x))^(1/3))/d + (3*(a + im*a*tan(c + d*x))^(4/3))/(4*d)]
    @test_int [(a + im*a*tan(c + d*x))^(4/3), x, 6, -((a^(4/3)*x)/2^(2/3)) - (im*2^(1/3)*sqrt(3)*a^(4/3)*atan((a^(1/3) + 2^(2/3)*(a + im*a*tan(c + d*x))^(1/3))/(sqrt(3)*a^(1/3))))/d + (im*a^(4/3)*log(cos(c + d*x)))/(2^(2/3)*d) + (3*im*a^(4/3)*log(2^(1/3)*a^(1/3) - (a + im*a*tan(c + d*x))^(1/3)))/(2^(2/3)*d) + (3*im*a*(a + im*a*tan(c + d*x))^(1/3))/d]
    @test_int [cot(c + d*x)*(a + im*a*tan(c + d*x))^(4/3), x, 13, -((im*a^(4/3)*x)/2^(2/3)) - (sqrt(3)*a^(4/3)*atan((a^(1/3) + 2*(a + im*a*tan(c + d*x))^(1/3))/(sqrt(3)*a^(1/3))))/d + (2^(1/3)*sqrt(3)*a^(4/3)*atan((a^(1/3) + 2^(2/3)*(a + im*a*tan(c + d*x))^(1/3))/(sqrt(3)*a^(1/3))))/d - (a^(4/3)*log(cos(c + d*x)))/(2^(2/3)*d) - (a^(4/3)*log(tan(c + d*x)))/(2*d) + (3*a^(4/3)*log(a^(1/3) - (a + im*a*tan(c + d*x))^(1/3)))/(2*d) - (3*a^(4/3)*log(2^(1/3)*a^(1/3) - (a + im*a*tan(c + d*x))^(1/3)))/(2^(2/3)*d)]
    @test_int [cot(c + d*x)^2*(a + im*a*tan(c + d*x))^(4/3), x, 13, (a^(4/3)*x)/2^(2/3) - (4*im*a^(4/3)*atan((a^(1/3) + 2*(a + im*a*tan(c + d*x))^(1/3))/(sqrt(3)*a^(1/3))))/(sqrt(3)*d) + (im*2^(1/3)*sqrt(3)*a^(4/3)*atan((a^(1/3) + 2^(2/3)*(a + im*a*tan(c + d*x))^(1/3))/(sqrt(3)*a^(1/3))))/d - (im*a^(4/3)*log(cos(c + d*x)))/(2^(2/3)*d) - (2*im*a^(4/3)*log(tan(c + d*x)))/(3*d) + (2*im*a^(4/3)*log(a^(1/3) - (a + im*a*tan(c + d*x))^(1/3)))/d - (3*im*a^(4/3)*log(2^(1/3)*a^(1/3) - (a + im*a*tan(c + d*x))^(1/3)))/(2^(2/3)*d) + (im*a*(a + im*a*tan(c + d*x))^(1/3))/d - (cot(c + d*x)*(a + im*a*tan(c + d*x))^(4/3))/d]
    @test_int [cot(c + d*x)^3*(a + im*a*tan(c + d*x))^(4/3), x, 13, (im*a^(4/3)*x)/2^(2/3) + (11*a^(4/3)*atan((a^(1/3) + 2*(a + im*a*tan(c + d*x))^(1/3))/(sqrt(3)*a^(1/3))))/(3*sqrt(3)*d) - (2^(1/3)*sqrt(3)*a^(4/3)*atan((a^(1/3) + 2^(2/3)*(a + im*a*tan(c + d*x))^(1/3))/(sqrt(3)*a^(1/3))))/d + (a^(4/3)*log(cos(c + d*x)))/(2^(2/3)*d) + (11*a^(4/3)*log(tan(c + d*x)))/(18*d) - (11*a^(4/3)*log(a^(1/3) - (a + im*a*tan(c + d*x))^(1/3)))/(6*d) + (3*a^(4/3)*log(2^(1/3)*a^(1/3) - (a + im*a*tan(c + d*x))^(1/3)))/(2^(2/3)*d) - (2*im*a*cot(c + d*x)*(a + im*a*tan(c + d*x))^(1/3))/(3*d) - (cot(c + d*x)^2*(a + im*a*tan(c + d*x))^(4/3))/(2*d)]


    @test_int [(a + im*a*tan(c + d*x))^(5/3), x, 6, -((a^(5/3)*x)/2^(1/3)) + (im*2^(2/3)*sqrt(3)*a^(5/3)*atan((a^(1/3) + 2^(2/3)*(a + im*a*tan(c + d*x))^(1/3))/(sqrt(3)*a^(1/3))))/d + (im*a^(5/3)*log(cos(c + d*x)))/(2^(1/3)*d) + (3*im*a^(5/3)*log(2^(1/3)*a^(1/3) - (a + im*a*tan(c + d*x))^(1/3)))/(2^(1/3)*d) + (3*im*a*(a + im*a*tan(c + d*x))^(2/3))/(2*d)]


    #= ::Subsubsection::Closed:: =#
    #=m<0=#


    @test_int [tan(c + d*x)^m/(a + im*a*tan(c + d*x))^(1/3), x, 3, (AppellF1(1 + m, 4/3, 1, 2 + m, (-im)*tan(c + d*x), im*tan(c + d*x))*(1 + im*tan(c + d*x))^(1/3)*tan(c + d*x)^(1 + m))/(d*(1 + m)*(a + im*a*tan(c + d*x))^(1/3))]
    @test_int [tan(c + d*x)^(1/2)/(a + im*a*tan(c + d*x))^(1/3), x, 4, (2*AppellF1(3/2, 4/3, 1, 5/2, (-im)*tan(c + d*x), im*tan(c + d*x))*(1 + im*tan(c + d*x))^(1/3)*tan(c + d*x)^(3/2))/(3*d*(a + im*a*tan(c + d*x))^(1/3))]

    @test_int [tan(c + d*x)^4/(a + im*a*tan(c + d*x))^(1/3), x, 9, -(x/(4*2^(1/3)*a^(1/3))) + (im*sqrt(3)*atan((a^(1/3) + 2^(2/3)*(a + im*a*tan(c + d*x))^(1/3))/(sqrt(3)*a^(1/3))))/(2*2^(1/3)*a^(1/3)*d) + (im*log(cos(c + d*x)))/(4*2^(1/3)*a^(1/3)*d) + (3*im*log(2^(1/3)*a^(1/3) - (a + im*a*tan(c + d*x))^(1/3)))/(4*2^(1/3)*a^(1/3)*d) - (15*im*tan(c + d*x)^2)/(8*d*(a + im*a*tan(c + d*x))^(1/3)) + (3*tan(c + d*x)^3)/(8*d*(a + im*a*tan(c + d*x))^(1/3)) + (45*im*(a + im*a*tan(c + d*x))^(2/3))/(8*a*d) - (39*im*(a + im*a*tan(c + d*x))^(5/3))/(20*a^2*d)]
    @test_int [tan(c + d*x)^3/(a + im*a*tan(c + d*x))^(1/3), x, 8, -((im*x)/(4*2^(1/3)*a^(1/3))) - (sqrt(3)*atan((a^(1/3) + 2^(2/3)*(a + im*a*tan(c + d*x))^(1/3))/(sqrt(3)*a^(1/3))))/(2*2^(1/3)*a^(1/3)*d) - log(cos(c + d*x))/(4*2^(1/3)*a^(1/3)*d) - (3*log(2^(1/3)*a^(1/3) - (a + im*a*tan(c + d*x))^(1/3)))/(4*2^(1/3)*a^(1/3)*d) + 21/(10*d*(a + im*a*tan(c + d*x))^(1/3)) + (3*tan(c + d*x)^2)/(5*d*(a + im*a*tan(c + d*x))^(1/3)) + (3*(a + im*a*tan(c + d*x))^(2/3))/(10*a*d)]
    @test_int [tan(c + d*x)^2/(a + im*a*tan(c + d*x))^(1/3), x, 7, x/(4*2^(1/3)*a^(1/3)) - (im*sqrt(3)*atan((a^(1/3) + 2^(2/3)*(a + im*a*tan(c + d*x))^(1/3))/(sqrt(3)*a^(1/3))))/(2*2^(1/3)*a^(1/3)*d) - (im*log(cos(c + d*x)))/(4*2^(1/3)*a^(1/3)*d) - (3*im*log(2^(1/3)*a^(1/3) - (a + im*a*tan(c + d*x))^(1/3)))/(4*2^(1/3)*a^(1/3)*d) - (3*im)/(2*d*(a + im*a*tan(c + d*x))^(1/3)) - (3*im*(a + im*a*tan(c + d*x))^(2/3))/(2*a*d)]
    @test_int [tan(c + d*x)^1/(a + im*a*tan(c + d*x))^(1/3), x, 6, (im*x)/(4*2^(1/3)*a^(1/3)) + (sqrt(3)*atan((a^(1/3) + 2^(2/3)*(a + im*a*tan(c + d*x))^(1/3))/(sqrt(3)*a^(1/3))))/(2*2^(1/3)*a^(1/3)*d) + log(cos(c + d*x))/(4*2^(1/3)*a^(1/3)*d) + (3*log(2^(1/3)*a^(1/3) - (a + im*a*tan(c + d*x))^(1/3)))/(4*2^(1/3)*a^(1/3)*d) - 3/(2*d*(a + im*a*tan(c + d*x))^(1/3))]
    @test_int [tan(c + d*x)^0/(a + im*a*tan(c + d*x))^(1/3), x, 6, -(x/(4*2^(1/3)*a^(1/3))) + (im*sqrt(3)*atan((a^(1/3) + 2^(2/3)*(a + im*a*tan(c + d*x))^(1/3))/(sqrt(3)*a^(1/3))))/(2*2^(1/3)*a^(1/3)*d) + (im*log(cos(c + d*x)))/(4*2^(1/3)*a^(1/3)*d) + (3*im*log(2^(1/3)*a^(1/3) - (a + im*a*tan(c + d*x))^(1/3)))/(4*2^(1/3)*a^(1/3)*d) + (3*im)/(2*d*(a + im*a*tan(c + d*x))^(1/3))]
    @test_int [cot(c + d*x)^1/(a + im*a*tan(c + d*x))^(1/3), x, 13, -((im*x)/(4*2^(1/3)*a^(1/3))) + (sqrt(3)*atan((a^(1/3) + 2*(a + im*a*tan(c + d*x))^(1/3))/(sqrt(3)*a^(1/3))))/(a^(1/3)*d) - (sqrt(3)*atan((a^(1/3) + 2^(2/3)*(a + im*a*tan(c + d*x))^(1/3))/(sqrt(3)*a^(1/3))))/(2*2^(1/3)*a^(1/3)*d) - log(cos(c + d*x))/(4*2^(1/3)*a^(1/3)*d) - log(tan(c + d*x))/(2*a^(1/3)*d) + (3*log(a^(1/3) - (a + im*a*tan(c + d*x))^(1/3)))/(2*a^(1/3)*d) - (3*log(2^(1/3)*a^(1/3) - (a + im*a*tan(c + d*x))^(1/3)))/(4*2^(1/3)*a^(1/3)*d) + 3/(2*d*(a + im*a*tan(c + d*x))^(1/3))]
    @test_int [cot(c + d*x)^2/(a + im*a*tan(c + d*x))^(1/3), x, 13, x/(4*2^(1/3)*a^(1/3)) - (im*atan((a^(1/3) + 2*(a + im*a*tan(c + d*x))^(1/3))/(sqrt(3)*a^(1/3))))/(sqrt(3)*a^(1/3)*d) - (im*sqrt(3)*atan((a^(1/3) + 2^(2/3)*(a + im*a*tan(c + d*x))^(1/3))/(sqrt(3)*a^(1/3))))/(2*2^(1/3)*a^(1/3)*d) - (im*log(cos(c + d*x)))/(4*2^(1/3)*a^(1/3)*d) + (im*log(tan(c + d*x)))/(6*a^(1/3)*d) - (im*log(a^(1/3) - (a + im*a*tan(c + d*x))^(1/3)))/(2*a^(1/3)*d) - (3*im*log(2^(1/3)*a^(1/3) - (a + im*a*tan(c + d*x))^(1/3)))/(4*2^(1/3)*a^(1/3)*d) - (5*im)/(2*d*(a + im*a*tan(c + d*x))^(1/3)) - cot(c + d*x)/(d*(a + im*a*tan(c + d*x))^(1/3))]


    @test_int [1/(a + im*a*tan(c + d*x))^(2/3), x, 6, -(x/(4*2^(2/3)*a^(2/3))) - (im*sqrt(3)*atan((a^(1/3) + 2^(2/3)*(a + im*a*tan(c + d*x))^(1/3))/(sqrt(3)*a^(1/3))))/(2*2^(2/3)*a^(2/3)*d) + (im*log(cos(c + d*x)))/(4*2^(2/3)*a^(2/3)*d) + (3*im*log(2^(1/3)*a^(1/3) - (a + im*a*tan(c + d*x))^(1/3)))/(4*2^(2/3)*a^(2/3)*d) + (3*im)/(4*d*(a + im*a*tan(c + d*x))^(2/3))]


    @test_int [tan(c + d*x)^m/(a + im*a*tan(c + d*x))^(4/3), x, 3, (AppellF1(1 + m, 7/3, 1, 2 + m, (-im)*tan(c + d*x), im*tan(c + d*x))*(1 + im*tan(c + d*x))^(1/3)*tan(c + d*x)^(1 + m))/(a*d*(1 + m)*(a + im*a*tan(c + d*x))^(1/3))]
    @test_int [tan(c + d*x)^(1/2)/(a + im*a*tan(c + d*x))^(4/3), x, 4, (2*AppellF1(3/2, 7/3, 1, 5/2, (-im)*tan(c + d*x), im*tan(c + d*x))*(1 + im*tan(c + d*x))^(1/3)*tan(c + d*x)^(3/2))/(3*a*d*(a + im*a*tan(c + d*x))^(1/3))]

    @test_int [tan(c + d*x)^4/(a + im*a*tan(c + d*x))^(4/3), x, 9,  -(x/(8*2^(1/3)*a^(4/3))) + (im*sqrt(3)*atan((a^(1/3) + 2^(2/3)*(a + im*a*tan(c + d*x))^(1/3))/(sqrt(3)*a^(1/3))))/(4*2^(1/3)*a^(4/3)*d) + (im*log(cos(c + d*x)))/(8*2^(1/3)*a^(4/3)*d) + (3*im*log(2^(1/3)*a^(1/3) - (a + im*a*tan(c + d*x))^(1/3)))/(8*2^(1/3)*a^(4/3)*d) - (39*im*tan(c + d*x)^2)/(40*d*(a + im*a*tan(c + d*x))^(4/3)) + (3*tan(c + d*x)^3)/(5*d*(a + im*a*tan(c + d*x))^(4/3)) - (51*im)/(10*a*d*(a + im*a*tan(c + d*x))^(1/3)) - (87*im*(a + im*a*tan(c + d*x))^(2/3))/(40*a^2*d)]
    @test_int [tan(c + d*x)^3/(a + im*a*tan(c + d*x))^(4/3), x, 8,  -((im*x)/(8*2^(1/3)*a^(4/3))) - (sqrt(3)*atan((a^(1/3) + 2^(2/3)*(a + im*a*tan(c + d*x))^(1/3))/(sqrt(3)*a^(1/3))))/(4*2^(1/3)*a^(4/3)*d) - log(cos(c + d*x))/(8*2^(1/3)*a^(4/3)*d) - (3*log(2^(1/3)*a^(1/3) - (a + im*a*tan(c + d*x))^(1/3)))/(8*2^(1/3)*a^(4/3)*d) + 15/(8*d*(a + im*a*tan(c + d*x))^(4/3)) + (3*tan(c + d*x)^2)/(2*d*(a + im*a*tan(c + d*x))^(4/3)) - 27/(4*a*d*(a + im*a*tan(c + d*x))^(1/3))]
    @test_int [tan(c + d*x)^2/(a + im*a*tan(c + d*x))^(4/3), x, 7, x/(8*2^(1/3)*a^(4/3)) - (im*sqrt(3)*atan((a^(1/3) + 2^(2/3)*(a + im*a*tan(c + d*x))^(1/3))/(sqrt(3)*a^(1/3))))/(4*2^(1/3)*a^(4/3)*d) - (im*log(cos(c + d*x)))/(8*2^(1/3)*a^(4/3)*d) - (3*im*log(2^(1/3)*a^(1/3) - (a + im*a*tan(c + d*x))^(1/3)))/(8*2^(1/3)*a^(4/3)*d) - (3*im)/(8*d*(a + im*a*tan(c + d*x))^(4/3)) + (9*im)/(4*a*d*(a + im*a*tan(c + d*x))^(1/3))]
    @test_int [tan(c + d*x)^1/(a + im*a*tan(c + d*x))^(4/3), x, 7, (im*x)/(8*2^(1/3)*a^(4/3)) + (sqrt(3)*atan((a^(1/3) + 2^(2/3)*(a + im*a*tan(c + d*x))^(1/3))/(sqrt(3)*a^(1/3))))/(4*2^(1/3)*a^(4/3)*d) + log(cos(c + d*x))/(8*2^(1/3)*a^(4/3)*d) + (3*log(2^(1/3)*a^(1/3) - (a + im*a*tan(c + d*x))^(1/3)))/(8*2^(1/3)*a^(4/3)*d) - 3/(8*d*(a + im*a*tan(c + d*x))^(4/3)) + 3/(4*a*d*(a + im*a*tan(c + d*x))^(1/3))]
    @test_int [tan(c + d*x)^0/(a + im*a*tan(c + d*x))^(4/3), x, 7, -(x/(8*2^(1/3)*a^(4/3))) + (im*sqrt(3)*atan((a^(1/3) + 2^(2/3)*(a + im*a*tan(c + d*x))^(1/3))/(sqrt(3)*a^(1/3))))/(4*2^(1/3)*a^(4/3)*d) + (im*log(cos(c + d*x)))/(8*2^(1/3)*a^(4/3)*d) + (3*im*log(2^(1/3)*a^(1/3) - (a + im*a*tan(c + d*x))^(1/3)))/(8*2^(1/3)*a^(4/3)*d) + (3*im)/(8*d*(a + im*a*tan(c + d*x))^(4/3)) + (3*im)/(4*a*d*(a + im*a*tan(c + d*x))^(1/3))]
    @test_int [cot(c + d*x)^1/(a + im*a*tan(c + d*x))^(4/3), x, 15, -((im*x)/(8*2^(1/3)*a^(4/3))) + (sqrt(3)*atan((a^(1/3) + 2*(a + im*a*tan(c + d*x))^(1/3))/(sqrt(3)*a^(1/3))))/(a^(4/3)*d) - (sqrt(3)*atan((a^(1/3) + 2^(2/3)*(a + im*a*tan(c + d*x))^(1/3))/(sqrt(3)*a^(1/3))))/(4*2^(1/3)*a^(4/3)*d) - log(cos(c + d*x))/(8*2^(1/3)*a^(4/3)*d) - log(tan(c + d*x))/(2*a^(4/3)*d) + (3*log(a^(1/3) - (a + im*a*tan(c + d*x))^(1/3)))/(2*a^(4/3)*d) - (3*log(2^(1/3)*a^(1/3) - (a + im*a*tan(c + d*x))^(1/3)))/(8*2^(1/3)*a^(4/3)*d) + 3/(8*d*(a + im*a*tan(c + d*x))^(4/3)) + 9/(4*a*d*(a + im*a*tan(c + d*x))^(1/3))]
    @test_int [cot(c + d*x)^2/(a + im*a*tan(c + d*x))^(4/3), x, 14, x/(8*2^(1/3)*a^(4/3)) - (4*im*atan((a^(1/3) + 2*(a + im*a*tan(c + d*x))^(1/3))/(sqrt(3)*a^(1/3))))/(sqrt(3)*a^(4/3)*d) - (im*sqrt(3)*atan((a^(1/3) + 2^(2/3)*(a + im*a*tan(c + d*x))^(1/3))/(sqrt(3)*a^(1/3))))/(4*2^(1/3)*a^(4/3)*d) - (im*log(cos(c + d*x)))/(8*2^(1/3)*a^(4/3)*d) + (2*im*log(tan(c + d*x)))/(3*a^(4/3)*d) - (2*im*log(a^(1/3) - (a + im*a*tan(c + d*x))^(1/3)))/(a^(4/3)*d) - (3*im*log(2^(1/3)*a^(1/3) - (a + im*a*tan(c + d*x))^(1/3)))/(8*2^(1/3)*a^(4/3)*d) - (11*im)/(8*d*(a + im*a*tan(c + d*x))^(4/3)) - cot(c + d*x)/(d*(a + im*a*tan(c + d*x))^(4/3)) - (19*im)/(4*a*d*(a + im*a*tan(c + d*x))^(1/3))]


    @test_int [(a + im*a*tan(c + d*x))^(-5/3), x, 7, -(x/(8*2^(2/3)*a^(5/3))) - (im*sqrt(3)*atan((a^(1/3) + 2^(2/3)*(a + im*a*tan(c + d*x))^(1/3))/(sqrt(3)*a^(1/3))))/(4*2^(2/3)*a^(5/3)*d) + (im*log(cos(c + d*x)))/(8*2^(2/3)*a^(5/3)*d) + (3*im*log(2^(1/3)*a^(1/3) - (a + im*a*tan(c + d*x))^(1/3)))/(8*2^(2/3)*a^(5/3)*d) + (3*im)/(10*d*(a + im*a*tan(c + d*x))^(5/3)) + (3*im)/(8*a*d*(a + im*a*tan(c + d*x))^(2/3))]


    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*(a+im*a*tan(e+f*x))^m*(d*tan(e+f*x))^n*with*n*synbolic=#


    @test_int [(e*tan(c + d*x))^m*(a + im*a*tan(c + d*x)), x, 2, (a*HypergeometricFunctions._₂F₁(1, 1 + m, 2 + m, im*tan(c + d*x))*(e*tan(c + d*x))^(1 + m))/(d*e*(1 + m))]
    @test_int [(e*tan(c + d*x))^m*(a - im*a*tan(c + d*x)), x, 2, (a*HypergeometricFunctions._₂F₁(1, 1 + m, 2 + m, (-im)*tan(c + d*x))*(e*tan(c + d*x))^(1 + m))/(d*e*(1 + m))]


    @test_int [(d*tan(e + f*x))^n*(a + im*a*tan(e + f*x))^4, x, 6, If(13>=8, -((2*a^4*(16 + 11*n + 2*n^2)*(d*tan(e + f*x))^(1 + n))/(d*f*(1 + n)*(2 + n)*(3 + n))) + (8*a^4*HypergeometricFunctions._₂F₁(1, 1 + n, 2 + n, im*tan(e + f*x))*(d*tan(e + f*x))^(1 + n))/(d*f*(1 + n)) - ((d*tan(e + f*x))^(1 + n)*(a^2 + im*a^2*tan(e + f*x))^2)/(d*f*(3 + n)) - (2*(4 + n)*(d*tan(e + f*x))^(1 + n)*(a^4 + im*a^4*tan(e + f*x)))/(d*f*(2 + n)*(3 + n)), -((2*a^4*(16 + 11*n + 2*n^2)*(d*tan(e + f*x))^(1 + n))/(d*f*(3 + n)*(2 + 3*n + n^2))) + (8*a^4*HypergeometricFunctions._₂F₁(1, 1 + n, 2 + n, im*tan(e + f*x))*(d*tan(e + f*x))^(1 + n))/(d*f*(1 + n)) - ((d*tan(e + f*x))^(1 + n)*(a^2 + im*a^2*tan(e + f*x))^2)/(d*f*(3 + n)) - (2*(4 + n)*(d*tan(e + f*x))^(1 + n)*(a^4 + im*a^4*tan(e + f*x)))/(d*f*(2 + n)*(3 + n)))]
    @test_int [(d*tan(e + f*x))^n*(a + im*a*tan(e + f*x))^3, x, 5, -((a^3*(5 + 2*n)*(d*tan(e + f*x))^(1 + n))/(d*f*(1 + n)*(2 + n))) + (4*a^3*HypergeometricFunctions._₂F₁(1, 1 + n, 2 + n, im*tan(e + f*x))*(d*tan(e + f*x))^(1 + n))/(d*f*(1 + n)) - ((d*tan(e + f*x))^(1 + n)*(a^3 + im*a^3*tan(e + f*x)))/(d*f*(2 + n))]
    @test_int [(d*tan(e + f*x))^n*(a + im*a*tan(e + f*x))^2, x, 4, -((a^2*(d*tan(e + f*x))^(1 + n))/(d*f*(1 + n))) + (2*a^2*HypergeometricFunctions._₂F₁(1, 1 + n, 2 + n, im*tan(e + f*x))*(d*tan(e + f*x))^(1 + n))/(d*f*(1 + n))]
    @test_int [(d*tan(e + f*x))^n*(a + im*a*tan(e + f*x))^1, x, 2, (a*HypergeometricFunctions._₂F₁(1, 1 + n, 2 + n, im*tan(e + f*x))*(d*tan(e + f*x))^(1 + n))/(d*f*(1 + n))]
    @test_int [(d*tan(e + f*x))^n/(a + im*a*tan(e + f*x))^1, x, 6, ((1 - n)*HypergeometricFunctions._₂F₁(1, (1 + n)/2, (3 + n)/2, -tan(e + f*x)^2)*(d*tan(e + f*x))^(1 + n))/(2*a*d*f*(1 + n)) + (im*n*HypergeometricFunctions._₂F₁(1, (2 + n)/2, (4 + n)/2, -tan(e + f*x)^2)*(d*tan(e + f*x))^(2 + n))/(2*a*d^2*f*(2 + n)) + (d*tan(e + f*x))^(1 + n)/(2*d*f*(a + im*a*tan(e + f*x)))]
    @test_int [(d*tan(e + f*x))^n/(a + im*a*tan(e + f*x))^2, x, 7, ((1 - n)^2*HypergeometricFunctions._₂F₁(1, (1 + n)/2, (3 + n)/2, -tan(e + f*x)^2)*(d*tan(e + f*x))^(1 + n))/(4*a^2*d*f*(1 + n)) + ((2 - n)*(d*tan(e + f*x))^(1 + n))/(4*a^2*d*f*(1 + im*tan(e + f*x))) + (im*(2 - n)*n*HypergeometricFunctions._₂F₁(1, (2 + n)/2, (4 + n)/2, -tan(e + f*x)^2)*(d*tan(e + f*x))^(2 + n))/(4*a^2*d^2*f*(2 + n)) + (d*tan(e + f*x))^(1 + n)/(4*d*f*(a + im*a*tan(e + f*x))^2)]
    @test_int [(d*tan(e + f*x))^n/(a + im*a*tan(e + f*x))^3, x, 8, ((1 - 2*n)*(1 - n)*(3 - n)*HypergeometricFunctions._₂F₁(1, (1 + n)/2, (3 + n)/2, -tan(e + f*x)^2)*(d*tan(e + f*x))^(1 + n))/(24*a^3*d*f*(1 + n)) + (im*(5 - 2*n)*(2 - n)*n*HypergeometricFunctions._₂F₁(1, (2 + n)/2, (4 + n)/2, -tan(e + f*x)^2)*(d*tan(e + f*x))^(2 + n))/(24*a^3*d^2*f*(2 + n)) + (d*tan(e + f*x))^(1 + n)/(6*d*f*(a + im*a*tan(e + f*x))^3) + ((7 - 2*n)*(d*tan(e + f*x))^(1 + n))/(24*a*d*f*(a + im*a*tan(e + f*x))^2) + ((5 - 2*n)*(2 - n)*(d*tan(e + f*x))^(1 + n))/(24*d*f*(a^3 + im*a^3*tan(e + f*x)))]
    @test_int [(d*tan(e + f*x))^n/(a + im*a*tan(e + f*x))^4, x, 9, ((1 - n)*(3 - n)*(1 - 4*n + n^2)*HypergeometricFunctions._₂F₁(1, (1 + n)/2, (3 + n)/2, -tan(e + f*x)^2)*(d*tan(e + f*x))^(1 + n))/(48*a^4*d*f*(1 + n)) + ((13 - 7*n + n^2)*(d*tan(e + f*x))^(1 + n))/(48*a^4*d*f*(1 + im*tan(e + f*x))^2) + ((2 - n)^2*(4 - n)*(d*tan(e + f*x))^(1 + n))/(48*a^4*d*f*(1 + im*tan(e + f*x))) + (im*(2 - n)^2*(4 - n)*n*HypergeometricFunctions._₂F₁(1, (2 + n)/2, (4 + n)/2, -tan(e + f*x)^2)*(d*tan(e + f*x))^(2 + n))/(48*a^4*d^2*f*(2 + n)) + (d*tan(e + f*x))^(1 + n)/(8*d*f*(a + im*a*tan(e + f*x))^4) + ((5 - n)*(d*tan(e + f*x))^(1 + n))/(24*a*d*f*(a + im*a*tan(e + f*x))^3)]

    @test_int [(d*tan(e + f*x))^n*(a - im*a*tan(e + f*x))^1, x, 2, (a*HypergeometricFunctions._₂F₁(1, 1 + n, 2 + n, (-im)*tan(e + f*x))*(d*tan(e + f*x))^(1 + n))/(d*f*(1 + n))]
    @test_int [(d*tan(e + f*x))^n/(a - im*a*tan(e + f*x))^1, x, 6, ((1 - n)*HypergeometricFunctions._₂F₁(1, (1 + n)/2, (3 + n)/2, -tan(e + f*x)^2)*(d*tan(e + f*x))^(1 + n))/(2*a*d*f*(1 + n)) - (im*n*HypergeometricFunctions._₂F₁(1, (2 + n)/2, (4 + n)/2, -tan(e + f*x)^2)*(d*tan(e + f*x))^(2 + n))/(2*a*d^2*f*(2 + n)) + (d*tan(e + f*x))^(1 + n)/(2*d*f*(a - im*a*tan(e + f*x)))]


    @test_int [(d*tan(e + f*x))^n*(a + im*a*tan(e + f*x))^(3/2), x, 3, (a*AppellF1(1 + n, -(1/2), 1, 2 + n, (-im)*tan(e + f*x), im*tan(e + f*x))*(d*tan(e + f*x))^(1 + n)*sqrt(a + im*a*tan(e + f*x)))/(d*f*(1 + n)*sqrt(1 + im*tan(e + f*x)))]
    @test_int [(d*tan(e + f*x))^n*(a + im*a*tan(e + f*x))^(1/2), x, 3, (a*AppellF1(1 + n, 1/2, 1, 2 + n, (-im)*tan(e + f*x), im*tan(e + f*x))*sqrt(1 + im*tan(e + f*x))*(d*tan(e + f*x))^(1 + n))/(d*f*(1 + n)*sqrt(a + im*a*tan(e + f*x)))]
    @test_int [(d*tan(e + f*x))^n/(a + im*a*tan(e + f*x))^(1/2), x, 3, (AppellF1(1 + n, 3/2, 1, 2 + n, (-im)*tan(e + f*x), im*tan(e + f*x))*sqrt(1 + im*tan(e + f*x))*(d*tan(e + f*x))^(1 + n))/(d*f*(1 + n)*sqrt(a + im*a*tan(e + f*x)))]
    @test_int [(d*tan(e + f*x))^n/(a + im*a*tan(e + f*x))^(3/2), x, 3, (AppellF1(1 + n, 5/2, 1, 2 + n, (-im)*tan(e + f*x), im*tan(e + f*x))*sqrt(1 + im*tan(e + f*x))*(d*tan(e + f*x))^(1 + n))/(a*d*f*(1 + n)*sqrt(a + im*a*tan(e + f*x)))]


    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*(a+im*a*tan(e+f*x))^m*(d*tan(e+f*x))^n*with*m*synbolic=#


    @test_int [(a + im*a*tan(e + f*x))^m*(d*tan(e + f*x))^n, x, 3, (AppellF1(1 + n, 1 - m, 1, 2 + n, (-im)*tan(e + f*x), im*tan(e + f*x))*(d*tan(e + f*x))^(1 + n)*(a + im*a*tan(e + f*x))^m)/((1 + im*tan(e + f*x))^m*(d*f*(1 + n)))]


    @test_int [tan(c + d*x)^4*(a + im*a*tan(c + d*x))^m, x, 6, If(13>=8, (2*im*(a + im*a*tan(c + d*x))^m)/(d*(6 + 5*m + m^2)) - (im*HypergeometricFunctions._₂F₁(1, m, 1 + m, (1/2)*(1 + im*tan(c + d*x)))*(a + im*a*tan(c + d*x))^m)/(2*d*m) - (im*m*tan(c + d*x)^2*(a + im*a*tan(c + d*x))^m)/(d*(6 + 5*m + m^2)) + (tan(c + d*x)^3*(a + im*a*tan(c + d*x))^m)/(d*(3 + m)) + (im*(6 + 3*m + m^2)*(a + im*a*tan(c + d*x))^(1 + m))/(a*d*(1 + m)*(2 + m)*(3 + m)), (2*im*(a + im*a*tan(c + d*x))^m)/(d*(6 + 5*m + m^2)) - (im*HypergeometricFunctions._₂F₁(1, m, 1 + m, (1/2)*(1 + im*tan(c + d*x)))*(a + im*a*tan(c + d*x))^m)/(2*d*m) - (im*m*tan(c + d*x)^2*(a + im*a*tan(c + d*x))^m)/(d*(6 + 5*m + m^2)) + (tan(c + d*x)^3*(a + im*a*tan(c + d*x))^m)/(d*(3 + m)) + (im*(6 + 3*m + m^2)*(a + im*a*tan(c + d*x))^(1 + m))/(a*d*(3 + m)*(2 + 3*m + m^2)))]
    @test_int [tan(c + d*x)^3*(a + im*a*tan(c + d*x))^m, x, 5, -((2*(a + im*a*tan(c + d*x))^m)/(d*m*(2 + m))) + (HypergeometricFunctions._₂F₁(1, m, 1 + m, (1/2)*(1 + im*tan(c + d*x)))*(a + im*a*tan(c + d*x))^m)/(2*d*m) + (tan(c + d*x)^2*(a + im*a*tan(c + d*x))^m)/(d*(2 + m)) - (m*(a + im*a*tan(c + d*x))^(1 + m))/(a*d*(2 + 3*m + m^2))]
    @test_int [tan(c + d*x)^2*(a + im*a*tan(c + d*x))^m, x, 3, (im*HypergeometricFunctions._₂F₁(1, m, 1 + m, (1/2)*(1 + im*tan(c + d*x)))*(a + im*a*tan(c + d*x))^m)/(2*d*m) - (im*(a + im*a*tan(c + d*x))^(1 + m))/(a*d*(1 + m))]
    @test_int [tan(c + d*x)^1*(a + im*a*tan(c + d*x))^m, x, 3, (a + im*a*tan(c + d*x))^m/(d*m) - (HypergeometricFunctions._₂F₁(1, m, 1 + m, (1/2)*(1 + im*tan(c + d*x)))*(a + im*a*tan(c + d*x))^m)/(2*d*m)]
    @test_int [tan(c + d*x)^0*(a + im*a*tan(c + d*x))^m, x, 2, -((im*HypergeometricFunctions._₂F₁(1, m, 1 + m, (1/2)*(1 + im*tan(c + d*x)))*(a + im*a*tan(c + d*x))^m)/(2*d*m))]
    @test_int [cot(c + d*x)^1*(a + im*a*tan(c + d*x))^m, x, 5, (HypergeometricFunctions._₂F₁(1, m, 1 + m, (1/2)*(1 + im*tan(c + d*x)))*(a + im*a*tan(c + d*x))^m)/(2*d*m) - (HypergeometricFunctions._₂F₁(1, m, 1 + m, 1 + im*tan(c + d*x))*(a + im*a*tan(c + d*x))^m)/(d*m)]
    @test_int [cot(c + d*x)^2*(a + im*a*tan(c + d*x))^m, x, 6, -((cot(c + d*x)*(a + im*a*tan(c + d*x))^m)/d) + (im*HypergeometricFunctions._₂F₁(1, m, 1 + m, (1/2)*(1 + im*tan(c + d*x)))*(a + im*a*tan(c + d*x))^m)/(2*d*m) - (im*HypergeometricFunctions._₂F₁(1, m, 1 + m, 1 + im*tan(c + d*x))*(a + im*a*tan(c + d*x))^m)/d]


    @test_int [tan(c + d*x)^(3/2)*(a + im*a*tan(c + d*x))^m, x, 4, (2*AppellF1(5/2, 1 - m, 1, 7/2, (-im)*tan(c + d*x), im*tan(c + d*x))*tan(c + d*x)^(5/2)*(a + im*a*tan(c + d*x))^m)/((1 + im*tan(c + d*x))^m*(5*d))]
    @test_int [tan(c + d*x)^(1/2)*(a + im*a*tan(c + d*x))^m, x, 4, (2*AppellF1(3/2, 1 - m, 1, 5/2, (-im)*tan(c + d*x), im*tan(c + d*x))*tan(c + d*x)^(3/2)*(a + im*a*tan(c + d*x))^m)/((1 + im*tan(c + d*x))^m*(3*d))]
    @test_int [(a + im*a*tan(c + d*x))^m/tan(c + d*x)^(1/2), x, 4, (2*AppellF1(1/2, 1 - m, 1, 3/2, (-im)*tan(c + d*x), im*tan(c + d*x))*sqrt(tan(c + d*x))*(a + im*a*tan(c + d*x))^m)/((1 + im*tan(c + d*x))^m*d)]
    @test_int [(a + im*a*tan(c + d*x))^m/tan(c + d*x)^(3/2), x, 4, -((2*AppellF1(-(1/2), 1 - m, 1, 1/2, (-im)*tan(c + d*x), im*tan(c + d*x))*(a + im*a*tan(c + d*x))^m)/((1 + im*tan(c + d*x))^m*(d*sqrt(tan(c + d*x)))))]


    #= ::Section::Closed:: =#
    #=Integrands*of*the*form*(a+a*tan(e+f*x))^m*(d*tan(e+f*x))^n=#


    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*(a+a*tan(e+f*x))^m*(d*tan(e+f*x))^(n/2)=#


    #= ::Subsubsection::Closed:: =#
    #=m>0=#


    @test_int [(a + a*tan(e + f*x))*(d*tan(e + f*x))^(5/2), x, 5, (sqrt(2)*a*d^(5/2)*atanh((sqrt(d) + sqrt(d)*tan(e + f*x))/(sqrt(2)*sqrt(d*tan(e + f*x)))))/f - (2*a*d^2*sqrt(d*tan(e + f*x)))/f + (2*a*d*(d*tan(e + f*x))^(3/2))/(3*f) + (2*a*(d*tan(e + f*x))^(5/2))/(5*f)]
    @test_int [(a + a*tan(e + f*x))*(d*tan(e + f*x))^(3/2), x, 4, (sqrt(2)*a*d^(3/2)*atan((sqrt(d) - sqrt(d)*tan(e + f*x))/(sqrt(2)*sqrt(d*tan(e + f*x)))))/f + (2*a*d*sqrt(d*tan(e + f*x)))/f + (2*a*(d*tan(e + f*x))^(3/2))/(3*f)]
    @test_int [(a + a*tan(e + f*x))*(d*tan(e + f*x))^(1/2), x, 3, -((sqrt(2)*a*sqrt(d)*atanh((sqrt(d) + sqrt(d)*tan(e + f*x))/(sqrt(2)*sqrt(d*tan(e + f*x)))))/f) + (2*a*sqrt(d*tan(e + f*x)))/f]
    @test_int [(a + a*tan(e + f*x))/(d*tan(e + f*x))^(1/2), x, 2, -((sqrt(2)*a*atan((sqrt(d)*(1 - tan(e + f*x)))/(sqrt(2)*sqrt(d*tan(e + f*x)))))/(sqrt(d)*f))]
    @test_int [(a + a*tan(e + f*x))/(d*tan(e + f*x))^(3/2), x, 3, (sqrt(2)*a*atanh((sqrt(d) + sqrt(d)*tan(e + f*x))/(sqrt(2)*sqrt(d*tan(e + f*x)))))/(d^(3/2)*f) - (2*a)/(d*f*sqrt(d*tan(e + f*x)))]
    @test_int [(a + a*tan(e + f*x))/(d*tan(e + f*x))^(5/2), x, 4, (sqrt(2)*a*atan((sqrt(d) - sqrt(d)*tan(e + f*x))/(sqrt(2)*sqrt(d*tan(e + f*x)))))/(d^(5/2)*f) - (2*a)/(3*d*f*(d*tan(e + f*x))^(3/2)) - (2*a)/(d^2*f*sqrt(d*tan(e + f*x)))]
    @test_int [(a + a*tan(e + f*x))/(d*tan(e + f*x))^(7/2), x, 5, -((sqrt(2)*a*atanh((sqrt(d) + sqrt(d)*tan(e + f*x))/(sqrt(2)*sqrt(d*tan(e + f*x)))))/(d^(7/2)*f)) - (2*a)/(5*d*f*(d*tan(e + f*x))^(5/2)) - (2*a)/(3*d^2*f*(d*tan(e + f*x))^(3/2)) + (2*a)/(d^3*f*sqrt(d*tan(e + f*x)))]


    @test_int [(a + a*tan(e + f*x))^2*(d*tan(e + f*x))^(5/2), x, 16, -((sqrt(2)*a^2*d^(5/2)*atan(1 - (sqrt(2)*sqrt(d*tan(e + f*x)))/sqrt(d)))/f) + (sqrt(2)*a^2*d^(5/2)*atan(1 + (sqrt(2)*sqrt(d*tan(e + f*x)))/sqrt(d)))/f - (a^2*d^(5/2)*log(sqrt(d) + sqrt(d)*tan(e + f*x) - sqrt(2)*sqrt(d*tan(e + f*x))))/(sqrt(2)*f) + (a^2*d^(5/2)*log(sqrt(d) + sqrt(d)*tan(e + f*x) + sqrt(2)*sqrt(d*tan(e + f*x))))/(sqrt(2)*f) - (4*a^2*d^2*sqrt(d*tan(e + f*x)))/f + (4*a^2*(d*tan(e + f*x))^(5/2))/(5*f) + (2*a^2*(d*tan(e + f*x))^(7/2))/(7*d*f)]
    @test_int [(a + a*tan(e + f*x))^2*(d*tan(e + f*x))^(3/2), x, 15, (sqrt(2)*a^2*d^(3/2)*atan(1 - (sqrt(2)*sqrt(d*tan(e + f*x)))/sqrt(d)))/f - (sqrt(2)*a^2*d^(3/2)*atan(1 + (sqrt(2)*sqrt(d*tan(e + f*x)))/sqrt(d)))/f - (a^2*d^(3/2)*log(sqrt(d) + sqrt(d)*tan(e + f*x) - sqrt(2)*sqrt(d*tan(e + f*x))))/(sqrt(2)*f) + (a^2*d^(3/2)*log(sqrt(d) + sqrt(d)*tan(e + f*x) + sqrt(2)*sqrt(d*tan(e + f*x))))/(sqrt(2)*f) + (4*a^2*(d*tan(e + f*x))^(3/2))/(3*f) + (2*a^2*(d*tan(e + f*x))^(5/2))/(5*d*f)]
    @test_int [(a + a*tan(e + f*x))^2*(d*tan(e + f*x))^(1/2), x, 15, (sqrt(2)*a^2*sqrt(d)*atan(1 - (sqrt(2)*sqrt(d*tan(e + f*x)))/sqrt(d)))/f - (sqrt(2)*a^2*sqrt(d)*atan(1 + (sqrt(2)*sqrt(d*tan(e + f*x)))/sqrt(d)))/f + (a^2*sqrt(d)*log(sqrt(d) + sqrt(d)*tan(e + f*x) - sqrt(2)*sqrt(d*tan(e + f*x))))/(sqrt(2)*f) - (a^2*sqrt(d)*log(sqrt(d) + sqrt(d)*tan(e + f*x) + sqrt(2)*sqrt(d*tan(e + f*x))))/(sqrt(2)*f) + (4*a^2*sqrt(d*tan(e + f*x)))/f + (2*a^2*(d*tan(e + f*x))^(3/2))/(3*d*f)]
    @test_int [(a + a*tan(e + f*x))^2/(d*tan(e + f*x))^(1/2), x, 14, -((sqrt(2)*a^2*atan(1 - (sqrt(2)*sqrt(d*tan(e + f*x)))/sqrt(d)))/(sqrt(d)*f)) + (sqrt(2)*a^2*atan(1 + (sqrt(2)*sqrt(d*tan(e + f*x)))/sqrt(d)))/(sqrt(d)*f) + (a^2*log(sqrt(d) + sqrt(d)*tan(e + f*x) - sqrt(2)*sqrt(d*tan(e + f*x))))/(sqrt(2)*sqrt(d)*f) - (a^2*log(sqrt(d) + sqrt(d)*tan(e + f*x) + sqrt(2)*sqrt(d*tan(e + f*x))))/(sqrt(2)*sqrt(d)*f) + (2*a^2*sqrt(d*tan(e + f*x)))/(d*f)]
    @test_int [(a + a*tan(e + f*x))^2/(d*tan(e + f*x))^(3/2), x, 13, -((sqrt(2)*a^2*atan(1 - (sqrt(2)*sqrt(d*tan(e + f*x)))/sqrt(d)))/(d^(3/2)*f)) + (sqrt(2)*a^2*atan(1 + (sqrt(2)*sqrt(d*tan(e + f*x)))/sqrt(d)))/(d^(3/2)*f) - (a^2*log(sqrt(d) + sqrt(d)*tan(e + f*x) - sqrt(2)*sqrt(d*tan(e + f*x))))/(sqrt(2)*d^(3/2)*f) + (a^2*log(sqrt(d) + sqrt(d)*tan(e + f*x) + sqrt(2)*sqrt(d*tan(e + f*x))))/(sqrt(2)*d^(3/2)*f) - (2*a^2)/(d*f*sqrt(d*tan(e + f*x)))]
    @test_int [(a + a*tan(e + f*x))^2/(d*tan(e + f*x))^(5/2), x, 14, (sqrt(2)*a^2*atan(1 - (sqrt(2)*sqrt(d*tan(e + f*x)))/sqrt(d)))/(d^(5/2)*f) - (sqrt(2)*a^2*atan(1 + (sqrt(2)*sqrt(d*tan(e + f*x)))/sqrt(d)))/(d^(5/2)*f) - (a^2*log(sqrt(d) + sqrt(d)*tan(e + f*x) - sqrt(2)*sqrt(d*tan(e + f*x))))/(sqrt(2)*d^(5/2)*f) + (a^2*log(sqrt(d) + sqrt(d)*tan(e + f*x) + sqrt(2)*sqrt(d*tan(e + f*x))))/(sqrt(2)*d^(5/2)*f) - (2*a^2)/(3*d*f*(d*tan(e + f*x))^(3/2)) - (4*a^2)/(d^2*f*sqrt(d*tan(e + f*x)))]


    @test_int [(a + a*tan(e + f*x))^3*(d*tan(e + f*x))^(7/2), x, 8, -((2*sqrt(2)*a^3*d^(7/2)*atanh((sqrt(d) + sqrt(d)*tan(e + f*x))/(sqrt(2)*sqrt(d*tan(e + f*x)))))/f) + (4*a^3*d^3*sqrt(d*tan(e + f*x)))/f - (4*a^3*d^2*(d*tan(e + f*x))^(3/2))/(3*f) - (4*a^3*d*(d*tan(e + f*x))^(5/2))/(5*f) + (4*a^3*(d*tan(e + f*x))^(7/2))/(7*f) + (16*a^3*(d*tan(e + f*x))^(9/2))/(33*d*f) + (2*(d*tan(e + f*x))^(9/2)*(a^3 + a^3*tan(e + f*x)))/(11*d*f)]
    @test_int [(a + a*tan(e + f*x))^3*(d*tan(e + f*x))^(5/2), x, 7, -((2*sqrt(2)*a^3*d^(5/2)*atan((sqrt(d) - sqrt(d)*tan(e + f*x))/(sqrt(2)*sqrt(d*tan(e + f*x)))))/f) - (4*a^3*d^2*sqrt(d*tan(e + f*x)))/f - (4*a^3*d*(d*tan(e + f*x))^(3/2))/(3*f) + (4*a^3*(d*tan(e + f*x))^(5/2))/(5*f) + (40*a^3*(d*tan(e + f*x))^(7/2))/(63*d*f) + (2*(d*tan(e + f*x))^(7/2)*(a^3 + a^3*tan(e + f*x)))/(9*d*f)]
    @test_int [(a + a*tan(e + f*x))^3*(d*tan(e + f*x))^(3/2), x, 6, (2*sqrt(2)*a^3*d^(3/2)*atanh((sqrt(d) + sqrt(d)*tan(e + f*x))/(sqrt(2)*sqrt(d*tan(e + f*x)))))/f - (4*a^3*d*sqrt(d*tan(e + f*x)))/f + (4*a^3*(d*tan(e + f*x))^(3/2))/(3*f) + (32*a^3*(d*tan(e + f*x))^(5/2))/(35*d*f) + (2*(d*tan(e + f*x))^(5/2)*(a^3 + a^3*tan(e + f*x)))/(7*d*f)]
    @test_int [(a + a*tan(e + f*x))^3*(d*tan(e + f*x))^(1/2), x, 5, (2*sqrt(2)*a^3*sqrt(d)*atan((sqrt(d) - sqrt(d)*tan(e + f*x))/(sqrt(2)*sqrt(d*tan(e + f*x)))))/f + (4*a^3*sqrt(d*tan(e + f*x)))/f + (8*a^3*(d*tan(e + f*x))^(3/2))/(5*d*f) + (2*(d*tan(e + f*x))^(3/2)*(a^3 + a^3*tan(e + f*x)))/(5*d*f)]
    @test_int [(a + a*tan(e + f*x))^3/(d*tan(e + f*x))^(1/2), x, 4, -((2*sqrt(2)*a^3*atanh((sqrt(d) + sqrt(d)*tan(e + f*x))/(sqrt(2)*sqrt(d*tan(e + f*x)))))/(sqrt(d)*f)) + (16*a^3*sqrt(d*tan(e + f*x)))/(3*d*f) + (2*sqrt(d*tan(e + f*x))*(a^3 + a^3*tan(e + f*x)))/(3*d*f)]
    @test_int [(a + a*tan(e + f*x))^3/(d*tan(e + f*x))^(3/2), x, 4, -((2*sqrt(2)*a^3*atan((sqrt(d) - sqrt(d)*tan(e + f*x))/(sqrt(2)*sqrt(d*tan(e + f*x)))))/(d^(3/2)*f)) + (4*a^3*sqrt(d*tan(e + f*x)))/(d^2*f) - (2*(a^3 + a^3*tan(e + f*x)))/(d*f*sqrt(d*tan(e + f*x)))]
    @test_int [(a + a*tan(e + f*x))^3/(d*tan(e + f*x))^(5/2), x, 4, (2*sqrt(2)*a^3*atanh((sqrt(d) + sqrt(d)*tan(e + f*x))/(sqrt(2)*sqrt(d*tan(e + f*x)))))/(d^(5/2)*f) - (16*a^3)/(3*d^2*f*sqrt(d*tan(e + f*x))) - (2*(a^3 + a^3*tan(e + f*x)))/(3*d*f*(d*tan(e + f*x))^(3/2))]
    @test_int [(a + a*tan(e + f*x))^3/(d*tan(e + f*x))^(7/2), x, 5, (2*sqrt(2)*a^3*atan((sqrt(d) - sqrt(d)*tan(e + f*x))/(sqrt(2)*sqrt(d*tan(e + f*x)))))/(d^(7/2)*f) - (8*a^3)/(5*d^2*f*(d*tan(e + f*x))^(3/2)) - (4*a^3)/(d^3*f*sqrt(d*tan(e + f*x))) - (2*(a^3 + a^3*tan(e + f*x)))/(5*d*f*(d*tan(e + f*x))^(5/2))]
    @test_int [(a + a*tan(e + f*x))^3/(d*tan(e + f*x))^(9/2), x, 6, -((2*sqrt(2)*a^3*atanh((sqrt(d) + sqrt(d)*tan(e + f*x))/(sqrt(2)*sqrt(d*tan(e + f*x)))))/(d^(9/2)*f)) - (32*a^3)/(35*d^2*f*(d*tan(e + f*x))^(5/2)) - (4*a^3)/(3*d^3*f*(d*tan(e + f*x))^(3/2)) + (4*a^3)/(d^4*f*sqrt(d*tan(e + f*x))) - (2*(a^3 + a^3*tan(e + f*x)))/(7*d*f*(d*tan(e + f*x))^(7/2))]


    #= ::Subsubsection::Closed:: =#
    #=m<0=#


    @test_int [1/(a + a*tan(e + f*x))*(d*tan(e + f*x))^(5/2), x, 7, -((d^(5/2)*atan(sqrt(d*tan(e + f*x))/sqrt(d)))/(a*f)) + (d^(5/2)*atan((sqrt(d) - sqrt(d)*tan(e + f*x))/(sqrt(2)*sqrt(d*tan(e + f*x)))))/(sqrt(2)*a*f) + (2*d^2*sqrt(d*tan(e + f*x)))/(a*f)]
    @test_int [1/(a + a*tan(e + f*x))*(d*tan(e + f*x))^(3/2), x, 6, (d^(3/2)*atan(sqrt(d*tan(e + f*x))/sqrt(d)))/(a*f) - (d^(3/2)*atanh((sqrt(d) + sqrt(d)*tan(e + f*x))/(sqrt(2)*sqrt(d*tan(e + f*x)))))/(sqrt(2)*a*f)]
    @test_int [1/(a + a*tan(e + f*x))*(d*tan(e + f*x))^(1/2), x, 6, -((sqrt(d)*atan(sqrt(d*tan(e + f*x))/sqrt(d)))/(a*f)) - (sqrt(d)*atan((sqrt(d) - sqrt(d)*tan(e + f*x))/(sqrt(2)*sqrt(d*tan(e + f*x)))))/(sqrt(2)*a*f)]
    @test_int [1/(a + a*tan(e + f*x))/(d*tan(e + f*x))^(1/2), x, 6, atan(sqrt(d*tan(e + f*x))/sqrt(d))/(a*sqrt(d)*f) + atanh((sqrt(d)*(1 + tan(e + f*x)))/(sqrt(2)*sqrt(d*tan(e + f*x))))/(sqrt(2)*a*sqrt(d)*f)]
    @test_int [1/(a + a*tan(e + f*x))/(d*tan(e + f*x))^(3/2), x, 7, -(atan(sqrt(d*tan(e + f*x))/sqrt(d))/(a*d^(3/2)*f)) + atan((sqrt(d) - sqrt(d)*tan(e + f*x))/(sqrt(2)*sqrt(d*tan(e + f*x))))/(sqrt(2)*a*d^(3/2)*f) - 2/(a*d*f*sqrt(d*tan(e + f*x)))]
    @test_int [1/(a + a*tan(e + f*x))/(d*tan(e + f*x))^(5/2), x, 10, atan(sqrt(d*tan(e + f*x))/sqrt(d))/(a*d^(5/2)*f) - atanh((sqrt(d) + sqrt(d)*tan(e + f*x))/(sqrt(2)*sqrt(d*tan(e + f*x))))/(sqrt(2)*a*d^(5/2)*f) - 2/(3*a*d*f*(d*tan(e + f*x))^(3/2)) + 2/(a*d^2*f*sqrt(d*tan(e + f*x)))]


    @test_int [1/(a + a*tan(e + f*x))^2*(d*tan(e + f*x))^(5/2), x, 17, (3*d^(5/2)*atan(sqrt(d*tan(e + f*x))/sqrt(d)))/(2*a^2*f) + (d^(5/2)*atan(1 - (sqrt(2)*sqrt(d*tan(e + f*x)))/sqrt(d)))/(2*sqrt(2)*a^2*f) - (d^(5/2)*atan(1 + (sqrt(2)*sqrt(d*tan(e + f*x)))/sqrt(d)))/(2*sqrt(2)*a^2*f) + (d^(5/2)*log(sqrt(d) + sqrt(d)*tan(e + f*x) - sqrt(2)*sqrt(d*tan(e + f*x))))/(4*sqrt(2)*a^2*f) - (d^(5/2)*log(sqrt(d) + sqrt(d)*tan(e + f*x) + sqrt(2)*sqrt(d*tan(e + f*x))))/(4*sqrt(2)*a^2*f) - (d^2*sqrt(d*tan(e + f*x)))/(2*f*(a^2 + a^2*tan(e + f*x)))]
    @test_int [1/(a + a*tan(e + f*x))^2*(d*tan(e + f*x))^(3/2), x, 18, -((d^(3/2)*atan(sqrt(d*tan(e + f*x))/sqrt(d)))/(2*a^2*f)) - (d^(3/2)*atan(1 - (sqrt(2)*sqrt(d*tan(e + f*x)))/sqrt(d)))/(2*sqrt(2)*a^2*f) + (d^(3/2)*atan(1 + (sqrt(2)*sqrt(d*tan(e + f*x)))/sqrt(d)))/(2*sqrt(2)*a^2*f) + (d^(3/2)*log(sqrt(d) + sqrt(d)*tan(e + f*x) - sqrt(2)*sqrt(d*tan(e + f*x))))/(4*sqrt(2)*a^2*f) - (d^(3/2)*log(sqrt(d) + sqrt(d)*tan(e + f*x) + sqrt(2)*sqrt(d*tan(e + f*x))))/(4*sqrt(2)*a^2*f) + (d*sqrt(d*tan(e + f*x)))/(2*f*(a^2 + a^2*tan(e + f*x)))]
    @test_int [1/(a + a*tan(e + f*x))^2*(d*tan(e + f*x))^(1/2), x, 17, -((sqrt(d)*atan(sqrt(d*tan(e + f*x))/sqrt(d)))/(2*a^2*f)) - (sqrt(d)*atan(1 - (sqrt(2)*sqrt(d*tan(e + f*x)))/sqrt(d)))/(2*sqrt(2)*a^2*f) + (sqrt(d)*atan(1 + (sqrt(2)*sqrt(d*tan(e + f*x)))/sqrt(d)))/(2*sqrt(2)*a^2*f) - (sqrt(d)*log(sqrt(d) + sqrt(d)*tan(e + f*x) - sqrt(2)*sqrt(d*tan(e + f*x))))/(4*sqrt(2)*a^2*f) + (sqrt(d)*log(sqrt(d) + sqrt(d)*tan(e + f*x) + sqrt(2)*sqrt(d*tan(e + f*x))))/(4*sqrt(2)*a^2*f) - sqrt(d*tan(e + f*x))/(2*f*(a^2 + a^2*tan(e + f*x)))]
    @test_int [1/(a + a*tan(e + f*x))^2/(d*tan(e + f*x))^(1/2), x, 18, (3*atan(sqrt(d*tan(e + f*x))/sqrt(d)))/(2*a^2*sqrt(d)*f) + atan(1 - (sqrt(2)*sqrt(d*tan(e + f*x)))/sqrt(d))/(2*sqrt(2)*a^2*sqrt(d)*f) - atan(1 + (sqrt(2)*sqrt(d*tan(e + f*x)))/sqrt(d))/(2*sqrt(2)*a^2*sqrt(d)*f) - log(sqrt(d) + sqrt(d)*tan(e + f*x) - sqrt(2)*sqrt(d*tan(e + f*x)))/(4*sqrt(2)*a^2*sqrt(d)*f) + log(sqrt(d) + sqrt(d)*tan(e + f*x) + sqrt(2)*sqrt(d*tan(e + f*x)))/(4*sqrt(2)*a^2*sqrt(d)*f) + sqrt(d*tan(e + f*x))/(2*d*f*(a^2 + a^2*tan(e + f*x)))]
    @test_int [1/(a + a*tan(e + f*x))^2/(d*tan(e + f*x))^(3/2), x, 18, -((5*atan(sqrt(d*tan(e + f*x))/sqrt(d)))/(2*a^2*d^(3/2)*f)) + atan(1 - (sqrt(2)*sqrt(d*tan(e + f*x)))/sqrt(d))/(2*sqrt(2)*a^2*d^(3/2)*f) - atan(1 + (sqrt(2)*sqrt(d*tan(e + f*x)))/sqrt(d))/(2*sqrt(2)*a^2*d^(3/2)*f) + log(sqrt(d) + sqrt(d)*tan(e + f*x) - sqrt(2)*sqrt(d*tan(e + f*x)))/(4*sqrt(2)*a^2*d^(3/2)*f) - log(sqrt(d) + sqrt(d)*tan(e + f*x) + sqrt(2)*sqrt(d*tan(e + f*x)))/(4*sqrt(2)*a^2*d^(3/2)*f) - 5/(2*a^2*d*f*sqrt(d*tan(e + f*x))) + 1/(2*d*f*sqrt(d*tan(e + f*x))*(a^2 + a^2*tan(e + f*x)))]
    @test_int [1/(a + a*tan(e + f*x))^2/(d*tan(e + f*x))^(5/2), x, 20, (7*atan(sqrt(d*tan(e + f*x))/sqrt(d)))/(2*a^2*d^(5/2)*f) - atan(1 - (sqrt(2)*sqrt(d*tan(e + f*x)))/sqrt(d))/(2*sqrt(2)*a^2*d^(5/2)*f) + atan(1 + (sqrt(2)*sqrt(d*tan(e + f*x)))/sqrt(d))/(2*sqrt(2)*a^2*d^(5/2)*f) + log(sqrt(d) + sqrt(d)*tan(e + f*x) - sqrt(2)*sqrt(d*tan(e + f*x)))/(4*sqrt(2)*a^2*d^(5/2)*f) - log(sqrt(d) + sqrt(d)*tan(e + f*x) + sqrt(2)*sqrt(d*tan(e + f*x)))/(4*sqrt(2)*a^2*d^(5/2)*f) - 7/(6*a^2*d*f*(d*tan(e + f*x))^(3/2)) + 9/(2*a^2*d^2*f*sqrt(d*tan(e + f*x))) + 1/(2*d*f*(d*tan(e + f*x))^(3/2)*(a^2 + a^2*tan(e + f*x)))]


    @test_int [1/(a + a*tan(e + f*x))^3*(d*tan(e + f*x))^(9/2), x, 9, -((31*d^(9/2)*atan(sqrt(d*tan(e + f*x))/sqrt(d)))/(8*a^3*f)) + (d^(9/2)*atanh((sqrt(d) + sqrt(d)*tan(e + f*x))/(sqrt(2)*sqrt(d*tan(e + f*x)))))/(2*sqrt(2)*a^3*f) + (27*d^4*sqrt(d*tan(e + f*x)))/(8*a^3*f) - (9*d^3*(d*tan(e + f*x))^(3/2))/(8*a^3*f*(1 + tan(e + f*x))) - (d^2*(d*tan(e + f*x))^(5/2))/(4*a*f*(a + a*tan(e + f*x))^2)]
    @test_int [1/(a + a*tan(e + f*x))^3*(d*tan(e + f*x))^(7/2), x, 8, (11*d^(7/2)*atan(sqrt(d*tan(e + f*x))/sqrt(d)))/(8*a^3*f) + (d^(7/2)*atan((sqrt(d) - sqrt(d)*tan(e + f*x))/(sqrt(2)*sqrt(d*tan(e + f*x)))))/(2*sqrt(2)*a^3*f) - (7*d^3*sqrt(d*tan(e + f*x)))/(8*a^3*f*(1 + tan(e + f*x))) - (d^2*(d*tan(e + f*x))^(3/2))/(4*a*f*(a + a*tan(e + f*x))^2)]
    @test_int [1/(a + a*tan(e + f*x))^3*(d*tan(e + f*x))^(5/2), x, 8, (d^(5/2)*atan(sqrt(d*tan(e + f*x))/sqrt(d)))/(8*a^3*f) - (d^(5/2)*atanh((sqrt(d) + sqrt(d)*tan(e + f*x))/(sqrt(2)*sqrt(d*tan(e + f*x)))))/(2*sqrt(2)*a^3*f) + (5*d^2*sqrt(d*tan(e + f*x)))/(8*a^3*f*(1 + tan(e + f*x))) - (d^2*sqrt(d*tan(e + f*x)))/(4*a*f*(a + a*tan(e + f*x))^2)]
    @test_int [1/(a + a*tan(e + f*x))^3*(d*tan(e + f*x))^(3/2), x, 8, -((5*d^(3/2)*atan(sqrt(d*tan(e + f*x))/sqrt(d)))/(8*a^3*f)) - (d^(3/2)*atan((sqrt(d) - sqrt(d)*tan(e + f*x))/(sqrt(2)*sqrt(d*tan(e + f*x)))))/(2*sqrt(2)*a^3*f) + (d*sqrt(d*tan(e + f*x)))/(4*a*f*(a + a*tan(e + f*x))^2) - (d*sqrt(d*tan(e + f*x)))/(8*f*(a^3 + a^3*tan(e + f*x)))]
    @test_int [1/(a + a*tan(e + f*x))^3*(d*tan(e + f*x))^(1/2), x, 8, (sqrt(d)*atan(sqrt(d*tan(e + f*x))/sqrt(d)))/(8*a^3*f) + (sqrt(d)*atanh((sqrt(d) + sqrt(d)*tan(e + f*x))/(sqrt(2)*sqrt(d*tan(e + f*x)))))/(2*sqrt(2)*a^3*f) - sqrt(d*tan(e + f*x))/(4*a*f*(a + a*tan(e + f*x))^2) - (3*sqrt(d*tan(e + f*x)))/(8*f*(a^3 + a^3*tan(e + f*x)))]
    @test_int [1/(a + a*tan(e + f*x))^3/(d*tan(e + f*x))^(1/2), x, 8, (11*atan(sqrt(d*tan(e + f*x))/sqrt(d)))/(8*a^3*sqrt(d)*f) + atan((sqrt(d) - sqrt(d)*tan(e + f*x))/(sqrt(2)*sqrt(d*tan(e + f*x))))/(2*sqrt(2)*a^3*sqrt(d)*f) + (7*sqrt(d*tan(e + f*x)))/(8*a^3*d*f*(1 + tan(e + f*x))) + sqrt(d*tan(e + f*x))/(4*a*d*f*(a + a*tan(e + f*x))^2)]
    @test_int [1/(a + a*tan(e + f*x))^3/(d*tan(e + f*x))^(3/2), x, 9, -((31*atan(sqrt(d*tan(e + f*x))/sqrt(d)))/(8*a^3*d^(3/2)*f)) - atanh((sqrt(d) + sqrt(d)*tan(e + f*x))/(sqrt(2)*sqrt(d*tan(e + f*x))))/(2*sqrt(2)*a^3*d^(3/2)*f) - 27/(8*a^3*d*f*sqrt(d*tan(e + f*x))) + 9/(8*a^3*d*f*sqrt(d*tan(e + f*x))*(1 + tan(e + f*x))) + 1/(4*a*d*f*sqrt(d*tan(e + f*x))*(a + a*tan(e + f*x))^2)]
    @test_int [1/(a + a*tan(e + f*x))^3/(d*tan(e + f*x))^(5/2), x, 10, (59*atan(sqrt(d*tan(e + f*x))/sqrt(d)))/(8*a^3*d^(5/2)*f) - atan((sqrt(d) - sqrt(d)*tan(e + f*x))/(sqrt(2)*sqrt(d*tan(e + f*x))))/(2*sqrt(2)*a^3*d^(5/2)*f) - 55/(24*a^3*d*f*(d*tan(e + f*x))^(3/2)) + 63/(8*a^3*d^2*f*sqrt(d*tan(e + f*x))) + 11/(8*a^3*d*f*(d*tan(e + f*x))^(3/2)*(1 + tan(e + f*x))) + 1/(4*a*d*f*(d*tan(e + f*x))^(3/2)*(a + a*tan(e + f*x))^2)]


    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*(a+a*tan(e+f*x))^(m/2)*tan(e+f*x))^n=#


    #= ::Subsubsection::Closed:: =#
    #=m>0=#


    @test_int [tan(e + f*x)^5*sqrt(1 + tan(e + f*x)), x, 11, -((sqrt((1/2)*(-1 + sqrt(2)))*atan((4 - 3*sqrt(2) + (2 - sqrt(2))*tan(e + f*x))/(2*sqrt(-7 + 5*sqrt(2))*sqrt(1 + tan(e + f*x)))))/f) - (sqrt((1/2)*(1 + sqrt(2)))*atanh((4 + 3*sqrt(2) + (2 + sqrt(2))*tan(e + f*x))/(2*sqrt(7 + 5*sqrt(2))*sqrt(1 + tan(e + f*x)))))/f + (2*sqrt(1 + tan(e + f*x)))/f + (52*(1 + tan(e + f*x))^(3/2))/(315*f) - (26*tan(e + f*x)*(1 + tan(e + f*x))^(3/2))/(105*f) - (4*tan(e + f*x)^2*(1 + tan(e + f*x))^(3/2))/(21*f) + (2*tan(e + f*x)^3*(1 + tan(e + f*x))^(3/2))/(9*f)]
    @test_int [tan(e + f*x)^3*sqrt(1 + tan(e + f*x)), x, 9, (sqrt((1/2)*(-1 + sqrt(2)))*atan((4 - 3*sqrt(2) + (2 - sqrt(2))*tan(e + f*x))/(2*sqrt(-7 + 5*sqrt(2))*sqrt(1 + tan(e + f*x)))))/f + (sqrt((1/2)*(1 + sqrt(2)))*atanh((4 + 3*sqrt(2) + (2 + sqrt(2))*tan(e + f*x))/(2*sqrt(7 + 5*sqrt(2))*sqrt(1 + tan(e + f*x)))))/f - (2*sqrt(1 + tan(e + f*x)))/f - (4*(1 + tan(e + f*x))^(3/2))/(15*f) + (2*tan(e + f*x)*(1 + tan(e + f*x))^(3/2))/(5*f)]
    @test_int [tan(e + f*x)^1*sqrt(1 + tan(e + f*x)), x, 6, -((sqrt((1/2)*(-1 + sqrt(2)))*atan((4 - 3*sqrt(2) + (2 - sqrt(2))*tan(e + f*x))/(2*sqrt(-7 + 5*sqrt(2))*sqrt(1 + tan(e + f*x)))))/f) - (sqrt((1/2)*(1 + sqrt(2)))*atanh((4 + 3*sqrt(2) + (2 + sqrt(2))*tan(e + f*x))/(2*sqrt(7 + 5*sqrt(2))*sqrt(1 + tan(e + f*x)))))/f + (2*sqrt(1 + tan(e + f*x)))/f]
    @test_int [cot(e + f*x)^1*sqrt(1 + tan(e + f*x)), x, 9, (sqrt((1/2)*(-1 + sqrt(2)))*atan((4 - 3*sqrt(2) + (2 - sqrt(2))*tan(e + f*x))/(2*sqrt(-7 + 5*sqrt(2))*sqrt(1 + tan(e + f*x)))))/f - (2*atanh(sqrt(1 + tan(e + f*x))))/f + (sqrt((1/2)*(1 + sqrt(2)))*atanh((4 + 3*sqrt(2) + (2 + sqrt(2))*tan(e + f*x))/(2*sqrt(7 + 5*sqrt(2))*sqrt(1 + tan(e + f*x)))))/f]
    @test_int [cot(e + f*x)^3*sqrt(1 + tan(e + f*x)), x, 11, -((sqrt((1/2)*(-1 + sqrt(2)))*atan((4 - 3*sqrt(2) + (2 - sqrt(2))*tan(e + f*x))/(2*sqrt(-7 + 5*sqrt(2))*sqrt(1 + tan(e + f*x)))))/f) + (9*atanh(sqrt(1 + tan(e + f*x))))/(4*f) - (sqrt((1/2)*(1 + sqrt(2)))*atanh((4 + 3*sqrt(2) + (2 + sqrt(2))*tan(e + f*x))/(2*sqrt(7 + 5*sqrt(2))*sqrt(1 + tan(e + f*x)))))/f - (cot(e + f*x)*sqrt(1 + tan(e + f*x)))/(4*f) - (cot(e + f*x)^2*sqrt(1 + tan(e + f*x)))/(2*f)]
    @test_int [cot(e + f*x)^5*sqrt(1 + tan(e + f*x)), x, 13, (sqrt((1/2)*(-1 + sqrt(2)))*atan((4 - 3*sqrt(2) + (2 - sqrt(2))*tan(e + f*x))/(2*sqrt(-7 + 5*sqrt(2))*sqrt(1 + tan(e + f*x)))))/f - (139*atanh(sqrt(1 + tan(e + f*x))))/(64*f) + (sqrt((1/2)*(1 + sqrt(2)))*atanh((4 + 3*sqrt(2) + (2 + sqrt(2))*tan(e + f*x))/(2*sqrt(7 + 5*sqrt(2))*sqrt(1 + tan(e + f*x)))))/f + (11*cot(e + f*x)*sqrt(1 + tan(e + f*x)))/(64*f) + (53*cot(e + f*x)^2*sqrt(1 + tan(e + f*x)))/(96*f) - (cot(e + f*x)^3*sqrt(1 + tan(e + f*x)))/(24*f) - (cot(e + f*x)^4*sqrt(1 + tan(e + f*x)))/(4*f)]

    @test_int [tan(e + f*x)^4*sqrt(1 + tan(e + f*x)), x, 14, -((sqrt((1/2)*(1 + sqrt(2)))*atan((sqrt(2*(1 + sqrt(2))) - 2*sqrt(1 + tan(e + f*x)))/sqrt(2*(-1 + sqrt(2)))))/f) + (sqrt((1/2)*(1 + sqrt(2)))*atan((sqrt(2*(1 + sqrt(2))) + 2*sqrt(1 + tan(e + f*x)))/sqrt(2*(-1 + sqrt(2)))))/f + log(1 + sqrt(2) + tan(e + f*x) - sqrt(2*(1 + sqrt(2)))*sqrt(1 + tan(e + f*x)))/(2*sqrt(2*(1 + sqrt(2)))*f) - log(1 + sqrt(2) + tan(e + f*x) + sqrt(2*(1 + sqrt(2)))*sqrt(1 + tan(e + f*x)))/(2*sqrt(2*(1 + sqrt(2)))*f) - (18*(1 + tan(e + f*x))^(3/2))/(35*f) - (8*tan(e + f*x)*(1 + tan(e + f*x))^(3/2))/(35*f) + (2*tan(e + f*x)^2*(1 + tan(e + f*x))^(3/2))/(7*f)]
    @test_int [tan(e + f*x)^2*sqrt(1 + tan(e + f*x)), x, 12, (sqrt((1/2)*(1 + sqrt(2)))*atan((sqrt(2*(1 + sqrt(2))) - 2*sqrt(1 + tan(e + f*x)))/sqrt(2*(-1 + sqrt(2)))))/f - (sqrt((1/2)*(1 + sqrt(2)))*atan((sqrt(2*(1 + sqrt(2))) + 2*sqrt(1 + tan(e + f*x)))/sqrt(2*(-1 + sqrt(2)))))/f - log(1 + sqrt(2) + tan(e + f*x) - sqrt(2*(1 + sqrt(2)))*sqrt(1 + tan(e + f*x)))/(2*sqrt(2*(1 + sqrt(2)))*f) + log(1 + sqrt(2) + tan(e + f*x) + sqrt(2*(1 + sqrt(2)))*sqrt(1 + tan(e + f*x)))/(2*sqrt(2*(1 + sqrt(2)))*f) + (2*(1 + tan(e + f*x))^(3/2))/(3*f)]
    @test_int [tan(e + f*x)^0*sqrt(1 + tan(e + f*x)), x, 11, -((sqrt((1/2)*(1 + sqrt(2)))*atan((sqrt(2*(1 + sqrt(2))) - 2*sqrt(1 + tan(e + f*x)))/sqrt(2*(-1 + sqrt(2)))))/f) + (sqrt((1/2)*(1 + sqrt(2)))*atan((sqrt(2*(1 + sqrt(2))) + 2*sqrt(1 + tan(e + f*x)))/sqrt(2*(-1 + sqrt(2)))))/f + log(1 + sqrt(2) + tan(e + f*x) - sqrt(2*(1 + sqrt(2)))*sqrt(1 + tan(e + f*x)))/(2*sqrt(2*(1 + sqrt(2)))*f) - log(1 + sqrt(2) + tan(e + f*x) + sqrt(2*(1 + sqrt(2)))*sqrt(1 + tan(e + f*x)))/(2*sqrt(2*(1 + sqrt(2)))*f)]
    @test_int [cot(e + f*x)^2*sqrt(1 + tan(e + f*x)), x, 16, (sqrt((1/2)*(1 + sqrt(2)))*atan((sqrt(2*(1 + sqrt(2))) - 2*sqrt(1 + tan(e + f*x)))/sqrt(2*(-1 + sqrt(2)))))/f - (sqrt((1/2)*(1 + sqrt(2)))*atan((sqrt(2*(1 + sqrt(2))) + 2*sqrt(1 + tan(e + f*x)))/sqrt(2*(-1 + sqrt(2)))))/f - atanh(sqrt(1 + tan(e + f*x)))/f - log(1 + sqrt(2) + tan(e + f*x) - sqrt(2*(1 + sqrt(2)))*sqrt(1 + tan(e + f*x)))/(2*sqrt(2*(1 + sqrt(2)))*f) + log(1 + sqrt(2) + tan(e + f*x) + sqrt(2*(1 + sqrt(2)))*sqrt(1 + tan(e + f*x)))/(2*sqrt(2*(1 + sqrt(2)))*f) - (cot(e + f*x)*sqrt(1 + tan(e + f*x)))/f]
    @test_int [cot(e + f*x)^4*sqrt(1 + tan(e + f*x)), x, 19, -((sqrt((1/2)*(1 + sqrt(2)))*atan((sqrt(2*(1 + sqrt(2))) - 2*sqrt(1 + tan(e + f*x)))/sqrt(2*(-1 + sqrt(2)))))/f) + (sqrt((1/2)*(1 + sqrt(2)))*atan((sqrt(2*(1 + sqrt(2))) + 2*sqrt(1 + tan(e + f*x)))/sqrt(2*(-1 + sqrt(2)))))/f + (7*atanh(sqrt(1 + tan(e + f*x))))/(8*f) + log(1 + sqrt(2) + tan(e + f*x) - sqrt(2*(1 + sqrt(2)))*sqrt(1 + tan(e + f*x)))/(2*sqrt(2*(1 + sqrt(2)))*f) - log(1 + sqrt(2) + tan(e + f*x) + sqrt(2*(1 + sqrt(2)))*sqrt(1 + tan(e + f*x)))/(2*sqrt(2*(1 + sqrt(2)))*f) + (9*cot(e + f*x)*sqrt(1 + tan(e + f*x)))/(8*f) - (cot(e + f*x)^2*sqrt(1 + tan(e + f*x)))/(12*f) - (cot(e + f*x)^3*sqrt(1 + tan(e + f*x)))/(3*f)]


    @test_int [tan(e + f*x)^5*(1 + tan(e + f*x))^(3/2), x, 19, (sqrt(1 + sqrt(2))*atan((sqrt(2*(1 + sqrt(2))) - 2*sqrt(1 + tan(e + f*x)))/sqrt(2*(-1 + sqrt(2)))))/f - (sqrt(1 + sqrt(2))*atan((sqrt(2*(1 + sqrt(2))) + 2*sqrt(1 + tan(e + f*x)))/sqrt(2*(-1 + sqrt(2)))))/f + log(1 + sqrt(2) + tan(e + f*x) - sqrt(2*(1 + sqrt(2)))*sqrt(1 + tan(e + f*x)))/(2*sqrt(1 + sqrt(2))*f) - log(1 + sqrt(2) + tan(e + f*x) + sqrt(2*(1 + sqrt(2)))*sqrt(1 + tan(e + f*x)))/(2*sqrt(1 + sqrt(2))*f) + (2*sqrt(1 + tan(e + f*x)))/f + (2*(1 + tan(e + f*x))^(3/2))/(3*f) + (20*(1 + tan(e + f*x))^(5/2))/(231*f) - (50*tan(e + f*x)*(1 + tan(e + f*x))^(5/2))/(231*f) - (4*tan(e + f*x)^2*(1 + tan(e + f*x))^(5/2))/(33*f) + (2*tan(e + f*x)^3*(1 + tan(e + f*x))^(5/2))/(11*f)]
    @test_int [tan(e + f*x)^3*(1 + tan(e + f*x))^(3/2), x, 17, -((sqrt(1 + sqrt(2))*atan((sqrt(2*(1 + sqrt(2))) - 2*sqrt(1 + tan(e + f*x)))/sqrt(2*(-1 + sqrt(2)))))/f) + (sqrt(1 + sqrt(2))*atan((sqrt(2*(1 + sqrt(2))) + 2*sqrt(1 + tan(e + f*x)))/sqrt(2*(-1 + sqrt(2)))))/f - log(1 + sqrt(2) + tan(e + f*x) - sqrt(2*(1 + sqrt(2)))*sqrt(1 + tan(e + f*x)))/(2*sqrt(1 + sqrt(2))*f) + log(1 + sqrt(2) + tan(e + f*x) + sqrt(2*(1 + sqrt(2)))*sqrt(1 + tan(e + f*x)))/(2*sqrt(1 + sqrt(2))*f) - (2*sqrt(1 + tan(e + f*x)))/f - (2*(1 + tan(e + f*x))^(3/2))/(3*f) - (4*(1 + tan(e + f*x))^(5/2))/(35*f) + (2*tan(e + f*x)*(1 + tan(e + f*x))^(5/2))/(7*f)]
    @test_int [tan(e + f*x)^1*(1 + tan(e + f*x))^(3/2), x, 14, (sqrt(1 + sqrt(2))*atan((sqrt(2*(1 + sqrt(2))) - 2*sqrt(1 + tan(e + f*x)))/sqrt(2*(-1 + sqrt(2)))))/f - (sqrt(1 + sqrt(2))*atan((sqrt(2*(1 + sqrt(2))) + 2*sqrt(1 + tan(e + f*x)))/sqrt(2*(-1 + sqrt(2)))))/f + log(1 + sqrt(2) + tan(e + f*x) - sqrt(2*(1 + sqrt(2)))*sqrt(1 + tan(e + f*x)))/(2*sqrt(1 + sqrt(2))*f) - log(1 + sqrt(2) + tan(e + f*x) + sqrt(2*(1 + sqrt(2)))*sqrt(1 + tan(e + f*x)))/(2*sqrt(1 + sqrt(2))*f) + (2*sqrt(1 + tan(e + f*x)))/f + (2*(1 + tan(e + f*x))^(3/2))/(3*f)]
    @test_int [cot(e + f*x)^1*(1 + tan(e + f*x))^(3/2), x, 16, -((sqrt(1 + sqrt(2))*atan((sqrt(2*(1 + sqrt(2))) - 2*sqrt(1 + tan(e + f*x)))/sqrt(2*(-1 + sqrt(2)))))/f) + (sqrt(1 + sqrt(2))*atan((sqrt(2*(1 + sqrt(2))) + 2*sqrt(1 + tan(e + f*x)))/sqrt(2*(-1 + sqrt(2)))))/f - (2*atanh(sqrt(1 + tan(e + f*x))))/f - log(1 + sqrt(2) + tan(e + f*x) - sqrt(2*(1 + sqrt(2)))*sqrt(1 + tan(e + f*x)))/(2*sqrt(1 + sqrt(2))*f) + log(1 + sqrt(2) + tan(e + f*x) + sqrt(2*(1 + sqrt(2)))*sqrt(1 + tan(e + f*x)))/(2*sqrt(1 + sqrt(2))*f)]
    @test_int [cot(e + f*x)^3*(1 + tan(e + f*x))^(3/2), x, 18, (sqrt(1 + sqrt(2))*atan((sqrt(2*(1 + sqrt(2))) - 2*sqrt(1 + tan(e + f*x)))/sqrt(2*(-1 + sqrt(2)))))/f - (sqrt(1 + sqrt(2))*atan((sqrt(2*(1 + sqrt(2))) + 2*sqrt(1 + tan(e + f*x)))/sqrt(2*(-1 + sqrt(2)))))/f + (5*atanh(sqrt(1 + tan(e + f*x))))/(4*f) + log(1 + sqrt(2) + tan(e + f*x) - sqrt(2*(1 + sqrt(2)))*sqrt(1 + tan(e + f*x)))/(2*sqrt(1 + sqrt(2))*f) - log(1 + sqrt(2) + tan(e + f*x) + sqrt(2*(1 + sqrt(2)))*sqrt(1 + tan(e + f*x)))/(2*sqrt(1 + sqrt(2))*f) - (5*cot(e + f*x)*sqrt(1 + tan(e + f*x)))/(4*f) - (cot(e + f*x)^2*sqrt(1 + tan(e + f*x)))/(2*f)]
    @test_int [cot(e + f*x)^5*(1 + tan(e + f*x))^(3/2), x, 20, -((sqrt(1 + sqrt(2))*atan((sqrt(2*(1 + sqrt(2))) - 2*sqrt(1 + tan(e + f*x)))/sqrt(2*(-1 + sqrt(2)))))/f) + (sqrt(1 + sqrt(2))*atan((sqrt(2*(1 + sqrt(2))) + 2*sqrt(1 + tan(e + f*x)))/sqrt(2*(-1 + sqrt(2)))))/f - (83*atanh(sqrt(1 + tan(e + f*x))))/(64*f) - log(1 + sqrt(2) + tan(e + f*x) - sqrt(2*(1 + sqrt(2)))*sqrt(1 + tan(e + f*x)))/(2*sqrt(1 + sqrt(2))*f) + log(1 + sqrt(2) + tan(e + f*x) + sqrt(2*(1 + sqrt(2)))*sqrt(1 + tan(e + f*x)))/(2*sqrt(1 + sqrt(2))*f) + (83*cot(e + f*x)*sqrt(1 + tan(e + f*x)))/(64*f) + (15*cot(e + f*x)^2*sqrt(1 + tan(e + f*x)))/(32*f) - (3*cot(e + f*x)^3*sqrt(1 + tan(e + f*x)))/(8*f) - (cot(e + f*x)^4*sqrt(1 + tan(e + f*x)))/(4*f)]

    @test_int [tan(e + f*x)^4*(1 + tan(e + f*x))^(3/2), x, 10, -((sqrt(-1 + sqrt(2))*atan((3 - 2*sqrt(2) + (1 - sqrt(2))*tan(e + f*x))/(sqrt(2*(-7 + 5*sqrt(2)))*sqrt(1 + tan(e + f*x)))))/f) - (sqrt(1 + sqrt(2))*atanh((3 + 2*sqrt(2) + (1 + sqrt(2))*tan(e + f*x))/(sqrt(2*(7 + 5*sqrt(2)))*sqrt(1 + tan(e + f*x)))))/f + (2*sqrt(1 + tan(e + f*x)))/f - (22*(1 + tan(e + f*x))^(5/2))/(63*f) - (8*tan(e + f*x)*(1 + tan(e + f*x))^(5/2))/(63*f) + (2*tan(e + f*x)^2*(1 + tan(e + f*x))^(5/2))/(9*f)]
    @test_int [tan(e + f*x)^2*(1 + tan(e + f*x))^(3/2), x, 8, (sqrt(-1 + sqrt(2))*atan((3 - 2*sqrt(2) + (1 - sqrt(2))*tan(e + f*x))/(sqrt(2*(-7 + 5*sqrt(2)))*sqrt(1 + tan(e + f*x)))))/f + (sqrt(1 + sqrt(2))*atanh((3 + 2*sqrt(2) + (1 + sqrt(2))*tan(e + f*x))/(sqrt(2*(7 + 5*sqrt(2)))*sqrt(1 + tan(e + f*x)))))/f - (2*sqrt(1 + tan(e + f*x)))/f + (2*(1 + tan(e + f*x))^(5/2))/(5*f)]
    @test_int [tan(e + f*x)^0*(1 + tan(e + f*x))^(3/2), x, 7, -((sqrt(-1 + sqrt(2))*atan((3 - 2*sqrt(2) + (1 - sqrt(2))*tan(e + f*x))/(sqrt(2*(-7 + 5*sqrt(2)))*sqrt(1 + tan(e + f*x)))))/f) - (sqrt(1 + sqrt(2))*atanh((3 + 2*sqrt(2) + (1 + sqrt(2))*tan(e + f*x))/(sqrt(2*(7 + 5*sqrt(2)))*sqrt(1 + tan(e + f*x)))))/f + (2*sqrt(1 + tan(e + f*x)))/f]
    @test_int [cot(e + f*x)^2*(1 + tan(e + f*x))^(3/2), x, 11, (sqrt(-1 + sqrt(2))*atan((3 - 2*sqrt(2) + (1 - sqrt(2))*tan(e + f*x))/(sqrt(2*(-7 + 5*sqrt(2)))*sqrt(1 + tan(e + f*x)))))/f - (3*atanh(sqrt(1 + tan(e + f*x))))/f + (sqrt(1 + sqrt(2))*atanh((3 + 2*sqrt(2) + (1 + sqrt(2))*tan(e + f*x))/(sqrt(2*(7 + 5*sqrt(2)))*sqrt(1 + tan(e + f*x)))))/f - (cot(e + f*x)*sqrt(1 + tan(e + f*x)))/f]
    @test_int [cot(e + f*x)^4*(1 + tan(e + f*x))^(3/2), x, 13, -((sqrt(-1 + sqrt(2))*atan((3 - 2*sqrt(2) + (1 - sqrt(2))*tan(e + f*x))/(sqrt(2*(-7 + 5*sqrt(2)))*sqrt(1 + tan(e + f*x)))))/f) + (25*atanh(sqrt(1 + tan(e + f*x))))/(8*f) - (sqrt(1 + sqrt(2))*atanh((3 + 2*sqrt(2) + (1 + sqrt(2))*tan(e + f*x))/(sqrt(2*(7 + 5*sqrt(2)))*sqrt(1 + tan(e + f*x)))))/f + (7*cot(e + f*x)*sqrt(1 + tan(e + f*x)))/(8*f) - (7*cot(e + f*x)^2*sqrt(1 + tan(e + f*x)))/(12*f) - (cot(e + f*x)^3*sqrt(1 + tan(e + f*x)))/(3*f)]


    #= ::Subsubsection::Closed:: =#
    #=m<0=#


    @test_int [tan(e + f*x)^5/sqrt(1 + tan(e + f*x)), x, 10, -((sqrt(-1 + sqrt(2))*atan((3 - 2*sqrt(2) + (1 - sqrt(2))*tan(e + f*x))/(sqrt(2*(-7 + 5*sqrt(2)))*sqrt(1 + tan(e + f*x)))))/(2*f)) - (sqrt(1 + sqrt(2))*atanh((3 + 2*sqrt(2) + (1 + sqrt(2))*tan(e + f*x))/(sqrt(2*(7 + 5*sqrt(2)))*sqrt(1 + tan(e + f*x)))))/(2*f) + (44*sqrt(1 + tan(e + f*x)))/(105*f) - (22*tan(e + f*x)*sqrt(1 + tan(e + f*x)))/(105*f) - (12*tan(e + f*x)^2*sqrt(1 + tan(e + f*x)))/(35*f) + (2*tan(e + f*x)^3*sqrt(1 + tan(e + f*x)))/(7*f)]
    @test_int [tan(e + f*x)^3/sqrt(1 + tan(e + f*x)), x, 8, (sqrt(-1 + sqrt(2))*atan((3 - 2*sqrt(2) + (1 - sqrt(2))*tan(e + f*x))/(sqrt(2*(-7 + 5*sqrt(2)))*sqrt(1 + tan(e + f*x)))))/(2*f) + (sqrt(1 + sqrt(2))*atanh((3 + 2*sqrt(2) + (1 + sqrt(2))*tan(e + f*x))/(sqrt(2*(7 + 5*sqrt(2)))*sqrt(1 + tan(e + f*x)))))/(2*f) - (4*sqrt(1 + tan(e + f*x)))/(3*f) + (2*tan(e + f*x)*sqrt(1 + tan(e + f*x)))/(3*f)]
    @test_int [tan(e + f*x)^1/sqrt(1 + tan(e + f*x)), x, 5, -((sqrt(-1 + sqrt(2))*atan((3 - 2*sqrt(2) + (1 - sqrt(2))*tan(e + f*x))/(sqrt(2*(-7 + 5*sqrt(2)))*sqrt(1 + tan(e + f*x)))))/(2*f)) - (sqrt(1 + sqrt(2))*atanh((3 + 2*sqrt(2) + (1 + sqrt(2))*tan(e + f*x))/(sqrt(2*(7 + 5*sqrt(2)))*sqrt(1 + tan(e + f*x)))))/(2*f)]
    @test_int [cot(e + f*x)^1/sqrt(1 + tan(e + f*x)), x, 9, (sqrt(-1 + sqrt(2))*atan((3 - 2*sqrt(2) + (1 - sqrt(2))*tan(e + f*x))/(sqrt(2*(-7 + 5*sqrt(2)))*sqrt(1 + tan(e + f*x)))))/(2*f) - (2*atanh(sqrt(1 + tan(e + f*x))))/f + (sqrt(1 + sqrt(2))*atanh((3 + 2*sqrt(2) + (1 + sqrt(2))*tan(e + f*x))/(sqrt(2*(7 + 5*sqrt(2)))*sqrt(1 + tan(e + f*x)))))/(2*f)]
    @test_int [cot(e + f*x)^3/sqrt(1 + tan(e + f*x)), x, 12, -((sqrt(-1 + sqrt(2))*atan((3 - 2*sqrt(2) + (1 - sqrt(2))*tan(e + f*x))/(sqrt(2*(-7 + 5*sqrt(2)))*sqrt(1 + tan(e + f*x)))))/(2*f)) + (5*atanh(sqrt(1 + tan(e + f*x))))/(4*f) - (sqrt(1 + sqrt(2))*atanh((3 + 2*sqrt(2) + (1 + sqrt(2))*tan(e + f*x))/(sqrt(2*(7 + 5*sqrt(2)))*sqrt(1 + tan(e + f*x)))))/(2*f) + (3*cot(e + f*x)*sqrt(1 + tan(e + f*x)))/(4*f) - (cot(e + f*x)^2*sqrt(1 + tan(e + f*x)))/(2*f)]
    @test_int [cot(e + f*x)^5/sqrt(1 + tan(e + f*x)), x, 14, (sqrt(-1 + sqrt(2))*atan((3 - 2*sqrt(2) + (1 - sqrt(2))*tan(e + f*x))/(sqrt(2*(-7 + 5*sqrt(2)))*sqrt(1 + tan(e + f*x)))))/(2*f) - (115*atanh(sqrt(1 + tan(e + f*x))))/(64*f) + (sqrt(1 + sqrt(2))*atanh((3 + 2*sqrt(2) + (1 + sqrt(2))*tan(e + f*x))/(sqrt(2*(7 + 5*sqrt(2)))*sqrt(1 + tan(e + f*x)))))/(2*f) - (13*cot(e + f*x)*sqrt(1 + tan(e + f*x)))/(64*f) + (13*cot(e + f*x)^2*sqrt(1 + tan(e + f*x)))/(96*f) + (7*cot(e + f*x)^3*sqrt(1 + tan(e + f*x)))/(24*f) - (cot(e + f*x)^4*sqrt(1 + tan(e + f*x)))/(4*f)]

    @test_int [tan(e + f*x)^4/sqrt(1 + tan(e + f*x)), x, 14, -((sqrt(1 + sqrt(2))*atan((sqrt(2*(1 + sqrt(2))) - 2*sqrt(1 + tan(e + f*x)))/sqrt(2*(-1 + sqrt(2)))))/(2*f)) + (sqrt(1 + sqrt(2))*atan((sqrt(2*(1 + sqrt(2))) + 2*sqrt(1 + tan(e + f*x)))/sqrt(2*(-1 + sqrt(2)))))/(2*f) - log(1 + sqrt(2) + tan(e + f*x) - sqrt(2*(1 + sqrt(2)))*sqrt(1 + tan(e + f*x)))/(4*sqrt(1 + sqrt(2))*f) + log(1 + sqrt(2) + tan(e + f*x) + sqrt(2*(1 + sqrt(2)))*sqrt(1 + tan(e + f*x)))/(4*sqrt(1 + sqrt(2))*f) - (14*sqrt(1 + tan(e + f*x)))/(15*f) - (8*tan(e + f*x)*sqrt(1 + tan(e + f*x)))/(15*f) + (2*tan(e + f*x)^2*sqrt(1 + tan(e + f*x)))/(5*f)]
    @test_int [tan(e + f*x)^2/sqrt(1 + tan(e + f*x)), x, 12, (sqrt(1 + sqrt(2))*atan((sqrt(2*(1 + sqrt(2))) - 2*sqrt(1 + tan(e + f*x)))/sqrt(2*(-1 + sqrt(2)))))/(2*f) - (sqrt(1 + sqrt(2))*atan((sqrt(2*(1 + sqrt(2))) + 2*sqrt(1 + tan(e + f*x)))/sqrt(2*(-1 + sqrt(2)))))/(2*f) + log(1 + sqrt(2) + tan(e + f*x) - sqrt(2*(1 + sqrt(2)))*sqrt(1 + tan(e + f*x)))/(4*sqrt(1 + sqrt(2))*f) - log(1 + sqrt(2) + tan(e + f*x) + sqrt(2*(1 + sqrt(2)))*sqrt(1 + tan(e + f*x)))/(4*sqrt(1 + sqrt(2))*f) + (2*sqrt(1 + tan(e + f*x)))/f]
    @test_int [tan(e + f*x)^0/sqrt(1 + tan(e + f*x)), x, 11, -((sqrt(1 + sqrt(2))*atan((sqrt(2*(1 + sqrt(2))) - 2*sqrt(1 + tan(e + f*x)))/sqrt(2*(-1 + sqrt(2)))))/(2*f)) + (sqrt(1 + sqrt(2))*atan((sqrt(2*(1 + sqrt(2))) + 2*sqrt(1 + tan(e + f*x)))/sqrt(2*(-1 + sqrt(2)))))/(2*f) - log(1 + sqrt(2) + tan(e + f*x) - sqrt(2*(1 + sqrt(2)))*sqrt(1 + tan(e + f*x)))/(4*sqrt(1 + sqrt(2))*f) + log(1 + sqrt(2) + tan(e + f*x) + sqrt(2*(1 + sqrt(2)))*sqrt(1 + tan(e + f*x)))/(4*sqrt(1 + sqrt(2))*f)]
    @test_int [cot(e + f*x)^2/sqrt(1 + tan(e + f*x)), x, 19, (sqrt(1 + sqrt(2))*atan((sqrt(2*(1 + sqrt(2))) - 2*sqrt(1 + tan(e + f*x)))/sqrt(2*(-1 + sqrt(2)))))/(2*f) - (sqrt(1 + sqrt(2))*atan((sqrt(2*(1 + sqrt(2))) + 2*sqrt(1 + tan(e + f*x)))/sqrt(2*(-1 + sqrt(2)))))/(2*f) + atanh(sqrt(1 + tan(e + f*x)))/f + log(1 + sqrt(2) + tan(e + f*x) - sqrt(2*(1 + sqrt(2)))*sqrt(1 + tan(e + f*x)))/(4*sqrt(1 + sqrt(2))*f) - log(1 + sqrt(2) + tan(e + f*x) + sqrt(2*(1 + sqrt(2)))*sqrt(1 + tan(e + f*x)))/(4*sqrt(1 + sqrt(2))*f) - (cot(e + f*x)*sqrt(1 + tan(e + f*x)))/f]
    @test_int [cot(e + f*x)^4/sqrt(1 + tan(e + f*x)), x, 19, -((sqrt(1 + sqrt(2))*atan((sqrt(2*(1 + sqrt(2))) - 2*sqrt(1 + tan(e + f*x)))/sqrt(2*(-1 + sqrt(2)))))/(2*f)) + (sqrt(1 + sqrt(2))*atan((sqrt(2*(1 + sqrt(2))) + 2*sqrt(1 + tan(e + f*x)))/sqrt(2*(-1 + sqrt(2)))))/(2*f) - (3*atanh(sqrt(1 + tan(e + f*x))))/(8*f) - log(1 + sqrt(2) + tan(e + f*x) - sqrt(2*(1 + sqrt(2)))*sqrt(1 + tan(e + f*x)))/(4*sqrt(1 + sqrt(2))*f) + log(1 + sqrt(2) + tan(e + f*x) + sqrt(2*(1 + sqrt(2)))*sqrt(1 + tan(e + f*x)))/(4*sqrt(1 + sqrt(2))*f) + (3*cot(e + f*x)*sqrt(1 + tan(e + f*x)))/(8*f) + (5*cot(e + f*x)^2*sqrt(1 + tan(e + f*x)))/(12*f) - (cot(e + f*x)^3*sqrt(1 + tan(e + f*x)))/(3*f)]


    #= ::Subsection:: =#
    #=Integrands*of*the*form*(a+a*tan(e+f*x))^m*(d*tan(e+f*x))^n*with*n*symbolic=#


    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*(a+a*tan(e+f*x))^m*(d*tan(e+f*x))^n*with*m*symbolic=#


    @test_int [(a + a*tan(e + f*x))^m*(d*tan(e + f*x))^n, x, 7, (AppellF1(1 + n, -m, 1, 2 + n, -tan(e + f*x), (-im)*tan(e + f*x))*(d*tan(e + f*x))^(1 + n)*(a + a*tan(e + f*x))^m)/((1 + tan(e + f*x))^m*(2*d*f*(1 + n))) + (AppellF1(1 + n, -m, 1, 2 + n, -tan(e + f*x), im*tan(e + f*x))*(d*tan(e + f*x))^(1 + n)*(a + a*tan(e + f*x))^m)/((1 + tan(e + f*x))^m*(2*d*f*(1 + n)))]


    #= ::Section::Closed:: =#
    #=Integrands*of*the*form*(a+b*tan(e+f*x))^m*(d*tan(e+f*x))^n=#


    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*(a+b*tan(e+f*x))^m*(d*tan(e+f*x))^n=#


    #= ::Subsubsection::Closed:: =#
    #=m>0=#


    @test_int [tan(c + d*x)^5*(a + b*tan(c + d*x)), x, 6, (-b)*x - (a*log(cos(c + d*x)))/d + (b*tan(c + d*x))/d - (a*tan(c + d*x)^2)/(2*d) - (b*tan(c + d*x)^3)/(3*d) + (a*tan(c + d*x)^4)/(4*d) + (b*tan(c + d*x)^5)/(5*d)]
    @test_int [tan(c + d*x)^4*(a + b*tan(c + d*x)), x, 5, a*x - (b*log(cos(c + d*x)))/d - (a*tan(c + d*x))/d - (b*tan(c + d*x)^2)/(2*d) + (a*tan(c + d*x)^3)/(3*d) + (b*tan(c + d*x)^4)/(4*d)]
    @test_int [tan(c + d*x)^3*(a + b*tan(c + d*x)), x, 4, b*x + (a*log(cos(c + d*x)))/d - (b*tan(c + d*x))/d + (a*tan(c + d*x)^2)/(2*d) + (b*tan(c + d*x)^3)/(3*d)]
    @test_int [tan(c + d*x)^2*(a + b*tan(c + d*x)), x, 3, (-a)*x + (b*log(cos(c + d*x)))/d + (a*tan(c + d*x))/d + (b*tan(c + d*x)^2)/(2*d)]
    @test_int [tan(c + d*x)^1*(a + b*tan(c + d*x)), x, 2, (-b)*x - (a*log(cos(c + d*x)))/d + (b*tan(c + d*x))/d]
    @test_int [tan(c + d*x)^0*(a + b*tan(c + d*x)), x, 2, a*x - (b*log(cos(c + d*x)))/d]
    @test_int [cot(c + d*x)^1*(a + b*tan(c + d*x)), x, 2, b*x + (a*log(sin(c + d*x)))/d]
    @test_int [cot(c + d*x)^2*(a + b*tan(c + d*x)), x, 3, (-a)*x - (a*cot(c + d*x))/d + (b*log(sin(c + d*x)))/d]
    @test_int [cot(c + d*x)^3*(a + b*tan(c + d*x)), x, 4, (-b)*x - (b*cot(c + d*x))/d - (a*cot(c + d*x)^2)/(2*d) - (a*log(sin(c + d*x)))/d]
    @test_int [cot(c + d*x)^4*(a + b*tan(c + d*x)), x, 5, a*x + (a*cot(c + d*x))/d - (b*cot(c + d*x)^2)/(2*d) - (a*cot(c + d*x)^3)/(3*d) - (b*log(sin(c + d*x)))/d]
    @test_int [cot(c + d*x)^5*(a + b*tan(c + d*x)), x, 6, b*x + (b*cot(c + d*x))/d + (a*cot(c + d*x)^2)/(2*d) - (b*cot(c + d*x)^3)/(3*d) - (a*cot(c + d*x)^4)/(4*d) + (a*log(sin(c + d*x)))/d]
    @test_int [cot(c + d*x)^6*(a + b*tan(c + d*x)), x, 7, (-a)*x - (a*cot(c + d*x))/d + (b*cot(c + d*x)^2)/(2*d) + (a*cot(c + d*x)^3)/(3*d) - (b*cot(c + d*x)^4)/(4*d) - (a*cot(c + d*x)^5)/(5*d) + (b*log(sin(c + d*x)))/d]


    @test_int [tan(c + d*x)^4*(a + b*tan(c + d*x))^2, x, 6, (a^2 - b^2)*x - (2*a*b*log(cos(c + d*x)))/d - ((a^2 - b^2)*tan(c + d*x))/d - (a*b*tan(c + d*x)^2)/d + ((a^2 - b^2)*tan(c + d*x)^3)/(3*d) + (a*b*tan(c + d*x)^4)/(2*d) + (b^2*tan(c + d*x)^5)/(5*d)]
    @test_int [tan(c + d*x)^3*(a + b*tan(c + d*x))^2, x, 5, 2*a*b*x + ((a^2 - b^2)*log(cos(c + d*x)))/d - (2*a*b*tan(c + d*x))/d + ((a^2 - b^2)*tan(c + d*x)^2)/(2*d) + (2*a*b*tan(c + d*x)^3)/(3*d) + (b^2*tan(c + d*x)^4)/(4*d)]
    @test_int [tan(c + d*x)^2*(a + b*tan(c + d*x))^2, x, 3, -((a^2 - b^2)*x) + (2*a*b*log(cos(c + d*x)))/d - (b^2*tan(c + d*x))/d + (a + b*tan(c + d*x))^3/(3*b*d)]
    @test_int [tan(c + d*x)^1*(a + b*tan(c + d*x))^2, x, 3, -2*a*b*x - ((a^2 - b^2)*log(cos(c + d*x)))/d + (a*b*tan(c + d*x))/d + (a + b*tan(c + d*x))^2/(2*d)]
    @test_int [tan(c + d*x)^0*(a + b*tan(c + d*x))^2, x, 2, (a^2 - b^2)*x - (2*a*b*log(cos(c + d*x)))/d + (b^2*tan(c + d*x))/d]
    @test_int [cot(c + d*x)^1*(a + b*tan(c + d*x))^2, x, 3, 2*a*b*x - (b^2*log(cos(c + d*x)))/d + (a^2*log(sin(c + d*x)))/d]
    @test_int [cot(c + d*x)^2*(a + b*tan(c + d*x))^2, x, 3, -((a^2 - b^2)*x) - (a^2*cot(c + d*x))/d + (2*a*b*log(sin(c + d*x)))/d]
    @test_int [cot(c + d*x)^3*(a + b*tan(c + d*x))^2, x, 4, -2*a*b*x - (2*a*b*cot(c + d*x))/d - (a^2*cot(c + d*x)^2)/(2*d) - ((a^2 - b^2)*log(sin(c + d*x)))/d]
    @test_int [cot(c + d*x)^4*(a + b*tan(c + d*x))^2, x, 5, (a^2 - b^2)*x + ((a^2 - b^2)*cot(c + d*x))/d - (a*b*cot(c + d*x)^2)/d - (a^2*cot(c + d*x)^3)/(3*d) - (2*a*b*log(sin(c + d*x)))/d]
    @test_int [cot(c + d*x)^5*(a + b*tan(c + d*x))^2, x, 6, 2*a*b*x + (2*a*b*cot(c + d*x))/d + ((a^2 - b^2)*cot(c + d*x)^2)/(2*d) - (2*a*b*cot(c + d*x)^3)/(3*d) - (a^2*cot(c + d*x)^4)/(4*d) + ((a^2 - b^2)*log(sin(c + d*x)))/d]
    @test_int [cot(c + d*x)^6*(a + b*tan(c + d*x))^2, x, 7, -((a^2 - b^2)*x) - ((a^2 - b^2)*cot(c + d*x))/d + (a*b*cot(c + d*x)^2)/d + ((a^2 - b^2)*cot(c + d*x)^3)/(3*d) - (a*b*cot(c + d*x)^4)/(2*d) - (a^2*cot(c + d*x)^5)/(5*d) + (2*a*b*log(sin(c + d*x)))/d]


    @test_int [tan(c + d*x)^3*(a + b*tan(c + d*x))^3, x, 7, b*(3*a^2 - b^2)*x + (a*(a^2 - 3*b^2)*log(cos(c + d*x)))/d - (b*(a^2 - b^2)*tan(c + d*x))/d - (a*(a + b*tan(c + d*x))^2)/(2*d) - (a + b*tan(c + d*x))^3/(3*d) - (a*(a + b*tan(c + d*x))^4)/(20*b^2*d) + (tan(c + d*x)*(a + b*tan(c + d*x))^4)/(5*b*d)]
    @test_int [tan(c + d*x)^2*(a + b*tan(c + d*x))^3, x, 4, (-a)*(a^2 - 3*b^2)*x + (b*(3*a^2 - b^2)*log(cos(c + d*x)))/d - (2*a*b^2*tan(c + d*x))/d - (b*(a + b*tan(c + d*x))^2)/(2*d) + (a + b*tan(c + d*x))^4/(4*b*d)]
    @test_int [tan(c + d*x)^1*(a + b*tan(c + d*x))^3, x, 4, (-b)*(3*a^2 - b^2)*x - (a*(a^2 - 3*b^2)*log(cos(c + d*x)))/d + (b*(a^2 - b^2)*tan(c + d*x))/d + (a*(a + b*tan(c + d*x))^2)/(2*d) + (a + b*tan(c + d*x))^3/(3*d)]
    @test_int [tan(c + d*x)^0*(a + b*tan(c + d*x))^3, x, 3, a*(a^2 - 3*b^2)*x - (b*(3*a^2 - b^2)*log(cos(c + d*x)))/d + (2*a*b^2*tan(c + d*x))/d + (b*(a + b*tan(c + d*x))^2)/(2*d)]
    @test_int [cot(c + d*x)^1*(a + b*tan(c + d*x))^3, x, 4, b*(3*a^2 - b^2)*x - (3*a*b^2*log(cos(c + d*x)))/d + (a^3*log(sin(c + d*x)))/d + (b^2*(a + b*tan(c + d*x)))/d]
    @test_int [cot(c + d*x)^2*(a + b*tan(c + d*x))^3, x, 4, (-a)*(a^2 - 3*b^2)*x - (b^3*log(cos(c + d*x)))/d + (3*a^2*b*log(sin(c + d*x)))/d - (a^2*cot(c + d*x)*(a + b*tan(c + d*x)))/d]
    @test_int [cot(c + d*x)^3*(a + b*tan(c + d*x))^3, x, 4, (-b)*(3*a^2 - b^2)*x - (5*a^2*b*cot(c + d*x))/(2*d) - (a*(a^2 - 3*b^2)*log(sin(c + d*x)))/d - (a^2*cot(c + d*x)^2*(a + b*tan(c + d*x)))/(2*d)]
    @test_int [cot(c + d*x)^4*(a + b*tan(c + d*x))^3, x, 5, a*(a^2 - 3*b^2)*x + (a*(a^2 - 3*b^2)*cot(c + d*x))/d - (7*a^2*b*cot(c + d*x)^2)/(6*d) - (b*(3*a^2 - b^2)*log(sin(c + d*x)))/d - (a^2*cot(c + d*x)^3*(a + b*tan(c + d*x)))/(3*d)]
    @test_int [cot(c + d*x)^5*(a + b*tan(c + d*x))^3, x, 6, b*(3*a^2 - b^2)*x + (b*(3*a^2 - b^2)*cot(c + d*x))/d + (a*(a^2 - 3*b^2)*cot(c + d*x)^2)/(2*d) - (3*a^2*b*cot(c + d*x)^3)/(4*d) + (a*(a^2 - 3*b^2)*log(sin(c + d*x)))/d - (a^2*cot(c + d*x)^4*(a + b*tan(c + d*x)))/(4*d)]
    @test_int [cot(c + d*x)^6*(a + b*tan(c + d*x))^3, x, 7, (-a)*(a^2 - 3*b^2)*x - (a*(a^2 - 3*b^2)*cot(c + d*x))/d + (b*(3*a^2 - b^2)*cot(c + d*x)^2)/(2*d) + (a*(a^2 - 3*b^2)*cot(c + d*x)^3)/(3*d) - (11*a^2*b*cot(c + d*x)^4)/(20*d) + (b*(3*a^2 - b^2)*log(sin(c + d*x)))/d - (a^2*cot(c + d*x)^5*(a + b*tan(c + d*x)))/(5*d)]


    @test_int [tan(c + d*x)^3*(a + b*tan(c + d*x))^4, x, 8, 4*a*b*(a^2 - b^2)*x + ((a^4 - 6*a^2*b^2 + b^4)*log(cos(c + d*x)))/d - (a*b*(a^2 - 3*b^2)*tan(c + d*x))/d - ((a^2 - b^2)*(a + b*tan(c + d*x))^2)/(2*d) - (a*(a + b*tan(c + d*x))^3)/(3*d) - (a + b*tan(c + d*x))^4/(4*d) - (a*(a + b*tan(c + d*x))^5)/(30*b^2*d) + (tan(c + d*x)*(a + b*tan(c + d*x))^5)/(6*b*d)]
    @test_int [tan(c + d*x)^2*(a + b*tan(c + d*x))^4, x, 5, -((a^4 - 6*a^2*b^2 + b^4)*x) + (4*a*b*(a^2 - b^2)*log(cos(c + d*x)))/d - (b^2*(3*a^2 - b^2)*tan(c + d*x))/d - (a*b*(a + b*tan(c + d*x))^2)/d - (b*(a + b*tan(c + d*x))^3)/(3*d) + (a + b*tan(c + d*x))^5/(5*b*d)]
    @test_int [tan(c + d*x)^1*(a + b*tan(c + d*x))^4, x, 5, -4*a*b*(a^2 - b^2)*x - ((a^4 - 6*a^2*b^2 + b^4)*log(cos(c + d*x)))/d + (a*b*(a^2 - 3*b^2)*tan(c + d*x))/d + ((a^2 - b^2)*(a + b*tan(c + d*x))^2)/(2*d) + (a*(a + b*tan(c + d*x))^3)/(3*d) + (a + b*tan(c + d*x))^4/(4*d)]
    @test_int [tan(c + d*x)^0*(a + b*tan(c + d*x))^4, x, 4, (a^4 - 6*a^2*b^2 + b^4)*x - (4*a*b*(a^2 - b^2)*log(cos(c + d*x)))/d + (b^2*(3*a^2 - b^2)*tan(c + d*x))/d + (a*b*(a + b*tan(c + d*x))^2)/d + (b*(a + b*tan(c + d*x))^3)/(3*d)]
    @test_int [cot(c + d*x)^1*(a + b*tan(c + d*x))^4, x, 5, 4*a*b*(a^2 - b^2)*x - (b^2*(6*a^2 - b^2)*log(cos(c + d*x)))/d + (a^4*log(sin(c + d*x)))/d + (3*a*b^3*tan(c + d*x))/d + (b^2*(a + b*tan(c + d*x))^2)/(2*d)]
    @test_int [cot(c + d*x)^2*(a + b*tan(c + d*x))^4, x, 5, -((a^4 - 6*a^2*b^2 + b^4)*x) - (4*a*b^3*log(cos(c + d*x)))/d + (4*a^3*b*log(sin(c + d*x)))/d + (b^2*(a^2 + b^2)*tan(c + d*x))/d - (a^2*cot(c + d*x)*(a + b*tan(c + d*x))^2)/d]
    @test_int [cot(c + d*x)^3*(a + b*tan(c + d*x))^4, x, 5, -4*a*b*(a^2 - b^2)*x - (3*a^3*b*cot(c + d*x))/d - (b^4*log(cos(c + d*x)))/d - (a^2*(a^2 - 6*b^2)*log(sin(c + d*x)))/d - (a^2*cot(c + d*x)^2*(a + b*tan(c + d*x))^2)/(2*d)]
    @test_int [cot(c + d*x)^4*(a + b*tan(c + d*x))^4, x, 5, (a^4 - 6*a^2*b^2 + b^4)*x + (a^2*(3*a^2 - 17*b^2)*cot(c + d*x))/(3*d) - (4*a^3*b*cot(c + d*x)^2)/(3*d) - (4*a*b*(a^2 - b^2)*log(sin(c + d*x)))/d - (a^2*cot(c + d*x)^3*(a + b*tan(c + d*x))^2)/(3*d)]
    @test_int [cot(c + d*x)^5*(a + b*tan(c + d*x))^4, x, 6, 4*a*b*(a^2 - b^2)*x + (4*a*b*(a^2 - b^2)*cot(c + d*x))/d + (a^2*(2*a^2 - 11*b^2)*cot(c + d*x)^2)/(4*d) - (5*a^3*b*cot(c + d*x)^3)/(6*d) + ((a^4 - 6*a^2*b^2 + b^4)*log(sin(c + d*x)))/d - (a^2*cot(c + d*x)^4*(a + b*tan(c + d*x))^2)/(4*d)]
    @test_int [cot(c + d*x)^6*(a + b*tan(c + d*x))^4, x, 7, -((a^4 - 6*a^2*b^2 + b^4)*x) - ((a^4 - 6*a^2*b^2 + b^4)*cot(c + d*x))/d + (2*a*b*(a^2 - b^2)*cot(c + d*x)^2)/d + (a^2*(5*a^2 - 27*b^2)*cot(c + d*x)^3)/(15*d) - (3*a^3*b*cot(c + d*x)^4)/(5*d) + (4*a*b*(a^2 - b^2)*log(sin(c + d*x)))/d - (a^2*cot(c + d*x)^5*(a + b*tan(c + d*x))^2)/(5*d)]
    @test_int [cot(c + d*x)^7*(a + b*tan(c + d*x))^4, x, 8, -4*a*b*(a^2 - b^2)*x - (4*a*b*(a^2 - b^2)*cot(c + d*x))/d - ((a^4 - 6*a^2*b^2 + b^4)*cot(c + d*x)^2)/(2*d) + (4*a*b*(a^2 - b^2)*cot(c + d*x)^3)/(3*d) + (a^2*(3*a^2 - 16*b^2)*cot(c + d*x)^4)/(12*d) - (7*a^3*b*cot(c + d*x)^5)/(15*d) - ((a^4 - 6*a^2*b^2 + b^4)*log(sin(c + d*x)))/d - (a^2*cot(c + d*x)^6*(a + b*tan(c + d*x))^2)/(6*d)]


    #= ::Subsubsection::Closed:: =#
    #=m<0=#


    @test_int [tan(c + d*x)^6/(a + b*tan(c + d*x)), x, 8, -((a*x)/(a^2 + b^2)) - (b*log(cos(c + d*x)))/((a^2 + b^2)*d) + (a^6*log(a + b*tan(c + d*x)))/(b^5*(a^2 + b^2)*d) - (a*(a^2 - b^2)*tan(c + d*x))/(b^4*d) + ((a^2 - b^2)*tan(c + d*x)^2)/(2*b^3*d) - (a*tan(c + d*x)^3)/(3*b^2*d) + tan(c + d*x)^4/(4*b*d)]
    @test_int [tan(c + d*x)^5/(a + b*tan(c + d*x)), x, 7, (b*x)/(a^2 + b^2) - (a^5*log(a + b*tan(c + d*x)))/(b^4*(a^2 + b^2)*d) - (a*log(cos(c + d*x)))/((a^2 + b^2)*d) + ((a^2 - b^2)*tan(c + d*x))/(b^3*d) - (a*tan(c + d*x)^2)/(2*b^2*d) + tan(c + d*x)^3/(3*b*d)]
    @test_int [tan(c + d*x)^4/(a + b*tan(c + d*x)), x, 6, (a*x)/(a^2 + b^2) + (a^4*log(a + b*tan(c + d*x)))/(b^3*(a^2 + b^2)*d) + (b*log(cos(c + d*x)))/((a^2 + b^2)*d) - (a*tan(c + d*x))/(b^2*d) + tan(c + d*x)^2/(2*b*d)]
    @test_int [tan(c + d*x)^3/(a + b*tan(c + d*x)), x, 5, -((b*x)/(a^2 + b^2)) - (a^3*log(a + b*tan(c + d*x)))/(b^2*(a^2 + b^2)*d) + (a*log(cos(c + d*x)))/((a^2 + b^2)*d) + tan(c + d*x)/(b*d)]
    @test_int [tan(c + d*x)^2/(a + b*tan(c + d*x)), x, 4, -((a*x)/(a^2 + b^2)) - log(cos(c + d*x))/(b*d) + (a^2*log(a*cos(c + d*x) + b*sin(c + d*x)))/(b*(a^2 + b^2)*d), -((a*x)/b^2) + (a^3*x)/(b^2*(a^2 + b^2)) - log(cos(c + d*x))/(b*d) + (a^2*log(a*cos(c + d*x) + b*sin(c + d*x)))/(b*(a^2 + b^2)*d)]
    @test_int [tan(c + d*x)^1/(a + b*tan(c + d*x)), x, 2, (b*x)/(a^2 + b^2) - (a*log(a*cos(c + d*x) + b*sin(c + d*x)))/((a^2 + b^2)*d)]
    @test_int [tan(c + d*x)^0/(a + b*tan(c + d*x)), x, 2, (a*x)/(a^2 + b^2) + (b*log(a*cos(c + d*x) + b*sin(c + d*x)))/((a^2 + b^2)*d)]
    @test_int [cot(c + d*x)^1/(a + b*tan(c + d*x)), x, 3, -((b*x)/(a^2 + b^2)) + log(sin(c + d*x))/(a*d) - (b^2*log(a*cos(c + d*x) + b*sin(c + d*x)))/(a*(a^2 + b^2)*d)]
    @test_int [cot(c + d*x)^2/(a + b*tan(c + d*x)), x, 4, -((a*x)/(a^2 + b^2)) - cot(c + d*x)/(a*d) - (b*log(sin(c + d*x)))/(a^2*d) + (b^3*log(a*cos(c + d*x) + b*sin(c + d*x)))/(a^2*(a^2 + b^2)*d)]
    @test_int [cot(c + d*x)^3/(a + b*tan(c + d*x)), x, 5, (b*x)/(a^2 + b^2) + (b*cot(c + d*x))/(a^2*d) - cot(c + d*x)^2/(2*a*d) - ((a^2 - b^2)*log(sin(c + d*x)))/(a^3*d) - (b^4*log(a*cos(c + d*x) + b*sin(c + d*x)))/(a^3*(a^2 + b^2)*d)]
    @test_int [cot(c + d*x)^4/(a + b*tan(c + d*x)), x, 6, (a*x)/(a^2 + b^2) + ((a^2 - b^2)*cot(c + d*x))/(a^3*d) + (b*cot(c + d*x)^2)/(2*a^2*d) - cot(c + d*x)^3/(3*a*d) + (b*(a^2 - b^2)*log(sin(c + d*x)))/(a^4*d) + (b^5*log(a*cos(c + d*x) + b*sin(c + d*x)))/(a^4*(a^2 + b^2)*d)]


    @test_int [tan(c + d*x)^6/(a + b*tan(c + d*x))^2, x, 8, -(((a^2 - b^2)*x)/(a^2 + b^2)^2) - (2*a*b*log(cos(c + d*x)))/((a^2 + b^2)^2*d) - (2*a^5*(2*a^2 + 3*b^2)*log(a + b*tan(c + d*x)))/(b^5*(a^2 + b^2)^2*d) + ((4*a^4 + 2*a^2*b^2 - b^4)*tan(c + d*x))/(b^4*(a^2 + b^2)*d) - (a*(2*a^2 + b^2)*tan(c + d*x)^2)/(b^3*(a^2 + b^2)*d) + ((4*a^2 + b^2)*tan(c + d*x)^3)/(3*b^2*(a^2 + b^2)*d) - (a^2*tan(c + d*x)^4)/(b*(a^2 + b^2)*d*(a + b*tan(c + d*x)))]
    @test_int [tan(c + d*x)^5/(a + b*tan(c + d*x))^2, x, 7, (2*a*b*x)/(a^2 + b^2)^2 - ((a^2 - b^2)*log(cos(c + d*x)))/((a^2 + b^2)^2*d) + (a^4*(3*a^2 + 5*b^2)*log(a + b*tan(c + d*x)))/(b^4*(a^2 + b^2)^2*d) - (a*(3*a^2 + 2*b^2)*tan(c + d*x))/(b^3*(a^2 + b^2)*d) + ((3*a^2 + b^2)*tan(c + d*x)^2)/(2*b^2*(a^2 + b^2)*d) - (a^2*tan(c + d*x)^3)/(b*(a^2 + b^2)*d*(a + b*tan(c + d*x)))]
    @test_int [tan(c + d*x)^4/(a + b*tan(c + d*x))^2, x, 6, ((a^2 - b^2)*x)/(a^2 + b^2)^2 + (2*a*b*log(cos(c + d*x)))/((a^2 + b^2)^2*d) - (2*a^3*(a^2 + 2*b^2)*log(a + b*tan(c + d*x)))/(b^3*(a^2 + b^2)^2*d) + ((2*a^2 + b^2)*tan(c + d*x))/(b^2*(a^2 + b^2)*d) - (a^2*tan(c + d*x)^2)/(b*(a^2 + b^2)*d*(a + b*tan(c + d*x)))]
    @test_int [tan(c + d*x)^3/(a + b*tan(c + d*x))^2, x, 5, -((2*a*b*x)/(a^2 + b^2)^2) + (a^2*(a^2 + 3*b^2)*log(a + b*tan(c + d*x)))/(b^2*(a^2 + b^2)^2*d) + ((a^2 - b^2)*log(cos(c + d*x)))/((a^2 + b^2)^2*d) + a^3/(b^2*(a^2 + b^2)*d*(a + b*tan(c + d*x))), -((2*a*b*x)/(a^2 + b^2)^2) + ((a^2 - b^2)*log(cos(c + d*x)))/((a^2 + b^2)^2*d) + (a^2*(a^2 + 3*b^2)*log(a + b*tan(c + d*x)))/(b^2*(a^2 + b^2)^2*d) - (a^2*tan(c + d*x))/(b*(a^2 + b^2)*d*(a + b*tan(c + d*x)))]
    @test_int [tan(c + d*x)^2/(a + b*tan(c + d*x))^2, x, 3, -(((a^2 - b^2)*x)/(a^2 + b^2)^2) - (2*a*b*log(a*cos(c + d*x) + b*sin(c + d*x)))/((a^2 + b^2)^2*d) - a^2/(b*(a^2 + b^2)*d*(a + b*tan(c + d*x)))]
    @test_int [tan(c + d*x)^1/(a + b*tan(c + d*x))^2, x, 3, (2*a*b*x)/(a^2 + b^2)^2 - ((a^2 - b^2)*log(a*cos(c + d*x) + b*sin(c + d*x)))/((a^2 + b^2)^2*d) + a/((a^2 + b^2)*d*(a + b*tan(c + d*x)))]
    @test_int [tan(c + d*x)^0/(a + b*tan(c + d*x))^2, x, 3, ((a^2 - b^2)*x)/(a^2 + b^2)^2 + (2*a*b*log(a*cos(c + d*x) + b*sin(c + d*x)))/((a^2 + b^2)^2*d) - b/((a^2 + b^2)*d*(a + b*tan(c + d*x)))]
    @test_int [cot(c + d*x)^1/(a + b*tan(c + d*x))^2, x, 4, -((2*a*b*x)/(a^2 + b^2)^2) + log(sin(c + d*x))/(a^2*d) - (b^2*(3*a^2 + b^2)*log(a*cos(c + d*x) + b*sin(c + d*x)))/(a^2*(a^2 + b^2)^2*d) + b^2/(a*(a^2 + b^2)*d*(a + b*tan(c + d*x)))]
    @test_int [cot(c + d*x)^2/(a + b*tan(c + d*x))^2, x, 5, -(((a^2 - b^2)*x)/(a^2 + b^2)^2) - (2*b*log(sin(c + d*x)))/(a^3*d) + (2*b^3*(2*a^2 + b^2)*log(a*cos(c + d*x) + b*sin(c + d*x)))/(a^3*(a^2 + b^2)^2*d) - (b*(a^2 + 2*b^2))/(a^2*(a^2 + b^2)*d*(a + b*tan(c + d*x))) - cot(c + d*x)/(a*d*(a + b*tan(c + d*x)))]
    @test_int [cot(c + d*x)^3/(a + b*tan(c + d*x))^2, x, 6, (2*a*b*x)/(a^2 + b^2)^2 - ((a^2 - 3*b^2)*log(sin(c + d*x)))/(a^4*d) - (b^4*(5*a^2 + 3*b^2)*log(a*cos(c + d*x) + b*sin(c + d*x)))/(a^4*(a^2 + b^2)^2*d) + (b^2*(2*a^2 + 3*b^2))/(a^3*(a^2 + b^2)*d*(a + b*tan(c + d*x))) + (3*b*cot(c + d*x))/(2*a^2*d*(a + b*tan(c + d*x))) - cot(c + d*x)^2/(2*a*d*(a + b*tan(c + d*x)))]


    @test_int [tan(c + d*x)^6/(a + b*tan(c + d*x))^3, x, 8, -((a*(a^2 - 3*b^2)*x)/(a^2 + b^2)^3) - (b*(3*a^2 - b^2)*log(cos(c + d*x)))/((a^2 + b^2)^3*d) + (a^4*(6*a^4 + 17*a^2*b^2 + 15*b^4)*log(a + b*tan(c + d*x)))/(b^5*(a^2 + b^2)^3*d) - (a*(6*a^4 + 11*a^2*b^2 + 3*b^4)*tan(c + d*x))/(b^4*(a^2 + b^2)^2*d) + ((6*a^4 + 11*a^2*b^2 + b^4)*tan(c + d*x)^2)/(2*b^3*(a^2 + b^2)^2*d) - (a^2*tan(c + d*x)^4)/(2*b*(a^2 + b^2)*d*(a + b*tan(c + d*x))^2) - (2*a^2*(a^2 + 2*b^2)*tan(c + d*x)^3)/(b^2*(a^2 + b^2)^2*d*(a + b*tan(c + d*x)))]
    @test_int [tan(c + d*x)^5/(a + b*tan(c + d*x))^3, x, 7, (b*(3*a^2 - b^2)*x)/(a^2 + b^2)^3 - (a*(a^2 - 3*b^2)*log(cos(c + d*x)))/((a^2 + b^2)^3*d) - (a^3*(3*a^4 + 9*a^2*b^2 + 10*b^4)*log(a + b*tan(c + d*x)))/(b^4*(a^2 + b^2)^3*d) + ((3*a^4 + 6*a^2*b^2 + b^4)*tan(c + d*x))/(b^3*(a^2 + b^2)^2*d) - (a^2*tan(c + d*x)^3)/(2*b*(a^2 + b^2)*d*(a + b*tan(c + d*x))^2) - (a^2*(3*a^2 + 7*b^2)*tan(c + d*x)^2)/(2*b^2*(a^2 + b^2)^2*d*(a + b*tan(c + d*x)))]
    @test_int [tan(c + d*x)^4/(a + b*tan(c + d*x))^3, x, 6, (a*(a^2 - 3*b^2)*x)/(a^2 + b^2)^3 + (b*(3*a^2 - b^2)*log(cos(c + d*x)))/((a^2 + b^2)^3*d) + (a^2*(a^4 + 3*a^2*b^2 + 6*b^4)*log(a + b*tan(c + d*x)))/(b^3*(a^2 + b^2)^3*d) - (a^2*tan(c + d*x)^2)/(2*b*(a^2 + b^2)*d*(a + b*tan(c + d*x))^2) + (a^3*(a^2 + 3*b^2))/(b^3*(a^2 + b^2)^2*d*(a + b*tan(c + d*x)))]
    @test_int [tan(c + d*x)^3/(a + b*tan(c + d*x))^3, x, 4, -((b*(3*a^2 - b^2)*x)/(a^2 + b^2)^3) + (a*(a^2 - 3*b^2)*log(a*cos(c + d*x) + b*sin(c + d*x)))/((a^2 + b^2)^3*d) - (a^2*tan(c + d*x))/(2*b*(a^2 + b^2)*d*(a + b*tan(c + d*x))^2) - (a^2*(a^2 + 5*b^2))/(2*b^2*(a^2 + b^2)^2*d*(a + b*tan(c + d*x)))]
    @test_int [tan(c + d*x)^2/(a + b*tan(c + d*x))^3, x, 4, -((a*(a^2 - 3*b^2)*x)/(a^2 + b^2)^3) - (b*(3*a^2 - b^2)*log(a*cos(c + d*x) + b*sin(c + d*x)))/((a^2 + b^2)^3*d) - a^2/(2*b*(a^2 + b^2)*d*(a + b*tan(c + d*x))^2) + (2*a*b)/((a^2 + b^2)^2*d*(a + b*tan(c + d*x)))]
    @test_int [tan(c + d*x)^1/(a + b*tan(c + d*x))^3, x, 4, (b*(3*a^2 - b^2)*x)/(a^2 + b^2)^3 - (a*(a^2 - 3*b^2)*log(a*cos(c + d*x) + b*sin(c + d*x)))/((a^2 + b^2)^3*d) + a/(2*(a^2 + b^2)*d*(a + b*tan(c + d*x))^2) + (a^2 - b^2)/((a^2 + b^2)^2*d*(a + b*tan(c + d*x)))]
    @test_int [tan(c + d*x)^0/(a + b*tan(c + d*x))^3, x, 4, (a*(a^2 - 3*b^2)*x)/(a^2 + b^2)^3 + (b*(3*a^2 - b^2)*log(a*cos(c + d*x) + b*sin(c + d*x)))/((a^2 + b^2)^3*d) - b/(2*(a^2 + b^2)*d*(a + b*tan(c + d*x))^2) - (2*a*b)/((a^2 + b^2)^2*d*(a + b*tan(c + d*x)))]
    @test_int [cot(c + d*x)^1/(a + b*tan(c + d*x))^3, x, 5, -((b*(3*a^2 - b^2)*x)/(a^2 + b^2)^3) + log(sin(c + d*x))/(a^3*d) - (b^2*(6*a^4 + 3*a^2*b^2 + b^4)*log(a*cos(c + d*x) + b*sin(c + d*x)))/(a^3*(a^2 + b^2)^3*d) + b^2/(2*a*(a^2 + b^2)*d*(a + b*tan(c + d*x))^2) + (b^2*(3*a^2 + b^2))/(a^2*(a^2 + b^2)^2*d*(a + b*tan(c + d*x)))]
    @test_int [cot(c + d*x)^2/(a + b*tan(c + d*x))^3, x, 6, -((a*(a^2 - 3*b^2)*x)/(a^2 + b^2)^3) - (3*b*log(sin(c + d*x)))/(a^4*d) + (b^3*(10*a^4 + 9*a^2*b^2 + 3*b^4)*log(a*cos(c + d*x) + b*sin(c + d*x)))/(a^4*(a^2 + b^2)^3*d) - (b*(2*a^2 + 3*b^2))/(2*a^2*(a^2 + b^2)*d*(a + b*tan(c + d*x))^2) - cot(c + d*x)/(a*d*(a + b*tan(c + d*x))^2) - (b*(a^4 + 6*a^2*b^2 + 3*b^4))/(a^3*(a^2 + b^2)^2*d*(a + b*tan(c + d*x)))]


    @test_int [tan(c + d*x)^6/(a + b*tan(c + d*x))^4, x, 8, -(((a^4 - 6*a^2*b^2 + b^4)*x)/(a^2 + b^2)^4) - (4*a*b*(a^2 - b^2)*log(cos(c + d*x)))/((a^2 + b^2)^4*d) - (4*a^3*(a^6 + 4*a^4*b^2 + 6*a^2*b^4 + 5*b^6)*log(a + b*tan(c + d*x)))/(b^5*(a^2 + b^2)^4*d) + ((4*a^6 + 12*a^4*b^2 + 13*a^2*b^4 + b^6)*tan(c + d*x))/(b^4*(a^2 + b^2)^3*d) - (a^2*tan(c + d*x)^4)/(3*b*(a^2 + b^2)*d*(a + b*tan(c + d*x))^3) - (a^2*(2*a^2 + 5*b^2)*tan(c + d*x)^3)/(3*b^2*(a^2 + b^2)^2*d*(a + b*tan(c + d*x))^2) - (2*a^2*(a^4 + 3*a^2*b^2 + 4*b^4)*tan(c + d*x)^2)/(b^3*(a^2 + b^2)^3*d*(a + b*tan(c + d*x)))]
    @test_int [tan(c + d*x)^5/(a + b*tan(c + d*x))^4, x, 7, (4*a*b*(a^2 - b^2)*x)/(a^2 + b^2)^4 - ((a^4 - 6*a^2*b^2 + b^4)*log(cos(c + d*x)))/((a^2 + b^2)^4*d) + (a^2*(a^6 + 4*a^4*b^2 + 5*a^2*b^4 + 10*b^6)*log(a + b*tan(c + d*x)))/(b^4*(a^2 + b^2)^4*d) - (a^2*tan(c + d*x)^3)/(3*b*(a^2 + b^2)*d*(a + b*tan(c + d*x))^3) - (a^2*(a^2 + 3*b^2)*tan(c + d*x)^2)/(2*b^2*(a^2 + b^2)^2*d*(a + b*tan(c + d*x))^2) + (a^3*(a^4 + 3*a^2*b^2 + 6*b^4))/(b^4*(a^2 + b^2)^3*d*(a + b*tan(c + d*x)))]
    @test_int [tan(c + d*x)^4/(a + b*tan(c + d*x))^4, x, 5, ((a^4 - 6*a^2*b^2 + b^4)*x)/(a^2 + b^2)^4 + (4*a*b*(a^2 - b^2)*log(a*cos(c + d*x) + b*sin(c + d*x)))/((a^2 + b^2)^4*d) - (a^2*tan(c + d*x)^2)/(3*b*(a^2 + b^2)*d*(a + b*tan(c + d*x))^3) + (a^3*(a^2 + 4*b^2))/(3*b^3*(a^2 + b^2)^2*d*(a + b*tan(c + d*x))^2) - (a^2*(2*a^4 + 7*a^2*b^2 + 17*b^4))/(3*b^3*(a^2 + b^2)^3*d*(a + b*tan(c + d*x)))]
    @test_int [tan(c + d*x)^3/(a + b*tan(c + d*x))^4, x, 5, -((4*a*b*(a^2 - b^2)*x)/(a^2 + b^2)^4) + ((a^4 - 6*a^2*b^2 + b^4)*log(a*cos(c + d*x) + b*sin(c + d*x)))/((a^2 + b^2)^4*d) - (a^2*tan(c + d*x))/(3*b*(a^2 + b^2)*d*(a + b*tan(c + d*x))^3) - (a^2*(a^2 + 7*b^2))/(6*b^2*(a^2 + b^2)^2*d*(a + b*tan(c + d*x))^2) - (a*(a^2 - 3*b^2))/((a^2 + b^2)^3*d*(a + b*tan(c + d*x)))]
    @test_int [tan(c + d*x)^2/(a + b*tan(c + d*x))^4, x, 5, -(((a^4 - 6*a^2*b^2 + b^4)*x)/(a^2 + b^2)^4) - (4*a*b*(a^2 - b^2)*log(a*cos(c + d*x) + b*sin(c + d*x)))/((a^2 + b^2)^4*d) - a^2/(3*b*(a^2 + b^2)*d*(a + b*tan(c + d*x))^3) + (a*b)/((a^2 + b^2)^2*d*(a + b*tan(c + d*x))^2) + (b*(3*a^2 - b^2))/((a^2 + b^2)^3*d*(a + b*tan(c + d*x)))]
    @test_int [tan(c + d*x)^1/(a + b*tan(c + d*x))^4, x, 5, (4*a*b*(a^2 - b^2)*x)/(a^2 + b^2)^4 - ((a^4 - 6*a^2*b^2 + b^4)*log(a*cos(c + d*x) + b*sin(c + d*x)))/((a^2 + b^2)^4*d) + a/(3*(a^2 + b^2)*d*(a + b*tan(c + d*x))^3) + (a^2 - b^2)/(2*(a^2 + b^2)^2*d*(a + b*tan(c + d*x))^2) + (a*(a^2 - 3*b^2))/((a^2 + b^2)^3*d*(a + b*tan(c + d*x)))]
    @test_int [tan(c + d*x)^0/(a + b*tan(c + d*x))^4, x, 5, ((a^4 - 6*a^2*b^2 + b^4)*x)/(a^2 + b^2)^4 + (4*a*b*(a^2 - b^2)*log(a*cos(c + d*x) + b*sin(c + d*x)))/((a^2 + b^2)^4*d) - b/(3*(a^2 + b^2)*d*(a + b*tan(c + d*x))^3) - (a*b)/((a^2 + b^2)^2*d*(a + b*tan(c + d*x))^2) - (b*(3*a^2 - b^2))/((a^2 + b^2)^3*d*(a + b*tan(c + d*x)))]
    @test_int [cot(c + d*x)^1/(a + b*tan(c + d*x))^4, x, 6, -((4*a*b*(a^2 - b^2)*x)/(a^2 + b^2)^4) + log(sin(c + d*x))/(a^4*d) - (b^2*(10*a^6 + 5*a^4*b^2 + 4*a^2*b^4 + b^6)*log(a*cos(c + d*x) + b*sin(c + d*x)))/(a^4*(a^2 + b^2)^4*d) + b^2/(3*a*(a^2 + b^2)*d*(a + b*tan(c + d*x))^3) + (b^2*(3*a^2 + b^2))/(2*a^2*(a^2 + b^2)^2*d*(a + b*tan(c + d*x))^2) + (b^2*(6*a^4 + 3*a^2*b^2 + b^4))/(a^3*(a^2 + b^2)^3*d*(a + b*tan(c + d*x)))]
    @test_int [cot(c + d*x)^2/(a + b*tan(c + d*x))^4, x, 7, -(((a^4 - 6*a^2*b^2 + b^4)*x)/(a^2 + b^2)^4) - (4*b*log(sin(c + d*x)))/(a^5*d) + (4*b^3*(5*a^6 + 6*a^4*b^2 + 4*a^2*b^4 + b^6)*log(a*cos(c + d*x) + b*sin(c + d*x)))/(a^5*(a^2 + b^2)^4*d) - (b*(3*a^2 + 4*b^2))/(3*a^2*(a^2 + b^2)*d*(a + b*tan(c + d*x))^3) - cot(c + d*x)/(a*d*(a + b*tan(c + d*x))^3) - (b*(a^4 + 4*a^2*b^2 + 2*b^4))/(a^3*(a^2 + b^2)^2*d*(a + b*tan(c + d*x))^2) - (b*(a^6 + 13*a^4*b^2 + 12*a^2*b^4 + 4*b^6))/(a^4*(a^2 + b^2)^3*d*(a + b*tan(c + d*x)))]


    @test_int [1/(3 + 5*tan(c + d*x)), x, 2, (3*x)/34 + (5*log(3*cos(c + d*x) + 5*sin(c + d*x)))/(34*d)]
    @test_int [1/(3 + 5*tan(c + d*x))^2, x, 3, -((4*x)/289) + (15*log(3*cos(c + d*x) + 5*sin(c + d*x)))/(578*d) - 5/(34*d*(3 + 5*tan(c + d*x)))]
    @test_int [1/(3 + 5*tan(c + d*x))^3, x, 4, -((99*x)/19652) + (5*log(3*cos(c + d*x) + 5*sin(c + d*x)))/(19652*d) - 5/(68*d*(3 + 5*tan(c + d*x))^2) - 15/(578*d*(3 + 5*tan(c + d*x)))]
    @test_int [1/(3 + 5*tan(c + d*x))^4, x, 5, -((161*x)/334084) - (60*log(3*cos(c + d*x) + 5*sin(c + d*x)))/(83521*d) - 5/(102*d*(3 + 5*tan(c + d*x))^3) - 15/(1156*d*(3 + 5*tan(c + d*x))^2) - 5/(19652*d*(3 + 5*tan(c + d*x)))]

    @test_int [1/(5 + 3*tan(c + d*x)), x, 2, (5*x)/34 + (3*log(5*cos(c + d*x) + 3*sin(c + d*x)))/(34*d)]
    @test_int [1/(5 + 3*tan(c + d*x))^2, x, 3, (4*x)/289 + (15*log(5*cos(c + d*x) + 3*sin(c + d*x)))/(578*d) - 3/(34*d*(5 + 3*tan(c + d*x)))]
    @test_int [1/(5 + 3*tan(c + d*x))^3, x, 4, -((5*x)/19652) + (99*log(5*cos(c + d*x) + 3*sin(c + d*x)))/(19652*d) - 3/(68*d*(5 + 3*tan(c + d*x))^2) - 15/(578*d*(5 + 3*tan(c + d*x)))]
    @test_int [1/(5 + 3*tan(c + d*x))^4, x, 5, -((161*x)/334084) + (60*log(5*cos(c + d*x) + 3*sin(c + d*x)))/(83521*d) - 1/(34*d*(5 + 3*tan(c + d*x))^3) - 15/(1156*d*(5 + 3*tan(c + d*x))^2) - 99/(19652*d*(5 + 3*tan(c + d*x)))]


    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*(a+b*tan(e+f*x))^(m/2)*(d*tan(e+f*x))^n=#


    #= ::Subsubsection::Closed:: =#
    #=m>0=#


    @test_int [tan(c + d*x)^4*sqrt(a + b*tan(c + d*x)), x, 14, (b*atanh((sqrt(a + sqrt(a^2 + b^2)) - sqrt(2)*sqrt(a + b*tan(c + d*x)))/sqrt(a - sqrt(a^2 + b^2))))/(sqrt(2)*sqrt(a - sqrt(a^2 + b^2))*d) - (b*atanh((sqrt(a + sqrt(a^2 + b^2)) + sqrt(2)*sqrt(a + b*tan(c + d*x)))/sqrt(a - sqrt(a^2 + b^2))))/(sqrt(2)*sqrt(a - sqrt(a^2 + b^2))*d) + (b*log(a + sqrt(a^2 + b^2) + b*tan(c + d*x) - sqrt(2)*sqrt(a + sqrt(a^2 + b^2))*sqrt(a + b*tan(c + d*x))))/(2*sqrt(2)*sqrt(a + sqrt(a^2 + b^2))*d) - (b*log(a + sqrt(a^2 + b^2) + b*tan(c + d*x) + sqrt(2)*sqrt(a + sqrt(a^2 + b^2))*sqrt(a + b*tan(c + d*x))))/(2*sqrt(2)*sqrt(a + sqrt(a^2 + b^2))*d) + (2*(8*a^2 - 35*b^2)*(a + b*tan(c + d*x))^(3/2))/(105*b^3*d) - (8*a*tan(c + d*x)*(a + b*tan(c + d*x))^(3/2))/(35*b^2*d) + (2*tan(c + d*x)^2*(a + b*tan(c + d*x))^(3/2))/(7*b*d)]
    @test_int [tan(c + d*x)^3*sqrt(a + b*tan(c + d*x)), x, 11, (sqrt(a - im*b)*atanh(sqrt(a + b*tan(c + d*x))/sqrt(a - im*b)))/d + (sqrt(a + im*b)*atanh(sqrt(a + b*tan(c + d*x))/sqrt(a + im*b)))/d - (2*sqrt(a + b*tan(c + d*x)))/d - (4*a*(a + b*tan(c + d*x))^(3/2))/(15*b^2*d) + (2*tan(c + d*x)*(a + b*tan(c + d*x))^(3/2))/(5*b*d)]
    @test_int [tan(c + d*x)^2*sqrt(a + b*tan(c + d*x)), x, 12, -((b*atanh((sqrt(a + sqrt(a^2 + b^2)) - sqrt(2)*sqrt(a + b*tan(c + d*x)))/sqrt(a - sqrt(a^2 + b^2))))/(sqrt(2)*sqrt(a - sqrt(a^2 + b^2))*d)) + (b*atanh((sqrt(a + sqrt(a^2 + b^2)) + sqrt(2)*sqrt(a + b*tan(c + d*x)))/sqrt(a - sqrt(a^2 + b^2))))/(sqrt(2)*sqrt(a - sqrt(a^2 + b^2))*d) - (b*log(a + sqrt(a^2 + b^2) + b*tan(c + d*x) - sqrt(2)*sqrt(a + sqrt(a^2 + b^2))*sqrt(a + b*tan(c + d*x))))/(2*sqrt(2)*sqrt(a + sqrt(a^2 + b^2))*d) + (b*log(a + sqrt(a^2 + b^2) + b*tan(c + d*x) + sqrt(2)*sqrt(a + sqrt(a^2 + b^2))*sqrt(a + b*tan(c + d*x))))/(2*sqrt(2)*sqrt(a + sqrt(a^2 + b^2))*d) + (2*(a + b*tan(c + d*x))^(3/2))/(3*b*d)]
    @test_int [tan(c + d*x)^1*sqrt(a + b*tan(c + d*x)), x, 8, -((sqrt(a - im*b)*atanh(sqrt(a + b*tan(c + d*x))/sqrt(a - im*b)))/d) - (sqrt(a + im*b)*atanh(sqrt(a + b*tan(c + d*x))/sqrt(a + im*b)))/d + (2*sqrt(a + b*tan(c + d*x)))/d]
    @test_int [tan(c + d*x)^0*sqrt(a + b*tan(c + d*x)), x, 11, (b*atanh((sqrt(a + sqrt(a^2 + b^2)) - sqrt(2)*sqrt(a + b*tan(c + d*x)))/sqrt(a - sqrt(a^2 + b^2))))/(sqrt(2)*sqrt(a - sqrt(a^2 + b^2))*d) - (b*atanh((sqrt(a + sqrt(a^2 + b^2)) + sqrt(2)*sqrt(a + b*tan(c + d*x)))/sqrt(a - sqrt(a^2 + b^2))))/(sqrt(2)*sqrt(a - sqrt(a^2 + b^2))*d) + (b*log(a + sqrt(a^2 + b^2) + b*tan(c + d*x) - sqrt(2)*sqrt(a + sqrt(a^2 + b^2))*sqrt(a + b*tan(c + d*x))))/(2*sqrt(2)*sqrt(a + sqrt(a^2 + b^2))*d) - (b*log(a + sqrt(a^2 + b^2) + b*tan(c + d*x) + sqrt(2)*sqrt(a + sqrt(a^2 + b^2))*sqrt(a + b*tan(c + d*x))))/(2*sqrt(2)*sqrt(a + sqrt(a^2 + b^2))*d)]
    @test_int [sqrt(a + b*tan(c + d*x))*cot(c + d*x)^1, x, 11, -((2*sqrt(a)*atanh(sqrt(a + b*tan(c + d*x))/sqrt(a)))/d) + (sqrt(a - im*b)*atanh(sqrt(a + b*tan(c + d*x))/sqrt(a - im*b)))/d + (sqrt(a + im*b)*atanh(sqrt(a + b*tan(c + d*x))/sqrt(a + im*b)))/d]
    @test_int [sqrt(a + b*tan(c + d*x))*cot(c + d*x)^2, x, 16, -((b*atanh(sqrt(a + b*tan(c + d*x))/sqrt(a)))/(sqrt(a)*d)) - (b*atanh((sqrt(a + sqrt(a^2 + b^2)) - sqrt(2)*sqrt(a + b*tan(c + d*x)))/sqrt(a - sqrt(a^2 + b^2))))/(sqrt(2)*sqrt(a - sqrt(a^2 + b^2))*d) + (b*atanh((sqrt(a + sqrt(a^2 + b^2)) + sqrt(2)*sqrt(a + b*tan(c + d*x)))/sqrt(a - sqrt(a^2 + b^2))))/(sqrt(2)*sqrt(a - sqrt(a^2 + b^2))*d) - (b*log(a + sqrt(a^2 + b^2) + b*tan(c + d*x) - sqrt(2)*sqrt(a + sqrt(a^2 + b^2))*sqrt(a + b*tan(c + d*x))))/(2*sqrt(2)*sqrt(a + sqrt(a^2 + b^2))*d) + (b*log(a + sqrt(a^2 + b^2) + b*tan(c + d*x) + sqrt(2)*sqrt(a + sqrt(a^2 + b^2))*sqrt(a + b*tan(c + d*x))))/(2*sqrt(2)*sqrt(a + sqrt(a^2 + b^2))*d) - (cot(c + d*x)*sqrt(a + b*tan(c + d*x)))/d]
    @test_int [sqrt(a + b*tan(c + d*x))*cot(c + d*x)^3, x, 13, ((8*a^2 + b^2)*atanh(sqrt(a + b*tan(c + d*x))/sqrt(a)))/(4*a^(3/2)*d) - (sqrt(a - im*b)*atanh(sqrt(a + b*tan(c + d*x))/sqrt(a - im*b)))/d - (sqrt(a + im*b)*atanh(sqrt(a + b*tan(c + d*x))/sqrt(a + im*b)))/d - (b*cot(c + d*x)*sqrt(a + b*tan(c + d*x)))/(4*a*d) - (cot(c + d*x)^2*sqrt(a + b*tan(c + d*x)))/(2*d)]


    @test_int [tan(c + d*x)^4*(a + b*tan(c + d*x))^(3/2), x, 11, -((im*(a - im*b)^(3/2)*atanh(sqrt(a + b*tan(c + d*x))/sqrt(a - im*b)))/d) + (im*(a + im*b)^(3/2)*atanh(sqrt(a + b*tan(c + d*x))/sqrt(a + im*b)))/d + (2*b*sqrt(a + b*tan(c + d*x)))/d + (2*(8*a^2 - 63*b^2)*(a + b*tan(c + d*x))^(5/2))/(315*b^3*d) - (8*a*tan(c + d*x)*(a + b*tan(c + d*x))^(5/2))/(63*b^2*d) + (2*tan(c + d*x)^2*(a + b*tan(c + d*x))^(5/2))/(9*b*d)]
    @test_int [tan(c + d*x)^3*(a + b*tan(c + d*x))^(3/2), x, 12, ((a - im*b)^(3/2)*atanh(sqrt(a + b*tan(c + d*x))/sqrt(a - im*b)))/d + ((a + im*b)^(3/2)*atanh(sqrt(a + b*tan(c + d*x))/sqrt(a + im*b)))/d - (2*a*sqrt(a + b*tan(c + d*x)))/d - (2*(a + b*tan(c + d*x))^(3/2))/(3*d) - (4*a*(a + b*tan(c + d*x))^(5/2))/(35*b^2*d) + (2*tan(c + d*x)*(a + b*tan(c + d*x))^(5/2))/(7*b*d)]
    @test_int [tan(c + d*x)^2*(a + b*tan(c + d*x))^(3/2), x, 9, (im*(a - im*b)^(3/2)*atanh(sqrt(a + b*tan(c + d*x))/sqrt(a - im*b)))/d - (im*(a + im*b)^(3/2)*atanh(sqrt(a + b*tan(c + d*x))/sqrt(a + im*b)))/d - (2*b*sqrt(a + b*tan(c + d*x)))/d + (2*(a + b*tan(c + d*x))^(5/2))/(5*b*d)]
    @test_int [tan(c + d*x)^1*(a + b*tan(c + d*x))^(3/2), x, 9, -(((a - im*b)^(3/2)*atanh(sqrt(a + b*tan(c + d*x))/sqrt(a - im*b)))/d) - ((a + im*b)^(3/2)*atanh(sqrt(a + b*tan(c + d*x))/sqrt(a + im*b)))/d + (2*a*sqrt(a + b*tan(c + d*x)))/d + (2*(a + b*tan(c + d*x))^(3/2))/(3*d)]
    @test_int [tan(c + d*x)^0*(a + b*tan(c + d*x))^(3/2), x, 8, -((im*(a - im*b)^(3/2)*atanh(sqrt(a + b*tan(c + d*x))/sqrt(a - im*b)))/d) + (im*(a + im*b)^(3/2)*atanh(sqrt(a + b*tan(c + d*x))/sqrt(a + im*b)))/d + (2*b*sqrt(a + b*tan(c + d*x)))/d]
    @test_int [(a + b*tan(c + d*x))^(3/2)*cot(c + d*x)^1, x, 11, -((2*a^(3/2)*atanh(sqrt(a + b*tan(c + d*x))/sqrt(a)))/d) + ((a - im*b)^(3/2)*atanh(sqrt(a + b*tan(c + d*x))/sqrt(a - im*b)))/d + ((a + im*b)^(3/2)*atanh(sqrt(a + b*tan(c + d*x))/sqrt(a + im*b)))/d]
    @test_int [(a + b*tan(c + d*x))^(3/2)*cot(c + d*x)^2, x, 12, -((3*sqrt(a)*b*atanh(sqrt(a + b*tan(c + d*x))/sqrt(a)))/d) + (im*(a - im*b)^(3/2)*atanh(sqrt(a + b*tan(c + d*x))/sqrt(a - im*b)))/d - (im*(a + im*b)^(3/2)*atanh(sqrt(a + b*tan(c + d*x))/sqrt(a + im*b)))/d - (a*cot(c + d*x)*sqrt(a + b*tan(c + d*x)))/d]
    @test_int [(a + b*tan(c + d*x))^(3/2)*cot(c + d*x)^3, x, 13, ((8*a^2 - 3*b^2)*atanh(sqrt(a + b*tan(c + d*x))/sqrt(a)))/(4*sqrt(a)*d) - ((a - im*b)^(3/2)*atanh(sqrt(a + b*tan(c + d*x))/sqrt(a - im*b)))/d - ((a + im*b)^(3/2)*atanh(sqrt(a + b*tan(c + d*x))/sqrt(a + im*b)))/d - (5*b*cot(c + d*x)*sqrt(a + b*tan(c + d*x)))/(4*d) - (a*cot(c + d*x)^2*sqrt(a + b*tan(c + d*x)))/(2*d)]


    @test_int [tan(c + d*x)^3*(a + b*tan(c + d*x))^(5/2), x, 13, ((a - im*b)^(5/2)*atanh(sqrt(a + b*tan(c + d*x))/sqrt(a - im*b)))/d + ((a + im*b)^(5/2)*atanh(sqrt(a + b*tan(c + d*x))/sqrt(a + im*b)))/d - (2*(a^2 - b^2)*sqrt(a + b*tan(c + d*x)))/d - (2*a*(a + b*tan(c + d*x))^(3/2))/(3*d) - (2*(a + b*tan(c + d*x))^(5/2))/(5*d) - (4*a*(a + b*tan(c + d*x))^(7/2))/(63*b^2*d) + (2*tan(c + d*x)*(a + b*tan(c + d*x))^(7/2))/(9*b*d)]
    @test_int [tan(c + d*x)^2*(a + b*tan(c + d*x))^(5/2), x, 10, (im*(a - im*b)^(5/2)*atanh(sqrt(a + b*tan(c + d*x))/sqrt(a - im*b)))/d - (im*(a + im*b)^(5/2)*atanh(sqrt(a + b*tan(c + d*x))/sqrt(a + im*b)))/d - (4*a*b*sqrt(a + b*tan(c + d*x)))/d - (2*b*(a + b*tan(c + d*x))^(3/2))/(3*d) + (2*(a + b*tan(c + d*x))^(7/2))/(7*b*d)]
    @test_int [tan(c + d*x)^1*(a + b*tan(c + d*x))^(5/2), x, 10, -(((a - im*b)^(5/2)*atanh(sqrt(a + b*tan(c + d*x))/sqrt(a - im*b)))/d) - ((a + im*b)^(5/2)*atanh(sqrt(a + b*tan(c + d*x))/sqrt(a + im*b)))/d + (2*(a^2 - b^2)*sqrt(a + b*tan(c + d*x)))/d + (2*a*(a + b*tan(c + d*x))^(3/2))/(3*d) + (2*(a + b*tan(c + d*x))^(5/2))/(5*d)]
    @test_int [tan(c + d*x)^0*(a + b*tan(c + d*x))^(5/2), x, 9, -((im*(a - im*b)^(5/2)*atanh(sqrt(a + b*tan(c + d*x))/sqrt(a - im*b)))/d) + (im*(a + im*b)^(5/2)*atanh(sqrt(a + b*tan(c + d*x))/sqrt(a + im*b)))/d + (4*a*b*sqrt(a + b*tan(c + d*x)))/d + (2*b*(a + b*tan(c + d*x))^(3/2))/(3*d)]
    @test_int [(a + b*tan(c + d*x))^(5/2)*cot(c + d*x)^1, x, 12, -((2*a^(5/2)*atanh(sqrt(a + b*tan(c + d*x))/sqrt(a)))/d) + ((a - im*b)^(5/2)*atanh(sqrt(a + b*tan(c + d*x))/sqrt(a - im*b)))/d + ((a + im*b)^(5/2)*atanh(sqrt(a + b*tan(c + d*x))/sqrt(a + im*b)))/d + (2*b^2*sqrt(a + b*tan(c + d*x)))/d]
    @test_int [(a + b*tan(c + d*x))^(5/2)*cot(c + d*x)^2, x, 12, -((5*a^(3/2)*b*atanh(sqrt(a + b*tan(c + d*x))/sqrt(a)))/d) + (im*(a - im*b)^(5/2)*atanh(sqrt(a + b*tan(c + d*x))/sqrt(a - im*b)))/d - (im*(a + im*b)^(5/2)*atanh(sqrt(a + b*tan(c + d*x))/sqrt(a + im*b)))/d - (a^2*cot(c + d*x)*sqrt(a + b*tan(c + d*x)))/d]
    @test_int [(a + b*tan(c + d*x))^(5/2)*cot(c + d*x)^3, x, 13, (sqrt(a)*(8*a^2 - 15*b^2)*atanh(sqrt(a + b*tan(c + d*x))/sqrt(a)))/(4*d) - ((a - im*b)^(5/2)*atanh(sqrt(a + b*tan(c + d*x))/sqrt(a - im*b)))/d - ((a + im*b)^(5/2)*atanh(sqrt(a + b*tan(c + d*x))/sqrt(a + im*b)))/d - (9*a*b*cot(c + d*x)*sqrt(a + b*tan(c + d*x)))/(4*d) - (a^2*cot(c + d*x)^2*sqrt(a + b*tan(c + d*x)))/(2*d)]
    @test_int [(a + b*tan(c + d*x))^(5/2)*cot(c + d*x)^4, x, 14, (5*b*(8*a^2 - b^2)*atanh(sqrt(a + b*tan(c + d*x))/sqrt(a)))/(8*sqrt(a)*d) - (im*(a - im*b)^(5/2)*atanh(sqrt(a + b*tan(c + d*x))/sqrt(a - im*b)))/d + (im*(a + im*b)^(5/2)*atanh(sqrt(a + b*tan(c + d*x))/sqrt(a + im*b)))/d + ((8*a^2 - 11*b^2)*cot(c + d*x)*sqrt(a + b*tan(c + d*x)))/(8*d) - (13*a*b*cot(c + d*x)^2*sqrt(a + b*tan(c + d*x)))/(12*d) - (a^2*cot(c + d*x)^3*sqrt(a + b*tan(c + d*x)))/(3*d)]


    @test_int [(a + b*tan(c + d*x))^(7/2), x, 10, -((im*(a - im*b)^(7/2)*atanh(sqrt(a + b*tan(c + d*x))/sqrt(a - im*b)))/d) + (im*(a + im*b)^(7/2)*atanh(sqrt(a + b*tan(c + d*x))/sqrt(a + im*b)))/d + (2*b*(3*a^2 - b^2)*sqrt(a + b*tan(c + d*x)))/d + (4*a*b*(a + b*tan(c + d*x))^(3/2))/(3*d) + (2*b*(a + b*tan(c + d*x))^(5/2))/(5*d)]


    #= ::Subsubsection::Closed:: =#
    #=m<0=#


    @test_int [tan(c + d*x)^5/sqrt(a + b*tan(c + d*x)), x, 12, -(atanh(sqrt(a + b*tan(c + d*x))/sqrt(a - im*b))/(sqrt(a - im*b)*d)) - atanh(sqrt(a + b*tan(c + d*x))/sqrt(a + im*b))/(sqrt(a + im*b)*d) - (4*a*(24*a^2 - 35*b^2)*sqrt(a + b*tan(c + d*x)))/(105*b^4*d) + (2*(24*a^2 - 35*b^2)*tan(c + d*x)*sqrt(a + b*tan(c + d*x)))/(105*b^3*d) - (12*a*tan(c + d*x)^2*sqrt(a + b*tan(c + d*x)))/(35*b^2*d) + (2*tan(c + d*x)^3*sqrt(a + b*tan(c + d*x)))/(7*b*d)]
    @test_int [tan(c + d*x)^4/sqrt(a + b*tan(c + d*x)), x, 14, (b*atanh((sqrt(a + sqrt(a^2 + b^2)) - sqrt(2)*sqrt(a + b*tan(c + d*x)))/sqrt(a - sqrt(a^2 + b^2))))/(sqrt(2)*sqrt(a^2 + b^2)*sqrt(a - sqrt(a^2 + b^2))*d) - (b*atanh((sqrt(a + sqrt(a^2 + b^2)) + sqrt(2)*sqrt(a + b*tan(c + d*x)))/sqrt(a - sqrt(a^2 + b^2))))/(sqrt(2)*sqrt(a^2 + b^2)*sqrt(a - sqrt(a^2 + b^2))*d) - (b*log(a + sqrt(a^2 + b^2) + b*tan(c + d*x) - sqrt(2)*sqrt(a + sqrt(a^2 + b^2))*sqrt(a + b*tan(c + d*x))))/(2*sqrt(2)*sqrt(a^2 + b^2)*sqrt(a + sqrt(a^2 + b^2))*d) + (b*log(a + sqrt(a^2 + b^2) + b*tan(c + d*x) + sqrt(2)*sqrt(a + sqrt(a^2 + b^2))*sqrt(a + b*tan(c + d*x))))/(2*sqrt(2)*sqrt(a^2 + b^2)*sqrt(a + sqrt(a^2 + b^2))*d) + (2*(8*a^2 - 15*b^2)*sqrt(a + b*tan(c + d*x)))/(15*b^3*d) - (8*a*tan(c + d*x)*sqrt(a + b*tan(c + d*x)))/(15*b^2*d) + (2*tan(c + d*x)^2*sqrt(a + b*tan(c + d*x)))/(5*b*d)]
    @test_int [tan(c + d*x)^3/sqrt(a + b*tan(c + d*x)), x, 10, atanh(sqrt(a + b*tan(c + d*x))/sqrt(a - im*b))/(sqrt(a - im*b)*d) + atanh(sqrt(a + b*tan(c + d*x))/sqrt(a + im*b))/(sqrt(a + im*b)*d) - (4*a*sqrt(a + b*tan(c + d*x)))/(3*b^2*d) + (2*tan(c + d*x)*sqrt(a + b*tan(c + d*x)))/(3*b*d)]
    @test_int [tan(c + d*x)^2/sqrt(a + b*tan(c + d*x)), x, 12, -((b*atanh((sqrt(a + sqrt(a^2 + b^2)) - sqrt(2)*sqrt(a + b*tan(c + d*x)))/sqrt(a - sqrt(a^2 + b^2))))/(sqrt(2)*sqrt(a^2 + b^2)*sqrt(a - sqrt(a^2 + b^2))*d)) + (b*atanh((sqrt(a + sqrt(a^2 + b^2)) + sqrt(2)*sqrt(a + b*tan(c + d*x)))/sqrt(a - sqrt(a^2 + b^2))))/(sqrt(2)*sqrt(a^2 + b^2)*sqrt(a - sqrt(a^2 + b^2))*d) + (b*log(a + sqrt(a^2 + b^2) + b*tan(c + d*x) - sqrt(2)*sqrt(a + sqrt(a^2 + b^2))*sqrt(a + b*tan(c + d*x))))/(2*sqrt(2)*sqrt(a^2 + b^2)*sqrt(a + sqrt(a^2 + b^2))*d) - (b*log(a + sqrt(a^2 + b^2) + b*tan(c + d*x) + sqrt(2)*sqrt(a + sqrt(a^2 + b^2))*sqrt(a + b*tan(c + d*x))))/(2*sqrt(2)*sqrt(a^2 + b^2)*sqrt(a + sqrt(a^2 + b^2))*d) + (2*sqrt(a + b*tan(c + d*x)))/(b*d)]
    @test_int [tan(c + d*x)^1/sqrt(a + b*tan(c + d*x)), x, 7, -(atanh(sqrt(a + b*tan(c + d*x))/sqrt(a - im*b))/(sqrt(a - im*b)*d)) - atanh(sqrt(a + b*tan(c + d*x))/sqrt(a + im*b))/(sqrt(a + im*b)*d)]
    @test_int [tan(c + d*x)^0/sqrt(a + b*tan(c + d*x)), x, 11, (b*atanh((sqrt(a + sqrt(a^2 + b^2)) - sqrt(2)*sqrt(a + b*tan(c + d*x)))/sqrt(a - sqrt(a^2 + b^2))))/(sqrt(2)*sqrt(a^2 + b^2)*sqrt(a - sqrt(a^2 + b^2))*d) - (b*atanh((sqrt(a + sqrt(a^2 + b^2)) + sqrt(2)*sqrt(a + b*tan(c + d*x)))/sqrt(a - sqrt(a^2 + b^2))))/(sqrt(2)*sqrt(a^2 + b^2)*sqrt(a - sqrt(a^2 + b^2))*d) - (b*log(a + sqrt(a^2 + b^2) + b*tan(c + d*x) - sqrt(2)*sqrt(a + sqrt(a^2 + b^2))*sqrt(a + b*tan(c + d*x))))/(2*sqrt(2)*sqrt(a^2 + b^2)*sqrt(a + sqrt(a^2 + b^2))*d) + (b*log(a + sqrt(a^2 + b^2) + b*tan(c + d*x) + sqrt(2)*sqrt(a + sqrt(a^2 + b^2))*sqrt(a + b*tan(c + d*x))))/(2*sqrt(2)*sqrt(a^2 + b^2)*sqrt(a + sqrt(a^2 + b^2))*d)]
    @test_int [cot(c + d*x)^1/sqrt(a + b*tan(c + d*x)), x, 11, -((2*atanh(sqrt(a + b*tan(c + d*x))/sqrt(a)))/(sqrt(a)*d)) + atanh(sqrt(a + b*tan(c + d*x))/sqrt(a - im*b))/(sqrt(a - im*b)*d) + atanh(sqrt(a + b*tan(c + d*x))/sqrt(a + im*b))/(sqrt(a + im*b)*d)]
    @test_int [cot(c + d*x)^2/sqrt(a + b*tan(c + d*x)), x, 17, (b*atanh(sqrt(a + b*tan(c + d*x))/sqrt(a)))/(a^(3/2)*d) - (b*atanh((sqrt(a + sqrt(a^2 + b^2)) - sqrt(2)*sqrt(a + b*tan(c + d*x)))/sqrt(a - sqrt(a^2 + b^2))))/(sqrt(2)*sqrt(a^2 + b^2)*sqrt(a - sqrt(a^2 + b^2))*d) + (b*atanh((sqrt(a + sqrt(a^2 + b^2)) + sqrt(2)*sqrt(a + b*tan(c + d*x)))/sqrt(a - sqrt(a^2 + b^2))))/(sqrt(2)*sqrt(a^2 + b^2)*sqrt(a - sqrt(a^2 + b^2))*d) + (b*log(a + sqrt(a^2 + b^2) + b*tan(c + d*x) - sqrt(2)*sqrt(a + sqrt(a^2 + b^2))*sqrt(a + b*tan(c + d*x))))/(2*sqrt(2)*sqrt(a^2 + b^2)*sqrt(a + sqrt(a^2 + b^2))*d) - (b*log(a + sqrt(a^2 + b^2) + b*tan(c + d*x) + sqrt(2)*sqrt(a + sqrt(a^2 + b^2))*sqrt(a + b*tan(c + d*x))))/(2*sqrt(2)*sqrt(a^2 + b^2)*sqrt(a + sqrt(a^2 + b^2))*d) - (cot(c + d*x)*sqrt(a + b*tan(c + d*x)))/(a*d)]
    @test_int [cot(c + d*x)^3/sqrt(a + b*tan(c + d*x)), x, 14, ((8*a^2 - 3*b^2)*atanh(sqrt(a + b*tan(c + d*x))/sqrt(a)))/(4*a^(5/2)*d) - atanh(sqrt(a + b*tan(c + d*x))/sqrt(a - im*b))/(sqrt(a - im*b)*d) - atanh(sqrt(a + b*tan(c + d*x))/sqrt(a + im*b))/(sqrt(a + im*b)*d) + (3*b*cot(c + d*x)*sqrt(a + b*tan(c + d*x)))/(4*a^2*d) - (cot(c + d*x)^2*sqrt(a + b*tan(c + d*x)))/(2*a*d)]


    @test_int [tan(c + d*x)^5/(a + b*tan(c + d*x))^(3/2), x, 11, -(atanh(sqrt(a + b*tan(c + d*x))/sqrt(a - im*b))/((a - im*b)^(3/2)*d)) - atanh(sqrt(a + b*tan(c + d*x))/sqrt(a + im*b))/((a + im*b)^(3/2)*d) - (2*a^2*tan(c + d*x)^3)/(b*(a^2 + b^2)*d*sqrt(a + b*tan(c + d*x))) + (2*(16*a^4 + 6*a^2*b^2 - 5*b^4)*sqrt(a + b*tan(c + d*x)))/(5*b^4*(a^2 + b^2)*d) - (2*a*(8*a^2 + 3*b^2)*tan(c + d*x)*sqrt(a + b*tan(c + d*x)))/(5*b^3*(a^2 + b^2)*d) + (2*(6*a^2 + b^2)*tan(c + d*x)^2*sqrt(a + b*tan(c + d*x)))/(5*b^2*(a^2 + b^2)*d)]
    @test_int [tan(c + d*x)^4/(a + b*tan(c + d*x))^(3/2), x, 10, -((im*atanh(sqrt(a + b*tan(c + d*x))/sqrt(a - im*b)))/((a - im*b)^(3/2)*d)) + (im*atanh(sqrt(a + b*tan(c + d*x))/sqrt(a + im*b)))/((a + im*b)^(3/2)*d) - (2*a^2*tan(c + d*x)^2)/(b*(a^2 + b^2)*d*sqrt(a + b*tan(c + d*x))) - (2*a*(8*a^2 + 5*b^2)*sqrt(a + b*tan(c + d*x)))/(3*b^3*(a^2 + b^2)*d) + (2*(4*a^2 + b^2)*tan(c + d*x)*sqrt(a + b*tan(c + d*x)))/(3*b^2*(a^2 + b^2)*d)]
    @test_int [tan(c + d*x)^3/(a + b*tan(c + d*x))^(3/2), x, 9, atanh(sqrt(a + b*tan(c + d*x))/sqrt(a - im*b))/((a - im*b)^(3/2)*d) + atanh(sqrt(a + b*tan(c + d*x))/sqrt(a + im*b))/((a + im*b)^(3/2)*d) - (2*a^2*tan(c + d*x))/(b*(a^2 + b^2)*d*sqrt(a + b*tan(c + d*x))) + (2*(2*a^2 + b^2)*sqrt(a + b*tan(c + d*x)))/(b^2*(a^2 + b^2)*d)]
    @test_int [tan(c + d*x)^2/(a + b*tan(c + d*x))^(3/2), x, 8, (im*atanh(sqrt(a + b*tan(c + d*x))/sqrt(a - im*b)))/((a - im*b)^(3/2)*d) - (im*atanh(sqrt(a + b*tan(c + d*x))/sqrt(a + im*b)))/((a + im*b)^(3/2)*d) - (2*a^2)/(b*(a^2 + b^2)*d*sqrt(a + b*tan(c + d*x)))]
    @test_int [tan(c + d*x)^1/(a + b*tan(c + d*x))^(3/2), x, 8, -(atanh(sqrt(a + b*tan(c + d*x))/sqrt(a - im*b))/((a - im*b)^(3/2)*d)) - atanh(sqrt(a + b*tan(c + d*x))/sqrt(a + im*b))/((a + im*b)^(3/2)*d) + (2*a)/((a^2 + b^2)*d*sqrt(a + b*tan(c + d*x)))]
    @test_int [tan(c + d*x)^0/(a + b*tan(c + d*x))^(3/2), x, 8, -((im*atanh(sqrt(a + b*tan(c + d*x))/sqrt(a - im*b)))/((a - im*b)^(3/2)*d)) + (im*atanh(sqrt(a + b*tan(c + d*x))/sqrt(a + im*b)))/((a + im*b)^(3/2)*d) - (2*b)/((a^2 + b^2)*d*sqrt(a + b*tan(c + d*x)))]
    @test_int [cot(c + d*x)^1/(a + b*tan(c + d*x))^(3/2), x, 12, -((2*atanh(sqrt(a + b*tan(c + d*x))/sqrt(a)))/(a^(3/2)*d)) + atanh(sqrt(a + b*tan(c + d*x))/sqrt(a - im*b))/((a - im*b)^(3/2)*d) + atanh(sqrt(a + b*tan(c + d*x))/sqrt(a + im*b))/((a + im*b)^(3/2)*d) + (2*b^2)/(a*(a^2 + b^2)*d*sqrt(a + b*tan(c + d*x)))]
    @test_int [cot(c + d*x)^2/(a + b*tan(c + d*x))^(3/2), x, 13, (3*b*atanh(sqrt(a + b*tan(c + d*x))/sqrt(a)))/(a^(5/2)*d) + (im*atanh(sqrt(a + b*tan(c + d*x))/sqrt(a - im*b)))/((a - im*b)^(3/2)*d) - (im*atanh(sqrt(a + b*tan(c + d*x))/sqrt(a + im*b)))/((a + im*b)^(3/2)*d) - (b*(a^2 + 3*b^2))/(a^2*(a^2 + b^2)*d*sqrt(a + b*tan(c + d*x))) - cot(c + d*x)/(a*d*sqrt(a + b*tan(c + d*x)))]
    @test_int [cot(c + d*x)^3/(a + b*tan(c + d*x))^(3/2), x, 14, ((8*a^2 - 15*b^2)*atanh(sqrt(a + b*tan(c + d*x))/sqrt(a)))/(4*a^(7/2)*d) - atanh(sqrt(a + b*tan(c + d*x))/sqrt(a - im*b))/((a - im*b)^(3/2)*d) - atanh(sqrt(a + b*tan(c + d*x))/sqrt(a + im*b))/((a + im*b)^(3/2)*d) + (b^2*(7*a^2 + 15*b^2))/(4*a^3*(a^2 + b^2)*d*sqrt(a + b*tan(c + d*x))) + (5*b*cot(c + d*x))/(4*a^2*d*sqrt(a + b*tan(c + d*x))) - cot(c + d*x)^2/(2*a*d*sqrt(a + b*tan(c + d*x)))]


    @test_int [tan(c + d*x)^5/(a + b*tan(c + d*x))^(5/2), x, 11, -(atanh(sqrt(a + b*tan(c + d*x))/sqrt(a - im*b))/((a - im*b)^(5/2)*d)) - atanh(sqrt(a + b*tan(c + d*x))/sqrt(a + im*b))/((a + im*b)^(5/2)*d) - (2*a^2*tan(c + d*x)^3)/(3*b*(a^2 + b^2)*d*(a + b*tan(c + d*x))^(3/2)) - (4*a^2*(a^2 + 2*b^2)*tan(c + d*x)^2)/(b^2*(a^2 + b^2)^2*d*sqrt(a + b*tan(c + d*x))) - (4*a*(8*a^4 + 15*a^2*b^2 + 4*b^4)*sqrt(a + b*tan(c + d*x)))/(3*b^4*(a^2 + b^2)^2*d) + (2*(8*a^4 + 15*a^2*b^2 + b^4)*tan(c + d*x)*sqrt(a + b*tan(c + d*x)))/(3*b^3*(a^2 + b^2)^2*d)]
    @test_int [tan(c + d*x)^4/(a + b*tan(c + d*x))^(5/2), x, 10, -((im*atanh(sqrt(a + b*tan(c + d*x))/sqrt(a - im*b)))/((a - im*b)^(5/2)*d)) + (im*atanh(sqrt(a + b*tan(c + d*x))/sqrt(a + im*b)))/((a + im*b)^(5/2)*d) - (2*a^2*tan(c + d*x)^2)/(3*b*(a^2 + b^2)*d*(a + b*tan(c + d*x))^(3/2)) + (4*a^3*(2*a^2 + 5*b^2))/(3*b^3*(a^2 + b^2)^2*d*sqrt(a + b*tan(c + d*x))) + (2*(4*a^2 + 3*b^2)*sqrt(a + b*tan(c + d*x)))/(3*b^3*(a^2 + b^2)*d)]
    @test_int [tan(c + d*x)^3/(a + b*tan(c + d*x))^(5/2), x, 9, atanh(sqrt(a + b*tan(c + d*x))/sqrt(a - im*b))/((a - im*b)^(5/2)*d) + atanh(sqrt(a + b*tan(c + d*x))/sqrt(a + im*b))/((a + im*b)^(5/2)*d) - (2*a^2*tan(c + d*x))/(3*b*(a^2 + b^2)*d*(a + b*tan(c + d*x))^(3/2)) - (4*a^2*(a^2 + 4*b^2))/(3*b^2*(a^2 + b^2)^2*d*sqrt(a + b*tan(c + d*x)))]
    @test_int [tan(c + d*x)^2/(a + b*tan(c + d*x))^(5/2), x, 9, (im*atanh(sqrt(a + b*tan(c + d*x))/sqrt(a - im*b)))/((a - im*b)^(5/2)*d) - (im*atanh(sqrt(a + b*tan(c + d*x))/sqrt(a + im*b)))/((a + im*b)^(5/2)*d) - (2*a^2)/(3*b*(a^2 + b^2)*d*(a + b*tan(c + d*x))^(3/2)) + (4*a*b)/((a^2 + b^2)^2*d*sqrt(a + b*tan(c + d*x)))]
    @test_int [tan(c + d*x)^1/(a + b*tan(c + d*x))^(5/2), x, 9, -(atanh(sqrt(a + b*tan(c + d*x))/sqrt(a - im*b))/((a - im*b)^(5/2)*d)) - atanh(sqrt(a + b*tan(c + d*x))/sqrt(a + im*b))/((a + im*b)^(5/2)*d) + (2*a)/(3*(a^2 + b^2)*d*(a + b*tan(c + d*x))^(3/2)) + (2*(a^2 - b^2))/((a^2 + b^2)^2*d*sqrt(a + b*tan(c + d*x)))]
    @test_int [tan(c + d*x)^0/(a + b*tan(c + d*x))^(5/2), x, 9, -((im*atanh(sqrt(a + b*tan(c + d*x))/sqrt(a - im*b)))/((a - im*b)^(5/2)*d)) + (im*atanh(sqrt(a + b*tan(c + d*x))/sqrt(a + im*b)))/((a + im*b)^(5/2)*d) - (2*b)/(3*(a^2 + b^2)*d*(a + b*tan(c + d*x))^(3/2)) - (4*a*b)/((a^2 + b^2)^2*d*sqrt(a + b*tan(c + d*x)))]
    @test_int [cot(c + d*x)^1/(a + b*tan(c + d*x))^(5/2), x, 13, -((2*atanh(sqrt(a + b*tan(c + d*x))/sqrt(a)))/(a^(5/2)*d)) + atanh(sqrt(a + b*tan(c + d*x))/sqrt(a - im*b))/((a - im*b)^(5/2)*d) + atanh(sqrt(a + b*tan(c + d*x))/sqrt(a + im*b))/((a + im*b)^(5/2)*d) + (2*b^2)/(3*a*(a^2 + b^2)*d*(a + b*tan(c + d*x))^(3/2)) + (2*b^2*(3*a^2 + b^2))/(a^2*(a^2 + b^2)^2*d*sqrt(a + b*tan(c + d*x)))]
    @test_int [cot(c + d*x)^2/(a + b*tan(c + d*x))^(5/2), x, 14, (5*b*atanh(sqrt(a + b*tan(c + d*x))/sqrt(a)))/(a^(7/2)*d) + (im*atanh(sqrt(a + b*tan(c + d*x))/sqrt(a - im*b)))/((a - im*b)^(5/2)*d) - (im*atanh(sqrt(a + b*tan(c + d*x))/sqrt(a + im*b)))/((a + im*b)^(5/2)*d) - (b*(3*a^2 + 5*b^2))/(3*a^2*(a^2 + b^2)*d*(a + b*tan(c + d*x))^(3/2)) - cot(c + d*x)/(a*d*(a + b*tan(c + d*x))^(3/2)) - (b*(a^4 + 10*a^2*b^2 + 5*b^4))/(a^3*(a^2 + b^2)^2*d*sqrt(a + b*tan(c + d*x)))]


    @test_int [1/(a + b*tan(c + d*x))^(7/2), x, 10, -((im*atanh(sqrt(a + b*tan(c + d*x))/sqrt(a - im*b)))/((a - im*b)^(7/2)*d)) + (im*atanh(sqrt(a + b*tan(c + d*x))/sqrt(a + im*b)))/((a + im*b)^(7/2)*d) - (2*b)/(5*(a^2 + b^2)*d*(a + b*tan(c + d*x))^(5/2)) - (4*a*b)/(3*(a^2 + b^2)^2*d*(a + b*tan(c + d*x))^(3/2)) - (2*b*(3*a^2 - b^2))/((a^2 + b^2)^3*d*sqrt(a + b*tan(c + d*x)))]


    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*(a+b*tan(e+f*x))^m*(d*tan(e+f*x))^(n/2)=#


    #= ::Subsubsection::Closed:: =#
    #=m>0=#


    @test_int [tan(c + d*x)^(5/2)*(a + b*tan(c + d*x)), x, 13, ((a - b)*atan(1 - sqrt(2)*sqrt(tan(c + d*x))))/(sqrt(2)*d) - ((a - b)*atan(1 + sqrt(2)*sqrt(tan(c + d*x))))/(sqrt(2)*d) - ((a + b)*log(1 - sqrt(2)*sqrt(tan(c + d*x)) + tan(c + d*x)))/(2*sqrt(2)*d) + ((a + b)*log(1 + sqrt(2)*sqrt(tan(c + d*x)) + tan(c + d*x)))/(2*sqrt(2)*d) - (2*b*sqrt(tan(c + d*x)))/d + (2*a*tan(c + d*x)^(3/2))/(3*d) + (2*b*tan(c + d*x)^(5/2))/(5*d)]
    @test_int [tan(c + d*x)^(3/2)*(a + b*tan(c + d*x)), x, 12, ((a + b)*atan(1 - sqrt(2)*sqrt(tan(c + d*x))))/(sqrt(2)*d) - ((a + b)*atan(1 + sqrt(2)*sqrt(tan(c + d*x))))/(sqrt(2)*d) + ((a - b)*log(1 - sqrt(2)*sqrt(tan(c + d*x)) + tan(c + d*x)))/(2*sqrt(2)*d) - ((a - b)*log(1 + sqrt(2)*sqrt(tan(c + d*x)) + tan(c + d*x)))/(2*sqrt(2)*d) + (2*a*sqrt(tan(c + d*x)))/d + (2*b*tan(c + d*x)^(3/2))/(3*d)]
    @test_int [sqrt(tan(c + d*x))*(a + b*tan(c + d*x)), x, 11, -(((a - b)*atan(1 - sqrt(2)*sqrt(tan(c + d*x))))/(sqrt(2)*d)) + ((a - b)*atan(1 + sqrt(2)*sqrt(tan(c + d*x))))/(sqrt(2)*d) + ((a + b)*log(1 - sqrt(2)*sqrt(tan(c + d*x)) + tan(c + d*x)))/(2*sqrt(2)*d) - ((a + b)*log(1 + sqrt(2)*sqrt(tan(c + d*x)) + tan(c + d*x)))/(2*sqrt(2)*d) + (2*b*sqrt(tan(c + d*x)))/d]
    @test_int [(a + b*tan(c + d*x))/sqrt(tan(c + d*x)), x, 10, -(((a + b)*atan(1 - sqrt(2)*sqrt(tan(c + d*x))))/(sqrt(2)*d)) + ((a + b)*atan(1 + sqrt(2)*sqrt(tan(c + d*x))))/(sqrt(2)*d) - ((a - b)*log(1 - sqrt(2)*sqrt(tan(c + d*x)) + tan(c + d*x)))/(2*sqrt(2)*d) + ((a - b)*log(1 + sqrt(2)*sqrt(tan(c + d*x)) + tan(c + d*x)))/(2*sqrt(2)*d)]
    @test_int [(a + b*tan(c + d*x))/tan(c + d*x)^(3/2), x, 11, ((a - b)*atan(1 - sqrt(2)*sqrt(tan(c + d*x))))/(sqrt(2)*d) - ((a - b)*atan(1 + sqrt(2)*sqrt(tan(c + d*x))))/(sqrt(2)*d) - ((a + b)*log(1 - sqrt(2)*sqrt(tan(c + d*x)) + tan(c + d*x)))/(2*sqrt(2)*d) + ((a + b)*log(1 + sqrt(2)*sqrt(tan(c + d*x)) + tan(c + d*x)))/(2*sqrt(2)*d) - (2*a)/(d*sqrt(tan(c + d*x)))]
    @test_int [(a + b*tan(c + d*x))/tan(c + d*x)^(5/2), x, 12, ((a + b)*atan(1 - sqrt(2)*sqrt(tan(c + d*x))))/(sqrt(2)*d) - ((a + b)*atan(1 + sqrt(2)*sqrt(tan(c + d*x))))/(sqrt(2)*d) + ((a - b)*log(1 - sqrt(2)*sqrt(tan(c + d*x)) + tan(c + d*x)))/(2*sqrt(2)*d) - ((a - b)*log(1 + sqrt(2)*sqrt(tan(c + d*x)) + tan(c + d*x)))/(2*sqrt(2)*d) - (2*a)/(3*d*tan(c + d*x)^(3/2)) - (2*b)/(d*sqrt(tan(c + d*x)))]
    @test_int [(a + b*tan(c + d*x))/tan(c + d*x)^(7/2), x, 13, -(((a - b)*atan(1 - sqrt(2)*sqrt(tan(c + d*x))))/(sqrt(2)*d)) + ((a - b)*atan(1 + sqrt(2)*sqrt(tan(c + d*x))))/(sqrt(2)*d) + ((a + b)*log(1 - sqrt(2)*sqrt(tan(c + d*x)) + tan(c + d*x)))/(2*sqrt(2)*d) - ((a + b)*log(1 + sqrt(2)*sqrt(tan(c + d*x)) + tan(c + d*x)))/(2*sqrt(2)*d) - (2*a)/(5*d*tan(c + d*x)^(5/2)) - (2*b)/(3*d*tan(c + d*x)^(3/2)) + (2*a)/(d*sqrt(tan(c + d*x)))]


    @test_int [tan(c + d*x)^(5/2)*(a + b*tan(c + d*x))^2, x, 14, ((a^2 - 2*a*b - b^2)*atan(1 - sqrt(2)*sqrt(tan(c + d*x))))/(sqrt(2)*d) - ((a^2 - 2*a*b - b^2)*atan(1 + sqrt(2)*sqrt(tan(c + d*x))))/(sqrt(2)*d) - ((a^2 + 2*a*b - b^2)*log(1 - sqrt(2)*sqrt(tan(c + d*x)) + tan(c + d*x)))/(2*sqrt(2)*d) + ((a^2 + 2*a*b - b^2)*log(1 + sqrt(2)*sqrt(tan(c + d*x)) + tan(c + d*x)))/(2*sqrt(2)*d) - (4*a*b*sqrt(tan(c + d*x)))/d + (2*(a^2 - b^2)*tan(c + d*x)^(3/2))/(3*d) + (4*a*b*tan(c + d*x)^(5/2))/(5*d) + (2*b^2*tan(c + d*x)^(7/2))/(7*d)]
    @test_int [tan(c + d*x)^(3/2)*(a + b*tan(c + d*x))^2, x, 13, ((a^2 + 2*a*b - b^2)*atan(1 - sqrt(2)*sqrt(tan(c + d*x))))/(sqrt(2)*d) - ((a^2 + 2*a*b - b^2)*atan(1 + sqrt(2)*sqrt(tan(c + d*x))))/(sqrt(2)*d) + ((a^2 - 2*a*b - b^2)*log(1 - sqrt(2)*sqrt(tan(c + d*x)) + tan(c + d*x)))/(2*sqrt(2)*d) - ((a^2 - 2*a*b - b^2)*log(1 + sqrt(2)*sqrt(tan(c + d*x)) + tan(c + d*x)))/(2*sqrt(2)*d) + (2*(a^2 - b^2)*sqrt(tan(c + d*x)))/d + (4*a*b*tan(c + d*x)^(3/2))/(3*d) + (2*b^2*tan(c + d*x)^(5/2))/(5*d)]
    @test_int [sqrt(tan(c + d*x))*(a + b*tan(c + d*x))^2, x, 12, -(((a^2 - 2*a*b - b^2)*atan(1 - sqrt(2)*sqrt(tan(c + d*x))))/(sqrt(2)*d)) + ((a^2 - 2*a*b - b^2)*atan(1 + sqrt(2)*sqrt(tan(c + d*x))))/(sqrt(2)*d) + ((a^2 + 2*a*b - b^2)*log(1 - sqrt(2)*sqrt(tan(c + d*x)) + tan(c + d*x)))/(2*sqrt(2)*d) - ((a^2 + 2*a*b - b^2)*log(1 + sqrt(2)*sqrt(tan(c + d*x)) + tan(c + d*x)))/(2*sqrt(2)*d) + (4*a*b*sqrt(tan(c + d*x)))/d + (2*b^2*tan(c + d*x)^(3/2))/(3*d)]
    @test_int [(a + b*tan(c + d*x))^2/sqrt(tan(c + d*x)), x, 11, -(((a^2 + 2*a*b - b^2)*atan(1 - sqrt(2)*sqrt(tan(c + d*x))))/(sqrt(2)*d)) + ((a^2 + 2*a*b - b^2)*atan(1 + sqrt(2)*sqrt(tan(c + d*x))))/(sqrt(2)*d) - ((a^2 - 2*a*b - b^2)*log(1 - sqrt(2)*sqrt(tan(c + d*x)) + tan(c + d*x)))/(2*sqrt(2)*d) + ((a^2 - 2*a*b - b^2)*log(1 + sqrt(2)*sqrt(tan(c + d*x)) + tan(c + d*x)))/(2*sqrt(2)*d) + (2*b^2*sqrt(tan(c + d*x)))/d]
    @test_int [(a + b*tan(c + d*x))^2/tan(c + d*x)^(3/2), x, 11, ((a^2 - 2*a*b - b^2)*atan(1 - sqrt(2)*sqrt(tan(c + d*x))))/(sqrt(2)*d) - ((a^2 - 2*a*b - b^2)*atan(1 + sqrt(2)*sqrt(tan(c + d*x))))/(sqrt(2)*d) - ((a^2 + 2*a*b - b^2)*log(1 - sqrt(2)*sqrt(tan(c + d*x)) + tan(c + d*x)))/(2*sqrt(2)*d) + ((a^2 + 2*a*b - b^2)*log(1 + sqrt(2)*sqrt(tan(c + d*x)) + tan(c + d*x)))/(2*sqrt(2)*d) - (2*a^2)/(d*sqrt(tan(c + d*x)))]
    @test_int [(a + b*tan(c + d*x))^2/tan(c + d*x)^(5/2), x, 12, ((a^2 + 2*a*b - b^2)*atan(1 - sqrt(2)*sqrt(tan(c + d*x))))/(sqrt(2)*d) - ((a^2 + 2*a*b - b^2)*atan(1 + sqrt(2)*sqrt(tan(c + d*x))))/(sqrt(2)*d) + ((a^2 - 2*a*b - b^2)*log(1 - sqrt(2)*sqrt(tan(c + d*x)) + tan(c + d*x)))/(2*sqrt(2)*d) - ((a^2 - 2*a*b - b^2)*log(1 + sqrt(2)*sqrt(tan(c + d*x)) + tan(c + d*x)))/(2*sqrt(2)*d) - (2*a^2)/(3*d*tan(c + d*x)^(3/2)) - (4*a*b)/(d*sqrt(tan(c + d*x)))]
    @test_int [(a + b*tan(c + d*x))^2/tan(c + d*x)^(7/2), x, 13, -(((a^2 - 2*a*b - b^2)*atan(1 - sqrt(2)*sqrt(tan(c + d*x))))/(sqrt(2)*d)) + ((a^2 - 2*a*b - b^2)*atan(1 + sqrt(2)*sqrt(tan(c + d*x))))/(sqrt(2)*d) + ((a^2 + 2*a*b - b^2)*log(1 - sqrt(2)*sqrt(tan(c + d*x)) + tan(c + d*x)))/(2*sqrt(2)*d) - ((a^2 + 2*a*b - b^2)*log(1 + sqrt(2)*sqrt(tan(c + d*x)) + tan(c + d*x)))/(2*sqrt(2)*d) - (2*a^2)/(5*d*tan(c + d*x)^(5/2)) - (4*a*b)/(3*d*tan(c + d*x)^(3/2)) + (2*(a^2 - b^2))/(d*sqrt(tan(c + d*x)))]


    @test_int [tan(c + d*x)^(5/2)*(a + b*tan(c + d*x))^3, x, 15, ((a + b)*(a^2 - 4*a*b + b^2)*atan(1 - sqrt(2)*sqrt(tan(c + d*x))))/(sqrt(2)*d) - ((a + b)*(a^2 - 4*a*b + b^2)*atan(1 + sqrt(2)*sqrt(tan(c + d*x))))/(sqrt(2)*d) - ((a - b)*(a^2 + 4*a*b + b^2)*log(1 - sqrt(2)*sqrt(tan(c + d*x)) + tan(c + d*x)))/(2*sqrt(2)*d) + ((a - b)*(a^2 + 4*a*b + b^2)*log(1 + sqrt(2)*sqrt(tan(c + d*x)) + tan(c + d*x)))/(2*sqrt(2)*d) - (2*b*(3*a^2 - b^2)*sqrt(tan(c + d*x)))/d + (2*a*(a^2 - 3*b^2)*tan(c + d*x)^(3/2))/(3*d) + (2*b*(3*a^2 - b^2)*tan(c + d*x)^(5/2))/(5*d) + (40*a*b^2*tan(c + d*x)^(7/2))/(63*d) + (2*b^2*tan(c + d*x)^(7/2)*(a + b*tan(c + d*x)))/(9*d)]
    @test_int [tan(c + d*x)^(3/2)*(a + b*tan(c + d*x))^3, x, 14, ((a - b)*(a^2 + 4*a*b + b^2)*atan(1 - sqrt(2)*sqrt(tan(c + d*x))))/(sqrt(2)*d) - ((a - b)*(a^2 + 4*a*b + b^2)*atan(1 + sqrt(2)*sqrt(tan(c + d*x))))/(sqrt(2)*d) + ((a + b)*(a^2 - 4*a*b + b^2)*log(1 - sqrt(2)*sqrt(tan(c + d*x)) + tan(c + d*x)))/(2*sqrt(2)*d) - ((a + b)*(a^2 - 4*a*b + b^2)*log(1 + sqrt(2)*sqrt(tan(c + d*x)) + tan(c + d*x)))/(2*sqrt(2)*d) + (2*a*(a^2 - 3*b^2)*sqrt(tan(c + d*x)))/d + (2*b*(3*a^2 - b^2)*tan(c + d*x)^(3/2))/(3*d) + (32*a*b^2*tan(c + d*x)^(5/2))/(35*d) + (2*b^2*tan(c + d*x)^(5/2)*(a + b*tan(c + d*x)))/(7*d)]
    @test_int [sqrt(tan(c + d*x))*(a + b*tan(c + d*x))^3, x, 13, -(((a + b)*(a^2 - 4*a*b + b^2)*atan(1 - sqrt(2)*sqrt(tan(c + d*x))))/(sqrt(2)*d)) + ((a + b)*(a^2 - 4*a*b + b^2)*atan(1 + sqrt(2)*sqrt(tan(c + d*x))))/(sqrt(2)*d) + ((a - b)*(a^2 + 4*a*b + b^2)*log(1 - sqrt(2)*sqrt(tan(c + d*x)) + tan(c + d*x)))/(2*sqrt(2)*d) - ((a - b)*(a^2 + 4*a*b + b^2)*log(1 + sqrt(2)*sqrt(tan(c + d*x)) + tan(c + d*x)))/(2*sqrt(2)*d) + (2*b*(3*a^2 - b^2)*sqrt(tan(c + d*x)))/d + (8*a*b^2*tan(c + d*x)^(3/2))/(5*d) + (2*b^2*tan(c + d*x)^(3/2)*(a + b*tan(c + d*x)))/(5*d)]
    @test_int [(a + b*tan(c + d*x))^3/sqrt(tan(c + d*x)), x, 12, -(((a - b)*(a^2 + 4*a*b + b^2)*atan(1 - sqrt(2)*sqrt(tan(c + d*x))))/(sqrt(2)*d)) + ((a - b)*(a^2 + 4*a*b + b^2)*atan(1 + sqrt(2)*sqrt(tan(c + d*x))))/(sqrt(2)*d) - ((a + b)*(a^2 - 4*a*b + b^2)*log(1 - sqrt(2)*sqrt(tan(c + d*x)) + tan(c + d*x)))/(2*sqrt(2)*d) + ((a + b)*(a^2 - 4*a*b + b^2)*log(1 + sqrt(2)*sqrt(tan(c + d*x)) + tan(c + d*x)))/(2*sqrt(2)*d) + (16*a*b^2*sqrt(tan(c + d*x)))/(3*d) + (2*b^2*sqrt(tan(c + d*x))*(a + b*tan(c + d*x)))/(3*d)]
    @test_int [(a + b*tan(c + d*x))^3/tan(c + d*x)^(3/2), x, 12, ((a + b)*(a^2 - 4*a*b + b^2)*atan(1 - sqrt(2)*sqrt(tan(c + d*x))))/(sqrt(2)*d) - ((a + b)*(a^2 - 4*a*b + b^2)*atan(1 + sqrt(2)*sqrt(tan(c + d*x))))/(sqrt(2)*d) - ((a - b)*(a^2 + 4*a*b + b^2)*log(1 - sqrt(2)*sqrt(tan(c + d*x)) + tan(c + d*x)))/(2*sqrt(2)*d) + ((a - b)*(a^2 + 4*a*b + b^2)*log(1 + sqrt(2)*sqrt(tan(c + d*x)) + tan(c + d*x)))/(2*sqrt(2)*d) + (2*b*(a^2 + b^2)*sqrt(tan(c + d*x)))/d - (2*a^2*(a + b*tan(c + d*x)))/(d*sqrt(tan(c + d*x)))]
    @test_int [(a + b*tan(c + d*x))^3/tan(c + d*x)^(5/2), x, 12, ((a - b)*(a^2 + 4*a*b + b^2)*atan(1 - sqrt(2)*sqrt(tan(c + d*x))))/(sqrt(2)*d) - ((a - b)*(a^2 + 4*a*b + b^2)*atan(1 + sqrt(2)*sqrt(tan(c + d*x))))/(sqrt(2)*d) + ((a + b)*(a^2 - 4*a*b + b^2)*log(1 - sqrt(2)*sqrt(tan(c + d*x)) + tan(c + d*x)))/(2*sqrt(2)*d) - ((a + b)*(a^2 - 4*a*b + b^2)*log(1 + sqrt(2)*sqrt(tan(c + d*x)) + tan(c + d*x)))/(2*sqrt(2)*d) - (16*a^2*b)/(3*d*sqrt(tan(c + d*x))) - (2*a^2*(a + b*tan(c + d*x)))/(3*d*tan(c + d*x)^(3/2))]
    @test_int [(a + b*tan(c + d*x))^3/tan(c + d*x)^(7/2), x, 13, -(((a + b)*(a^2 - 4*a*b + b^2)*atan(1 - sqrt(2)*sqrt(tan(c + d*x))))/(sqrt(2)*d)) + ((a + b)*(a^2 - 4*a*b + b^2)*atan(1 + sqrt(2)*sqrt(tan(c + d*x))))/(sqrt(2)*d) + ((a - b)*(a^2 + 4*a*b + b^2)*log(1 - sqrt(2)*sqrt(tan(c + d*x)) + tan(c + d*x)))/(2*sqrt(2)*d) - ((a - b)*(a^2 + 4*a*b + b^2)*log(1 + sqrt(2)*sqrt(tan(c + d*x)) + tan(c + d*x)))/(2*sqrt(2)*d) - (8*a^2*b)/(5*d*tan(c + d*x)^(3/2)) + (2*a*(a^2 - 3*b^2))/(d*sqrt(tan(c + d*x))) - (2*a^2*(a + b*tan(c + d*x)))/(5*d*tan(c + d*x)^(5/2))]
    @test_int [(a + b*tan(c + d*x))^3/tan(c + d*x)^(9/2), x, 14, -(((a - b)*(a^2 + 4*a*b + b^2)*atan(1 - sqrt(2)*sqrt(tan(c + d*x))))/(sqrt(2)*d)) + ((a - b)*(a^2 + 4*a*b + b^2)*atan(1 + sqrt(2)*sqrt(tan(c + d*x))))/(sqrt(2)*d) - ((a + b)*(a^2 - 4*a*b + b^2)*log(1 - sqrt(2)*sqrt(tan(c + d*x)) + tan(c + d*x)))/(2*sqrt(2)*d) + ((a + b)*(a^2 - 4*a*b + b^2)*log(1 + sqrt(2)*sqrt(tan(c + d*x)) + tan(c + d*x)))/(2*sqrt(2)*d) - (32*a^2*b)/(35*d*tan(c + d*x)^(5/2)) + (2*a*(a^2 - 3*b^2))/(3*d*tan(c + d*x)^(3/2)) + (2*b*(3*a^2 - b^2))/(d*sqrt(tan(c + d*x))) - (2*a^2*(a + b*tan(c + d*x)))/(7*d*tan(c + d*x)^(7/2))]
    @test_int [(a + b*tan(c + d*x))^3/tan(c + d*x)^(11/2), x, 15, ((a + b)*(a^2 - 4*a*b + b^2)*atan(1 - sqrt(2)*sqrt(tan(c + d*x))))/(sqrt(2)*d) - ((a + b)*(a^2 - 4*a*b + b^2)*atan(1 + sqrt(2)*sqrt(tan(c + d*x))))/(sqrt(2)*d) - ((a - b)*(a^2 + 4*a*b + b^2)*log(1 - sqrt(2)*sqrt(tan(c + d*x)) + tan(c + d*x)))/(2*sqrt(2)*d) + ((a - b)*(a^2 + 4*a*b + b^2)*log(1 + sqrt(2)*sqrt(tan(c + d*x)) + tan(c + d*x)))/(2*sqrt(2)*d) - (40*a^2*b)/(63*d*tan(c + d*x)^(7/2)) + (2*a*(a^2 - 3*b^2))/(5*d*tan(c + d*x)^(5/2)) + (2*b*(3*a^2 - b^2))/(3*d*tan(c + d*x)^(3/2)) - (2*a*(a^2 - 3*b^2))/(d*sqrt(tan(c + d*x))) - (2*a^2*(a + b*tan(c + d*x)))/(9*d*tan(c + d*x)^(9/2))]


    @test_int [(a + b*tan(c + d*x))/sqrt(tan(c + d*x)), x, 10, -(((a + b)*atan(1 - sqrt(2)*sqrt(tan(c + d*x))))/(sqrt(2)*d)) + ((a + b)*atan(1 + sqrt(2)*sqrt(tan(c + d*x))))/(sqrt(2)*d) - ((a - b)*log(1 - sqrt(2)*sqrt(tan(c + d*x)) + tan(c + d*x)))/(2*sqrt(2)*d) + ((a - b)*log(1 + sqrt(2)*sqrt(tan(c + d*x)) + tan(c + d*x)))/(2*sqrt(2)*d)]
    @test_int [(a + b*tan(c + d*x))/sqrt(-tan(c + d*x)), x, 10, ((a - b)*atan(1 - sqrt(2)*sqrt(-tan(c + d*x))))/(sqrt(2)*d) - ((a - b)*atan(1 + sqrt(2)*sqrt(-tan(c + d*x))))/(sqrt(2)*d) + ((a + b)*log(1 - sqrt(2)*sqrt(-tan(c + d*x)) - tan(c + d*x)))/(2*sqrt(2)*d) - ((a + b)*log(1 + sqrt(2)*sqrt(-tan(c + d*x)) - tan(c + d*x)))/(2*sqrt(2)*d)]
    @test_int [(a + b*tan(c + d*x))/sqrt(e*tan(c + d*x)), x, 10, -(((a + b)*atan(1 - (sqrt(2)*sqrt(e*tan(c + d*x)))/sqrt(e)))/(sqrt(2)*d*sqrt(e))) + ((a + b)*atan(1 + (sqrt(2)*sqrt(e*tan(c + d*x)))/sqrt(e)))/(sqrt(2)*d*sqrt(e)) - ((a - b)*log(sqrt(e) + sqrt(e)*tan(c + d*x) - sqrt(2)*sqrt(e*tan(c + d*x))))/(2*sqrt(2)*d*sqrt(e)) + ((a - b)*log(sqrt(e) + sqrt(e)*tan(c + d*x) + sqrt(2)*sqrt(e*tan(c + d*x))))/(2*sqrt(2)*d*sqrt(e))]
    @test_int [(a + b*tan(c + d*x))/sqrt(-e*tan(c + d*x)), x, 10, ((a - b)*atan(1 - (sqrt(2)*sqrt((-e)*tan(c + d*x)))/sqrt(e)))/(sqrt(2)*d*sqrt(e)) - ((a - b)*atan(1 + (sqrt(2)*sqrt((-e)*tan(c + d*x)))/sqrt(e)))/(sqrt(2)*d*sqrt(e)) + ((a + b)*log(sqrt(e) - sqrt(e)*tan(c + d*x) - sqrt(2)*sqrt((-e)*tan(c + d*x))))/(2*sqrt(2)*d*sqrt(e)) - ((a + b)*log(sqrt(e) - sqrt(e)*tan(c + d*x) + sqrt(2)*sqrt((-e)*tan(c + d*x))))/(2*sqrt(2)*d*sqrt(e))]


    #= ::Subsubsection::Closed:: =#
    #=m<0=#


    @test_int [tan(c + d*x)^(9/2)/(a + b*tan(c + d*x)), x, 17, -(((a + b)*atan(1 - sqrt(2)*sqrt(tan(c + d*x))))/(sqrt(2)*(a^2 + b^2)*d)) + ((a + b)*atan(1 + sqrt(2)*sqrt(tan(c + d*x))))/(sqrt(2)*(a^2 + b^2)*d) - (2*a^(9/2)*atan((sqrt(b)*sqrt(tan(c + d*x)))/sqrt(a)))/(b^(7/2)*(a^2 + b^2)*d) + ((a - b)*log(1 - sqrt(2)*sqrt(tan(c + d*x)) + tan(c + d*x)))/(2*sqrt(2)*(a^2 + b^2)*d) - ((a - b)*log(1 + sqrt(2)*sqrt(tan(c + d*x)) + tan(c + d*x)))/(2*sqrt(2)*(a^2 + b^2)*d) + (2*(a^2 - b^2)*sqrt(tan(c + d*x)))/(b^3*d) - (2*a*tan(c + d*x)^(3/2))/(3*b^2*d) + (2*tan(c + d*x)^(5/2))/(5*b*d)]
    @test_int [tan(c + d*x)^(7/2)/(a + b*tan(c + d*x)), x, 16, -(((a - b)*atan(1 - sqrt(2)*sqrt(tan(c + d*x))))/(sqrt(2)*(a^2 + b^2)*d)) + ((a - b)*atan(1 + sqrt(2)*sqrt(tan(c + d*x))))/(sqrt(2)*(a^2 + b^2)*d) + (2*a^(7/2)*atan((sqrt(b)*sqrt(tan(c + d*x)))/sqrt(a)))/(b^(5/2)*(a^2 + b^2)*d) - ((a + b)*log(1 - sqrt(2)*sqrt(tan(c + d*x)) + tan(c + d*x)))/(2*sqrt(2)*(a^2 + b^2)*d) + ((a + b)*log(1 + sqrt(2)*sqrt(tan(c + d*x)) + tan(c + d*x)))/(2*sqrt(2)*(a^2 + b^2)*d) - (2*a*sqrt(tan(c + d*x)))/(b^2*d) + (2*tan(c + d*x)^(3/2))/(3*b*d)]
    @test_int [tan(c + d*x)^(5/2)/(a + b*tan(c + d*x)), x, 15, ((a + b)*atan(1 - sqrt(2)*sqrt(tan(c + d*x))))/(sqrt(2)*(a^2 + b^2)*d) - ((a + b)*atan(1 + sqrt(2)*sqrt(tan(c + d*x))))/(sqrt(2)*(a^2 + b^2)*d) - (2*a^(5/2)*atan((sqrt(b)*sqrt(tan(c + d*x)))/sqrt(a)))/(b^(3/2)*(a^2 + b^2)*d) - ((a - b)*log(1 - sqrt(2)*sqrt(tan(c + d*x)) + tan(c + d*x)))/(2*sqrt(2)*(a^2 + b^2)*d) + ((a - b)*log(1 + sqrt(2)*sqrt(tan(c + d*x)) + tan(c + d*x)))/(2*sqrt(2)*(a^2 + b^2)*d) + (2*sqrt(tan(c + d*x)))/(b*d)]
    @test_int [tan(c + d*x)^(3/2)/(a + b*tan(c + d*x)), x, 14, ((a - b)*atan(1 - sqrt(2)*sqrt(tan(c + d*x))))/(sqrt(2)*(a^2 + b^2)*d) - ((a - b)*atan(1 + sqrt(2)*sqrt(tan(c + d*x))))/(sqrt(2)*(a^2 + b^2)*d) + (2*a^(3/2)*atan((sqrt(b)*sqrt(tan(c + d*x)))/sqrt(a)))/(sqrt(b)*(a^2 + b^2)*d) + ((a + b)*log(1 - sqrt(2)*sqrt(tan(c + d*x)) + tan(c + d*x)))/(2*sqrt(2)*(a^2 + b^2)*d) - ((a + b)*log(1 + sqrt(2)*sqrt(tan(c + d*x)) + tan(c + d*x)))/(2*sqrt(2)*(a^2 + b^2)*d)]
    @test_int [sqrt(tan(c + d*x))/(a + b*tan(c + d*x)), x, 14, -(((a + b)*atan(1 - sqrt(2)*sqrt(tan(c + d*x))))/(sqrt(2)*(a^2 + b^2)*d)) + ((a + b)*atan(1 + sqrt(2)*sqrt(tan(c + d*x))))/(sqrt(2)*(a^2 + b^2)*d) - (2*sqrt(a)*sqrt(b)*atan((sqrt(b)*sqrt(tan(c + d*x)))/sqrt(a)))/((a^2 + b^2)*d) + ((a - b)*log(1 - sqrt(2)*sqrt(tan(c + d*x)) + tan(c + d*x)))/(2*sqrt(2)*(a^2 + b^2)*d) - ((a - b)*log(1 + sqrt(2)*sqrt(tan(c + d*x)) + tan(c + d*x)))/(2*sqrt(2)*(a^2 + b^2)*d)]
    @test_int [1/(sqrt(tan(c + d*x))*(a + b*tan(c + d*x))), x, 14, -(((a - b)*atan(1 - sqrt(2)*sqrt(tan(c + d*x))))/(sqrt(2)*(a^2 + b^2)*d)) + ((a - b)*atan(1 + sqrt(2)*sqrt(tan(c + d*x))))/(sqrt(2)*(a^2 + b^2)*d) + (2*b^(3/2)*atan((sqrt(b)*sqrt(tan(c + d*x)))/sqrt(a)))/(sqrt(a)*(a^2 + b^2)*d) - ((a + b)*log(1 - sqrt(2)*sqrt(tan(c + d*x)) + tan(c + d*x)))/(2*sqrt(2)*(a^2 + b^2)*d) + ((a + b)*log(1 + sqrt(2)*sqrt(tan(c + d*x)) + tan(c + d*x)))/(2*sqrt(2)*(a^2 + b^2)*d)]
    @test_int [1/(tan(c + d*x)^(3/2)*(a + b*tan(c + d*x))), x, 15, ((a + b)*atan(1 - sqrt(2)*sqrt(tan(c + d*x))))/(sqrt(2)*(a^2 + b^2)*d) - ((a + b)*atan(1 + sqrt(2)*sqrt(tan(c + d*x))))/(sqrt(2)*(a^2 + b^2)*d) - (2*b^(5/2)*atan((sqrt(b)*sqrt(tan(c + d*x)))/sqrt(a)))/(a^(3/2)*(a^2 + b^2)*d) - ((a - b)*log(1 - sqrt(2)*sqrt(tan(c + d*x)) + tan(c + d*x)))/(2*sqrt(2)*(a^2 + b^2)*d) + ((a - b)*log(1 + sqrt(2)*sqrt(tan(c + d*x)) + tan(c + d*x)))/(2*sqrt(2)*(a^2 + b^2)*d) - 2/(a*d*sqrt(tan(c + d*x)))]
    @test_int [1/(tan(c + d*x)^(5/2)*(a + b*tan(c + d*x))), x, 16, ((a - b)*atan(1 - sqrt(2)*sqrt(tan(c + d*x))))/(sqrt(2)*(a^2 + b^2)*d) - ((a - b)*atan(1 + sqrt(2)*sqrt(tan(c + d*x))))/(sqrt(2)*(a^2 + b^2)*d) + (2*b^(7/2)*atan((sqrt(b)*sqrt(tan(c + d*x)))/sqrt(a)))/(a^(5/2)*(a^2 + b^2)*d) + ((a + b)*log(1 - sqrt(2)*sqrt(tan(c + d*x)) + tan(c + d*x)))/(2*sqrt(2)*(a^2 + b^2)*d) - ((a + b)*log(1 + sqrt(2)*sqrt(tan(c + d*x)) + tan(c + d*x)))/(2*sqrt(2)*(a^2 + b^2)*d) - 2/(3*a*d*tan(c + d*x)^(3/2)) + (2*b)/(a^2*d*sqrt(tan(c + d*x)))]
    @test_int [1/(tan(c + d*x)^(7/2)*(a + b*tan(c + d*x))), x, 17, -(((a + b)*atan(1 - sqrt(2)*sqrt(tan(c + d*x))))/(sqrt(2)*(a^2 + b^2)*d)) + ((a + b)*atan(1 + sqrt(2)*sqrt(tan(c + d*x))))/(sqrt(2)*(a^2 + b^2)*d) - (2*b^(9/2)*atan((sqrt(b)*sqrt(tan(c + d*x)))/sqrt(a)))/(a^(7/2)*(a^2 + b^2)*d) + ((a - b)*log(1 - sqrt(2)*sqrt(tan(c + d*x)) + tan(c + d*x)))/(2*sqrt(2)*(a^2 + b^2)*d) - ((a - b)*log(1 + sqrt(2)*sqrt(tan(c + d*x)) + tan(c + d*x)))/(2*sqrt(2)*(a^2 + b^2)*d) - 2/(5*a*d*tan(c + d*x)^(5/2)) + (2*b)/(3*a^2*d*tan(c + d*x)^(3/2)) + (2*(a^2 - b^2))/(a^3*d*sqrt(tan(c + d*x)))]


    @test_int [tan(c + d*x)^(9/2)/(a + b*tan(c + d*x))^2, x, 17, -(((a^2 + 2*a*b - b^2)*atan(1 - sqrt(2)*sqrt(tan(c + d*x))))/(sqrt(2)*(a^2 + b^2)^2*d)) + ((a^2 + 2*a*b - b^2)*atan(1 + sqrt(2)*sqrt(tan(c + d*x))))/(sqrt(2)*(a^2 + b^2)^2*d) + (a^(7/2)*(5*a^2 + 9*b^2)*atan((sqrt(b)*sqrt(tan(c + d*x)))/sqrt(a)))/(b^(7/2)*(a^2 + b^2)^2*d) + ((a^2 - 2*a*b - b^2)*log(1 - sqrt(2)*sqrt(tan(c + d*x)) + tan(c + d*x)))/(2*sqrt(2)*(a^2 + b^2)^2*d) - ((a^2 - 2*a*b - b^2)*log(1 + sqrt(2)*sqrt(tan(c + d*x)) + tan(c + d*x)))/(2*sqrt(2)*(a^2 + b^2)^2*d) - (a*(5*a^2 + 4*b^2)*sqrt(tan(c + d*x)))/(b^3*(a^2 + b^2)*d) + ((5*a^2 + 2*b^2)*tan(c + d*x)^(3/2))/(3*b^2*(a^2 + b^2)*d) - (a^2*tan(c + d*x)^(5/2))/(b*(a^2 + b^2)*d*(a + b*tan(c + d*x)))]
    @test_int [tan(c + d*x)^(7/2)/(a + b*tan(c + d*x))^2, x, 16, -(((a^2 - 2*a*b - b^2)*atan(1 - sqrt(2)*sqrt(tan(c + d*x))))/(sqrt(2)*(a^2 + b^2)^2*d)) + ((a^2 - 2*a*b - b^2)*atan(1 + sqrt(2)*sqrt(tan(c + d*x))))/(sqrt(2)*(a^2 + b^2)^2*d) - (a^(5/2)*(3*a^2 + 7*b^2)*atan((sqrt(b)*sqrt(tan(c + d*x)))/sqrt(a)))/(b^(5/2)*(a^2 + b^2)^2*d) - ((a^2 + 2*a*b - b^2)*log(1 - sqrt(2)*sqrt(tan(c + d*x)) + tan(c + d*x)))/(2*sqrt(2)*(a^2 + b^2)^2*d) + ((a^2 + 2*a*b - b^2)*log(1 + sqrt(2)*sqrt(tan(c + d*x)) + tan(c + d*x)))/(2*sqrt(2)*(a^2 + b^2)^2*d) + ((3*a^2 + 2*b^2)*sqrt(tan(c + d*x)))/(b^2*(a^2 + b^2)*d) - (a^2*tan(c + d*x)^(3/2))/(b*(a^2 + b^2)*d*(a + b*tan(c + d*x)))]
    @test_int [tan(c + d*x)^(5/2)/(a + b*tan(c + d*x))^2, x, 15, ((a^2 + 2*a*b - b^2)*atan(1 - sqrt(2)*sqrt(tan(c + d*x))))/(sqrt(2)*(a^2 + b^2)^2*d) - ((a^2 + 2*a*b - b^2)*atan(1 + sqrt(2)*sqrt(tan(c + d*x))))/(sqrt(2)*(a^2 + b^2)^2*d) + (a^(3/2)*(a^2 + 5*b^2)*atan((sqrt(b)*sqrt(tan(c + d*x)))/sqrt(a)))/(b^(3/2)*(a^2 + b^2)^2*d) - ((a^2 - 2*a*b - b^2)*log(1 - sqrt(2)*sqrt(tan(c + d*x)) + tan(c + d*x)))/(2*sqrt(2)*(a^2 + b^2)^2*d) + ((a^2 - 2*a*b - b^2)*log(1 + sqrt(2)*sqrt(tan(c + d*x)) + tan(c + d*x)))/(2*sqrt(2)*(a^2 + b^2)^2*d) - (a^2*sqrt(tan(c + d*x)))/(b*(a^2 + b^2)*d*(a + b*tan(c + d*x)))]
    @test_int [tan(c + d*x)^(3/2)/(a + b*tan(c + d*x))^2, x, 15, ((a^2 - 2*a*b - b^2)*atan(1 - sqrt(2)*sqrt(tan(c + d*x))))/(sqrt(2)*(a^2 + b^2)^2*d) - ((a^2 - 2*a*b - b^2)*atan(1 + sqrt(2)*sqrt(tan(c + d*x))))/(sqrt(2)*(a^2 + b^2)^2*d) + (sqrt(a)*(a^2 - 3*b^2)*atan((sqrt(b)*sqrt(tan(c + d*x)))/sqrt(a)))/(sqrt(b)*(a^2 + b^2)^2*d) + ((a^2 + 2*a*b - b^2)*log(1 - sqrt(2)*sqrt(tan(c + d*x)) + tan(c + d*x)))/(2*sqrt(2)*(a^2 + b^2)^2*d) - ((a^2 + 2*a*b - b^2)*log(1 + sqrt(2)*sqrt(tan(c + d*x)) + tan(c + d*x)))/(2*sqrt(2)*(a^2 + b^2)^2*d) + (a*sqrt(tan(c + d*x)))/((a^2 + b^2)*d*(a + b*tan(c + d*x)))]
    @test_int [sqrt(tan(c + d*x))/(a + b*tan(c + d*x))^2, x, 15, -(((a^2 + 2*a*b - b^2)*atan(1 - sqrt(2)*sqrt(tan(c + d*x))))/(sqrt(2)*(a^2 + b^2)^2*d)) + ((a^2 + 2*a*b - b^2)*atan(1 + sqrt(2)*sqrt(tan(c + d*x))))/(sqrt(2)*(a^2 + b^2)^2*d) - (sqrt(b)*(3*a^2 - b^2)*atan((sqrt(b)*sqrt(tan(c + d*x)))/sqrt(a)))/(sqrt(a)*(a^2 + b^2)^2*d) + ((a^2 - 2*a*b - b^2)*log(1 - sqrt(2)*sqrt(tan(c + d*x)) + tan(c + d*x)))/(2*sqrt(2)*(a^2 + b^2)^2*d) - ((a^2 - 2*a*b - b^2)*log(1 + sqrt(2)*sqrt(tan(c + d*x)) + tan(c + d*x)))/(2*sqrt(2)*(a^2 + b^2)^2*d) - (b*sqrt(tan(c + d*x)))/((a^2 + b^2)*d*(a + b*tan(c + d*x)))]
    @test_int [1/(sqrt(tan(c + d*x))*(a + b*tan(c + d*x))^2), x, 15, -(((a^2 - 2*a*b - b^2)*atan(1 - sqrt(2)*sqrt(tan(c + d*x))))/(sqrt(2)*(a^2 + b^2)^2*d)) + ((a^2 - 2*a*b - b^2)*atan(1 + sqrt(2)*sqrt(tan(c + d*x))))/(sqrt(2)*(a^2 + b^2)^2*d) + (b^(3/2)*(5*a^2 + b^2)*atan((sqrt(b)*sqrt(tan(c + d*x)))/sqrt(a)))/(a^(3/2)*(a^2 + b^2)^2*d) - ((a^2 + 2*a*b - b^2)*log(1 - sqrt(2)*sqrt(tan(c + d*x)) + tan(c + d*x)))/(2*sqrt(2)*(a^2 + b^2)^2*d) + ((a^2 + 2*a*b - b^2)*log(1 + sqrt(2)*sqrt(tan(c + d*x)) + tan(c + d*x)))/(2*sqrt(2)*(a^2 + b^2)^2*d) + (b^2*sqrt(tan(c + d*x)))/(a*(a^2 + b^2)*d*(a + b*tan(c + d*x)))]
    @test_int [1/(tan(c + d*x)^(3/2)*(a + b*tan(c + d*x))^2), x, 16, ((a^2 + 2*a*b - b^2)*atan(1 - sqrt(2)*sqrt(tan(c + d*x))))/(sqrt(2)*(a^2 + b^2)^2*d) - ((a^2 + 2*a*b - b^2)*atan(1 + sqrt(2)*sqrt(tan(c + d*x))))/(sqrt(2)*(a^2 + b^2)^2*d) - (b^(5/2)*(7*a^2 + 3*b^2)*atan((sqrt(b)*sqrt(tan(c + d*x)))/sqrt(a)))/(a^(5/2)*(a^2 + b^2)^2*d) - ((a^2 - 2*a*b - b^2)*log(1 - sqrt(2)*sqrt(tan(c + d*x)) + tan(c + d*x)))/(2*sqrt(2)*(a^2 + b^2)^2*d) + ((a^2 - 2*a*b - b^2)*log(1 + sqrt(2)*sqrt(tan(c + d*x)) + tan(c + d*x)))/(2*sqrt(2)*(a^2 + b^2)^2*d) - (2*a^2 + 3*b^2)/(a^2*(a^2 + b^2)*d*sqrt(tan(c + d*x))) + b^2/(a*(a^2 + b^2)*d*sqrt(tan(c + d*x))*(a + b*tan(c + d*x)))]
    @test_int [1/(tan(c + d*x)^(5/2)*(a + b*tan(c + d*x))^2), x, 17, ((a^2 - 2*a*b - b^2)*atan(1 - sqrt(2)*sqrt(tan(c + d*x))))/(sqrt(2)*(a^2 + b^2)^2*d) - ((a^2 - 2*a*b - b^2)*atan(1 + sqrt(2)*sqrt(tan(c + d*x))))/(sqrt(2)*(a^2 + b^2)^2*d) + (b^(7/2)*(9*a^2 + 5*b^2)*atan((sqrt(b)*sqrt(tan(c + d*x)))/sqrt(a)))/(a^(7/2)*(a^2 + b^2)^2*d) + ((a^2 + 2*a*b - b^2)*log(1 - sqrt(2)*sqrt(tan(c + d*x)) + tan(c + d*x)))/(2*sqrt(2)*(a^2 + b^2)^2*d) - ((a^2 + 2*a*b - b^2)*log(1 + sqrt(2)*sqrt(tan(c + d*x)) + tan(c + d*x)))/(2*sqrt(2)*(a^2 + b^2)^2*d) - (2*a^2 + 5*b^2)/(3*a^2*(a^2 + b^2)*d*tan(c + d*x)^(3/2)) + (b*(4*a^2 + 5*b^2))/(a^3*(a^2 + b^2)*d*sqrt(tan(c + d*x))) + b^2/(a*(a^2 + b^2)*d*tan(c + d*x)^(3/2)*(a + b*tan(c + d*x)))]


    @test_int [tan(c + d*x)^(11/2)/(a + b*tan(c + d*x))^3, x, 18, ((a + b)*(a^2 - 4*a*b + b^2)*atan(1 - sqrt(2)*sqrt(tan(c + d*x))))/(sqrt(2)*(a^2 + b^2)^3*d) - ((a + b)*(a^2 - 4*a*b + b^2)*atan(1 + sqrt(2)*sqrt(tan(c + d*x))))/(sqrt(2)*(a^2 + b^2)^3*d) + (a^(7/2)*(35*a^4 + 102*a^2*b^2 + 99*b^4)*atan((sqrt(b)*sqrt(tan(c + d*x)))/sqrt(a)))/(4*b^(9/2)*(a^2 + b^2)^3*d) + ((a - b)*(a^2 + 4*a*b + b^2)*log(1 - sqrt(2)*sqrt(tan(c + d*x)) + tan(c + d*x)))/(2*sqrt(2)*(a^2 + b^2)^3*d) - ((a - b)*(a^2 + 4*a*b + b^2)*log(1 + sqrt(2)*sqrt(tan(c + d*x)) + tan(c + d*x)))/(2*sqrt(2)*(a^2 + b^2)^3*d) - (a*(35*a^4 + 67*a^2*b^2 + 24*b^4)*sqrt(tan(c + d*x)))/(4*b^4*(a^2 + b^2)^2*d) + ((35*a^4 + 67*a^2*b^2 + 8*b^4)*tan(c + d*x)^(3/2))/(12*b^3*(a^2 + b^2)^2*d) - (a^2*tan(c + d*x)^(7/2))/(2*b*(a^2 + b^2)*d*(a + b*tan(c + d*x))^2) - (a^2*(7*a^2 + 15*b^2)*tan(c + d*x)^(5/2))/(4*b^2*(a^2 + b^2)^2*d*(a + b*tan(c + d*x)))]
    @test_int [tan(c + d*x)^(9/2)/(a + b*tan(c + d*x))^3, x, 17, -(((a - b)*(a^2 + 4*a*b + b^2)*atan(1 - sqrt(2)*sqrt(tan(c + d*x))))/(sqrt(2)*(a^2 + b^2)^3*d)) + ((a - b)*(a^2 + 4*a*b + b^2)*atan(1 + sqrt(2)*sqrt(tan(c + d*x))))/(sqrt(2)*(a^2 + b^2)^3*d) - (a^(5/2)*(15*a^4 + 46*a^2*b^2 + 63*b^4)*atan((sqrt(b)*sqrt(tan(c + d*x)))/sqrt(a)))/(4*b^(7/2)*(a^2 + b^2)^3*d) + ((a + b)*(a^2 - 4*a*b + b^2)*log(1 - sqrt(2)*sqrt(tan(c + d*x)) + tan(c + d*x)))/(2*sqrt(2)*(a^2 + b^2)^3*d) - ((a + b)*(a^2 - 4*a*b + b^2)*log(1 + sqrt(2)*sqrt(tan(c + d*x)) + tan(c + d*x)))/(2*sqrt(2)*(a^2 + b^2)^3*d) + ((15*a^4 + 31*a^2*b^2 + 8*b^4)*sqrt(tan(c + d*x)))/(4*b^3*(a^2 + b^2)^2*d) - (a^2*tan(c + d*x)^(5/2))/(2*b*(a^2 + b^2)*d*(a + b*tan(c + d*x))^2) - (a^2*(5*a^2 + 13*b^2)*tan(c + d*x)^(3/2))/(4*b^2*(a^2 + b^2)^2*d*(a + b*tan(c + d*x)))]
    @test_int [tan(c + d*x)^(7/2)/(a + b*tan(c + d*x))^3, x, 16, -(((a + b)*(a^2 - 4*a*b + b^2)*atan(1 - sqrt(2)*sqrt(tan(c + d*x))))/(sqrt(2)*(a^2 + b^2)^3*d)) + ((a + b)*(a^2 - 4*a*b + b^2)*atan(1 + sqrt(2)*sqrt(tan(c + d*x))))/(sqrt(2)*(a^2 + b^2)^3*d) + (a^(3/2)*(3*a^4 + 6*a^2*b^2 + 35*b^4)*atan((sqrt(b)*sqrt(tan(c + d*x)))/sqrt(a)))/(4*b^(5/2)*(a^2 + b^2)^3*d) - ((a - b)*(a^2 + 4*a*b + b^2)*log(1 - sqrt(2)*sqrt(tan(c + d*x)) + tan(c + d*x)))/(2*sqrt(2)*(a^2 + b^2)^3*d) + ((a - b)*(a^2 + 4*a*b + b^2)*log(1 + sqrt(2)*sqrt(tan(c + d*x)) + tan(c + d*x)))/(2*sqrt(2)*(a^2 + b^2)^3*d) - (a^2*tan(c + d*x)^(3/2))/(2*b*(a^2 + b^2)*d*(a + b*tan(c + d*x))^2) - (a^2*(3*a^2 + 11*b^2)*sqrt(tan(c + d*x)))/(4*b^2*(a^2 + b^2)^2*d*(a + b*tan(c + d*x)))]
    @test_int [tan(c + d*x)^(5/2)/(a + b*tan(c + d*x))^3, x, 16, ((a - b)*(a^2 + 4*a*b + b^2)*atan(1 - sqrt(2)*sqrt(tan(c + d*x))))/(sqrt(2)*(a^2 + b^2)^3*d) - ((a - b)*(a^2 + 4*a*b + b^2)*atan(1 + sqrt(2)*sqrt(tan(c + d*x))))/(sqrt(2)*(a^2 + b^2)^3*d) + (sqrt(a)*(a^4 + 18*a^2*b^2 - 15*b^4)*atan((sqrt(b)*sqrt(tan(c + d*x)))/sqrt(a)))/(4*b^(3/2)*(a^2 + b^2)^3*d) - ((a + b)*(a^2 - 4*a*b + b^2)*log(1 - sqrt(2)*sqrt(tan(c + d*x)) + tan(c + d*x)))/(2*sqrt(2)*(a^2 + b^2)^3*d) + ((a + b)*(a^2 - 4*a*b + b^2)*log(1 + sqrt(2)*sqrt(tan(c + d*x)) + tan(c + d*x)))/(2*sqrt(2)*(a^2 + b^2)^3*d) - (a^2*sqrt(tan(c + d*x)))/(2*b*(a^2 + b^2)*d*(a + b*tan(c + d*x))^2) + (a*(a^2 + 9*b^2)*sqrt(tan(c + d*x)))/(4*b*(a^2 + b^2)^2*d*(a + b*tan(c + d*x)))]
    @test_int [tan(c + d*x)^(3/2)/(a + b*tan(c + d*x))^3, x, 16, ((a + b)*(a^2 - 4*a*b + b^2)*atan(1 - sqrt(2)*sqrt(tan(c + d*x))))/(sqrt(2)*(a^2 + b^2)^3*d) - ((a + b)*(a^2 - 4*a*b + b^2)*atan(1 + sqrt(2)*sqrt(tan(c + d*x))))/(sqrt(2)*(a^2 + b^2)^3*d) + ((3*a^4 - 26*a^2*b^2 + 3*b^4)*atan((sqrt(b)*sqrt(tan(c + d*x)))/sqrt(a)))/(4*sqrt(a)*sqrt(b)*(a^2 + b^2)^3*d) + ((a - b)*(a^2 + 4*a*b + b^2)*log(1 - sqrt(2)*sqrt(tan(c + d*x)) + tan(c + d*x)))/(2*sqrt(2)*(a^2 + b^2)^3*d) - ((a - b)*(a^2 + 4*a*b + b^2)*log(1 + sqrt(2)*sqrt(tan(c + d*x)) + tan(c + d*x)))/(2*sqrt(2)*(a^2 + b^2)^3*d) + (a*sqrt(tan(c + d*x)))/(2*(a^2 + b^2)*d*(a + b*tan(c + d*x))^2) + ((3*a^2 - 5*b^2)*sqrt(tan(c + d*x)))/(4*(a^2 + b^2)^2*d*(a + b*tan(c + d*x)))]
    @test_int [sqrt(tan(c + d*x))/(a + b*tan(c + d*x))^3, x, 16, -(((a - b)*(a^2 + 4*a*b + b^2)*atan(1 - sqrt(2)*sqrt(tan(c + d*x))))/(sqrt(2)*(a^2 + b^2)^3*d)) + ((a - b)*(a^2 + 4*a*b + b^2)*atan(1 + sqrt(2)*sqrt(tan(c + d*x))))/(sqrt(2)*(a^2 + b^2)^3*d) - (sqrt(b)*(15*a^4 - 18*a^2*b^2 - b^4)*atan((sqrt(b)*sqrt(tan(c + d*x)))/sqrt(a)))/(4*a^(3/2)*(a^2 + b^2)^3*d) + ((a + b)*(a^2 - 4*a*b + b^2)*log(1 - sqrt(2)*sqrt(tan(c + d*x)) + tan(c + d*x)))/(2*sqrt(2)*(a^2 + b^2)^3*d) - ((a + b)*(a^2 - 4*a*b + b^2)*log(1 + sqrt(2)*sqrt(tan(c + d*x)) + tan(c + d*x)))/(2*sqrt(2)*(a^2 + b^2)^3*d) - (b*sqrt(tan(c + d*x)))/(2*(a^2 + b^2)*d*(a + b*tan(c + d*x))^2) - (b*(7*a^2 - b^2)*sqrt(tan(c + d*x)))/(4*a*(a^2 + b^2)^2*d*(a + b*tan(c + d*x)))]
    @test_int [1/(sqrt(tan(c + d*x))*(a + b*tan(c + d*x))^3), x, 16, -(((a + b)*(a^2 - 4*a*b + b^2)*atan(1 - sqrt(2)*sqrt(tan(c + d*x))))/(sqrt(2)*(a^2 + b^2)^3*d)) + ((a + b)*(a^2 - 4*a*b + b^2)*atan(1 + sqrt(2)*sqrt(tan(c + d*x))))/(sqrt(2)*(a^2 + b^2)^3*d) + (b^(3/2)*(35*a^4 + 6*a^2*b^2 + 3*b^4)*atan((sqrt(b)*sqrt(tan(c + d*x)))/sqrt(a)))/(4*a^(5/2)*(a^2 + b^2)^3*d) - ((a - b)*(a^2 + 4*a*b + b^2)*log(1 - sqrt(2)*sqrt(tan(c + d*x)) + tan(c + d*x)))/(2*sqrt(2)*(a^2 + b^2)^3*d) + ((a - b)*(a^2 + 4*a*b + b^2)*log(1 + sqrt(2)*sqrt(tan(c + d*x)) + tan(c + d*x)))/(2*sqrt(2)*(a^2 + b^2)^3*d) + (b^2*sqrt(tan(c + d*x)))/(2*a*(a^2 + b^2)*d*(a + b*tan(c + d*x))^2) + (b^2*(11*a^2 + 3*b^2)*sqrt(tan(c + d*x)))/(4*a^2*(a^2 + b^2)^2*d*(a + b*tan(c + d*x)))]
    @test_int [1/(tan(c + d*x)^(3/2)*(a + b*tan(c + d*x))^3), x, 17, ((a - b)*(a^2 + 4*a*b + b^2)*atan(1 - sqrt(2)*sqrt(tan(c + d*x))))/(sqrt(2)*(a^2 + b^2)^3*d) - ((a - b)*(a^2 + 4*a*b + b^2)*atan(1 + sqrt(2)*sqrt(tan(c + d*x))))/(sqrt(2)*(a^2 + b^2)^3*d) - (b^(5/2)*(63*a^4 + 46*a^2*b^2 + 15*b^4)*atan((sqrt(b)*sqrt(tan(c + d*x)))/sqrt(a)))/(4*a^(7/2)*(a^2 + b^2)^3*d) - ((a + b)*(a^2 - 4*a*b + b^2)*log(1 - sqrt(2)*sqrt(tan(c + d*x)) + tan(c + d*x)))/(2*sqrt(2)*(a^2 + b^2)^3*d) + ((a + b)*(a^2 - 4*a*b + b^2)*log(1 + sqrt(2)*sqrt(tan(c + d*x)) + tan(c + d*x)))/(2*sqrt(2)*(a^2 + b^2)^3*d) - (8*a^4 + 31*a^2*b^2 + 15*b^4)/(4*a^3*(a^2 + b^2)^2*d*sqrt(tan(c + d*x))) + b^2/(2*a*(a^2 + b^2)*d*sqrt(tan(c + d*x))*(a + b*tan(c + d*x))^2) + (b^2*(13*a^2 + 5*b^2))/(4*a^2*(a^2 + b^2)^2*d*sqrt(tan(c + d*x))*(a + b*tan(c + d*x)))]
    @test_int [1/(tan(c + d*x)^(5/2)*(a + b*tan(c + d*x))^3), x, 18, ((a + b)*(a^2 - 4*a*b + b^2)*atan(1 - sqrt(2)*sqrt(tan(c + d*x))))/(sqrt(2)*(a^2 + b^2)^3*d) - ((a + b)*(a^2 - 4*a*b + b^2)*atan(1 + sqrt(2)*sqrt(tan(c + d*x))))/(sqrt(2)*(a^2 + b^2)^3*d) + (b^(7/2)*(99*a^4 + 102*a^2*b^2 + 35*b^4)*atan((sqrt(b)*sqrt(tan(c + d*x)))/sqrt(a)))/(4*a^(9/2)*(a^2 + b^2)^3*d) + ((a - b)*(a^2 + 4*a*b + b^2)*log(1 - sqrt(2)*sqrt(tan(c + d*x)) + tan(c + d*x)))/(2*sqrt(2)*(a^2 + b^2)^3*d) - ((a - b)*(a^2 + 4*a*b + b^2)*log(1 + sqrt(2)*sqrt(tan(c + d*x)) + tan(c + d*x)))/(2*sqrt(2)*(a^2 + b^2)^3*d) - (8*a^4 + 67*a^2*b^2 + 35*b^4)/(12*a^3*(a^2 + b^2)^2*d*tan(c + d*x)^(3/2)) + (b*(24*a^4 + 67*a^2*b^2 + 35*b^4))/(4*a^4*(a^2 + b^2)^2*d*sqrt(tan(c + d*x))) + b^2/(2*a*(a^2 + b^2)*d*tan(c + d*x)^(3/2)*(a + b*tan(c + d*x))^2) + (b^2*(15*a^2 + 7*b^2))/(4*a^2*(a^2 + b^2)^2*d*tan(c + d*x)^(3/2)*(a + b*tan(c + d*x)))]


    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*(a+b*tan(e+f*x))^(m/2)*(d*tan(e+f*x))^(n/2)=#


    #= ::Subsubsection::Closed:: =#
    #=m>0=#


    @test_int [tan(c + d*x)^(5/2)*(a + b*tan(c + d*x))^(1/2), x, 14, -((sqrt(im*a - b)*atan((sqrt(im*a - b)*sqrt(tan(c + d*x)))/sqrt(a + b*tan(c + d*x))))/d) - ((a^2 + 8*b^2)*atanh((sqrt(b)*sqrt(tan(c + d*x)))/sqrt(a + b*tan(c + d*x))))/(4*b^(3/2)*d) + (sqrt(im*a + b)*atanh((sqrt(im*a + b)*sqrt(tan(c + d*x)))/sqrt(a + b*tan(c + d*x))))/d - (a*sqrt(tan(c + d*x))*sqrt(a + b*tan(c + d*x)))/(4*b*d) + (sqrt(tan(c + d*x))*(a + b*tan(c + d*x))^(3/2))/(2*b*d)]
    @test_int [tan(c + d*x)^(3/2)*(a + b*tan(c + d*x))^(1/2), x, 13, (im*sqrt(im*a - b)*atan((sqrt(im*a - b)*sqrt(tan(c + d*x)))/sqrt(a + b*tan(c + d*x))))/d + (a*atanh((sqrt(b)*sqrt(tan(c + d*x)))/sqrt(a + b*tan(c + d*x))))/(sqrt(b)*d) + (im*sqrt(im*a + b)*atanh((sqrt(im*a + b)*sqrt(tan(c + d*x)))/sqrt(a + b*tan(c + d*x))))/d + (sqrt(tan(c + d*x))*sqrt(a + b*tan(c + d*x)))/d]
    @test_int [tan(c + d*x)^(1/2)*(a + b*tan(c + d*x))^(1/2), x, 11, (sqrt(im*a - b)*atan((sqrt(im*a - b)*sqrt(tan(c + d*x)))/sqrt(a + b*tan(c + d*x))))/d + (2*sqrt(b)*atanh((sqrt(b)*sqrt(tan(c + d*x)))/sqrt(a + b*tan(c + d*x))))/d - (sqrt(im*a + b)*atanh((sqrt(im*a + b)*sqrt(tan(c + d*x)))/sqrt(a + b*tan(c + d*x))))/d]
    @test_int [tan(c + d*x)^(-1/2)*(a + b*tan(c + d*x))^(1/2), x, 7, -((im*sqrt(im*a - b)*atan((sqrt(im*a - b)*sqrt(tan(c + d*x)))/sqrt(a + b*tan(c + d*x))))/d) - (im*sqrt(im*a + b)*atanh((sqrt(im*a + b)*sqrt(tan(c + d*x)))/sqrt(a + b*tan(c + d*x))))/d]
    @test_int [tan(c + d*x)^(-3/2)*(a + b*tan(c + d*x))^(1/2), x, 8, -((sqrt(im*a - b)*atan((sqrt(im*a - b)*sqrt(tan(c + d*x)))/sqrt(a + b*tan(c + d*x))))/d) + (sqrt(im*a + b)*atanh((sqrt(im*a + b)*sqrt(tan(c + d*x)))/sqrt(a + b*tan(c + d*x))))/d - (2*sqrt(a + b*tan(c + d*x)))/(d*sqrt(tan(c + d*x)))]
    @test_int [tan(c + d*x)^(-5/2)*(a + b*tan(c + d*x))^(1/2), x, 10, (im*sqrt(im*a - b)*atan((sqrt(im*a - b)*sqrt(tan(c + d*x)))/sqrt(a + b*tan(c + d*x))))/d + (im*sqrt(im*a + b)*atanh((sqrt(im*a + b)*sqrt(tan(c + d*x)))/sqrt(a + b*tan(c + d*x))))/d - (2*sqrt(a + b*tan(c + d*x)))/(3*d*tan(c + d*x)^(3/2)) - (2*b*sqrt(a + b*tan(c + d*x)))/(3*a*d*sqrt(tan(c + d*x)))]
    @test_int [tan(c + d*x)^(-7/2)*(a + b*tan(c + d*x))^(1/2), x, 10, (sqrt(im*a - b)*atan((sqrt(im*a - b)*sqrt(tan(c + d*x)))/sqrt(a + b*tan(c + d*x))))/d - (sqrt(im*a + b)*atanh((sqrt(im*a + b)*sqrt(tan(c + d*x)))/sqrt(a + b*tan(c + d*x))))/d - (2*sqrt(a + b*tan(c + d*x)))/(5*d*tan(c + d*x)^(5/2)) - (2*b*sqrt(a + b*tan(c + d*x)))/(15*a*d*tan(c + d*x)^(3/2)) + (2*(15*a^2 + 2*b^2)*sqrt(a + b*tan(c + d*x)))/(15*a^2*d*sqrt(tan(c + d*x)))]


    @test_int [tan(c + d*x)^(5/2)*(a + b*tan(c + d*x))^(3/2), x, 15, (im*(im*a - b)^(3/2)*atan((sqrt(im*a - b)*sqrt(tan(c + d*x)))/sqrt(a + b*tan(c + d*x))))/d - (a*(a^2 + 24*b^2)*atanh((sqrt(b)*sqrt(tan(c + d*x)))/sqrt(a + b*tan(c + d*x))))/(8*b^(3/2)*d) - (im*(im*a + b)^(3/2)*atanh((sqrt(im*a + b)*sqrt(tan(c + d*x)))/sqrt(a + b*tan(c + d*x))))/d - ((a^2 + 8*b^2)*sqrt(tan(c + d*x))*sqrt(a + b*tan(c + d*x)))/(8*b*d) - (a*sqrt(tan(c + d*x))*(a + b*tan(c + d*x))^(3/2))/(12*b*d) + (sqrt(tan(c + d*x))*(a + b*tan(c + d*x))^(5/2))/(3*b*d)]
    @test_int [tan(c + d*x)^(3/2)*(a + b*tan(c + d*x))^(3/2), x, 14, ((im*a - b)^(3/2)*atan((sqrt(im*a - b)*sqrt(tan(c + d*x)))/sqrt(a + b*tan(c + d*x))))/d + ((3*a^2 - 8*b^2)*atanh((sqrt(b)*sqrt(tan(c + d*x)))/sqrt(a + b*tan(c + d*x))))/(4*sqrt(b)*d) + ((im*a + b)^(3/2)*atanh((sqrt(im*a + b)*sqrt(tan(c + d*x)))/sqrt(a + b*tan(c + d*x))))/d + (3*a*sqrt(tan(c + d*x))*sqrt(a + b*tan(c + d*x)))/(4*d) + (sqrt(tan(c + d*x))*(a + b*tan(c + d*x))^(3/2))/(2*d)]
    @test_int [tan(c + d*x)^(1/2)*(a + b*tan(c + d*x))^(3/2), x, 13, -((im*(im*a - b)^(3/2)*atan((sqrt(im*a - b)*sqrt(tan(c + d*x)))/sqrt(a + b*tan(c + d*x))))/d) + (3*a*sqrt(b)*atanh((sqrt(b)*sqrt(tan(c + d*x)))/sqrt(a + b*tan(c + d*x))))/d + (im*(im*a + b)^(3/2)*atanh((sqrt(im*a + b)*sqrt(tan(c + d*x)))/sqrt(a + b*tan(c + d*x))))/d + (b*sqrt(tan(c + d*x))*sqrt(a + b*tan(c + d*x)))/d]
    @test_int [tan(c + d*x)^(-1/2)*(a + b*tan(c + d*x))^(3/2), x, 12, -(((im*a - b)^(3/2)*atan((sqrt(im*a - b)*sqrt(tan(c + d*x)))/sqrt(a + b*tan(c + d*x))))/d) + (2*b^(3/2)*atanh((sqrt(b)*sqrt(tan(c + d*x)))/sqrt(a + b*tan(c + d*x))))/d - ((im*a + b)^(3/2)*atanh((sqrt(im*a + b)*sqrt(tan(c + d*x)))/sqrt(a + b*tan(c + d*x))))/d]
    @test_int [tan(c + d*x)^(-3/2)*(a + b*tan(c + d*x))^(3/2), x, 8, (im*(im*a - b)^(3/2)*atan((sqrt(im*a - b)*sqrt(tan(c + d*x)))/sqrt(a + b*tan(c + d*x))))/d - (im*(im*a + b)^(3/2)*atanh((sqrt(im*a + b)*sqrt(tan(c + d*x)))/sqrt(a + b*tan(c + d*x))))/d - (2*a*sqrt(a + b*tan(c + d*x)))/(d*sqrt(tan(c + d*x)))]
    @test_int [tan(c + d*x)^(-5/2)*(a + b*tan(c + d*x))^(3/2), x, 9, ((im*a - b)^(3/2)*atan((sqrt(im*a - b)*sqrt(tan(c + d*x)))/sqrt(a + b*tan(c + d*x))))/d + ((im*a + b)^(3/2)*atanh((sqrt(im*a + b)*sqrt(tan(c + d*x)))/sqrt(a + b*tan(c + d*x))))/d - (2*a*sqrt(a + b*tan(c + d*x)))/(3*d*tan(c + d*x)^(3/2)) - (8*b*sqrt(a + b*tan(c + d*x)))/(3*d*sqrt(tan(c + d*x)))]
    @test_int [tan(c + d*x)^(-7/2)*(a + b*tan(c + d*x))^(3/2), x, 10, -((im*(im*a - b)^(3/2)*atan((sqrt(im*a - b)*sqrt(tan(c + d*x)))/sqrt(a + b*tan(c + d*x))))/d) + (im*(im*a + b)^(3/2)*atanh((sqrt(im*a + b)*sqrt(tan(c + d*x)))/sqrt(a + b*tan(c + d*x))))/d - (2*a*sqrt(a + b*tan(c + d*x)))/(5*d*tan(c + d*x)^(5/2)) - (4*b*sqrt(a + b*tan(c + d*x)))/(5*d*tan(c + d*x)^(3/2)) + (2*(5*a^2 - b^2)*sqrt(a + b*tan(c + d*x)))/(5*a*d*sqrt(tan(c + d*x)))]
    @test_int [tan(c + d*x)^(-9/2)*(a + b*tan(c + d*x))^(3/2), x, 11, -(((im*a - b)^(3/2)*atan((sqrt(im*a - b)*sqrt(tan(c + d*x)))/sqrt(a + b*tan(c + d*x))))/d) - ((im*a + b)^(3/2)*atanh((sqrt(im*a + b)*sqrt(tan(c + d*x)))/sqrt(a + b*tan(c + d*x))))/d - (2*a*sqrt(a + b*tan(c + d*x)))/(7*d*tan(c + d*x)^(7/2)) - (16*b*sqrt(a + b*tan(c + d*x)))/(35*d*tan(c + d*x)^(5/2)) + (2*(35*a^2 - 3*b^2)*sqrt(a + b*tan(c + d*x)))/(105*a*d*tan(c + d*x)^(3/2)) + (4*b*(70*a^2 + 3*b^2)*sqrt(a + b*tan(c + d*x)))/(105*a^2*d*sqrt(tan(c + d*x)))]


    @test_int [tan(c + d*x)^(5/2)*(a + b*tan(c + d*x))^(5/2), x, 16, ((im*a - b)^(5/2)*atan((sqrt(im*a - b)*sqrt(tan(c + d*x)))/sqrt(a + b*tan(c + d*x))))/d - ((5*a^4 + 240*a^2*b^2 - 128*b^4)*atanh((sqrt(b)*sqrt(tan(c + d*x)))/sqrt(a + b*tan(c + d*x))))/(64*b^(3/2)*d) - ((im*a + b)^(5/2)*atanh((sqrt(im*a + b)*sqrt(tan(c + d*x)))/sqrt(a + b*tan(c + d*x))))/d - (a*(5*a^2 + 112*b^2)*sqrt(tan(c + d*x))*sqrt(a + b*tan(c + d*x)))/(64*b*d) - ((5*a^2 + 48*b^2)*sqrt(tan(c + d*x))*(a + b*tan(c + d*x))^(3/2))/(96*b*d) - (a*sqrt(tan(c + d*x))*(a + b*tan(c + d*x))^(5/2))/(24*b*d) + (sqrt(tan(c + d*x))*(a + b*tan(c + d*x))^(7/2))/(4*b*d)]
    @test_int [tan(c + d*x)^(3/2)*(a + b*tan(c + d*x))^(5/2), x, 15, -((im*(im*a - b)^(5/2)*atan((sqrt(im*a - b)*sqrt(tan(c + d*x)))/sqrt(a + b*tan(c + d*x))))/d) + (5*a*(a^2 - 8*b^2)*atanh((sqrt(b)*sqrt(tan(c + d*x)))/sqrt(a + b*tan(c + d*x))))/(8*sqrt(b)*d) - (im*(im*a + b)^(5/2)*atanh((sqrt(im*a + b)*sqrt(tan(c + d*x)))/sqrt(a + b*tan(c + d*x))))/d + ((11*a^2 - 8*b^2)*sqrt(tan(c + d*x))*sqrt(a + b*tan(c + d*x)))/(8*d) + (13*a*b*tan(c + d*x)^(3/2)*sqrt(a + b*tan(c + d*x)))/(12*d) + (b^2*tan(c + d*x)^(5/2)*sqrt(a + b*tan(c + d*x)))/(3*d)]
    @test_int [tan(c + d*x)^(1/2)*(a + b*tan(c + d*x))^(5/2), x, 14, -(((im*a - b)^(5/2)*atan((sqrt(im*a - b)*sqrt(tan(c + d*x)))/sqrt(a + b*tan(c + d*x))))/d) + (sqrt(b)*(15*a^2 - 8*b^2)*atanh((sqrt(b)*sqrt(tan(c + d*x)))/sqrt(a + b*tan(c + d*x))))/(4*d) + ((im*a + b)^(5/2)*atanh((sqrt(im*a + b)*sqrt(tan(c + d*x)))/sqrt(a + b*tan(c + d*x))))/d + (9*a*b*sqrt(tan(c + d*x))*sqrt(a + b*tan(c + d*x)))/(4*d) + (b^2*tan(c + d*x)^(3/2)*sqrt(a + b*tan(c + d*x)))/(2*d)]
    @test_int [tan(c + d*x)^(-1/2)*(a + b*tan(c + d*x))^(5/2), x, 13, (im*(im*a - b)^(5/2)*atan((sqrt(im*a - b)*sqrt(tan(c + d*x)))/sqrt(a + b*tan(c + d*x))))/d + (5*a*b^(3/2)*atanh((sqrt(b)*sqrt(tan(c + d*x)))/sqrt(a + b*tan(c + d*x))))/d + (im*(im*a + b)^(5/2)*atanh((sqrt(im*a + b)*sqrt(tan(c + d*x)))/sqrt(a + b*tan(c + d*x))))/d + (b^2*sqrt(tan(c + d*x))*sqrt(a + b*tan(c + d*x)))/d]
    @test_int [tan(c + d*x)^(-3/2)*(a + b*tan(c + d*x))^(5/2), x, 13, ((im*a - b)^(5/2)*atan((sqrt(im*a - b)*sqrt(tan(c + d*x)))/sqrt(a + b*tan(c + d*x))))/d + (2*b^(5/2)*atanh((sqrt(b)*sqrt(tan(c + d*x)))/sqrt(a + b*tan(c + d*x))))/d - ((im*a + b)^(5/2)*atanh((sqrt(im*a + b)*sqrt(tan(c + d*x)))/sqrt(a + b*tan(c + d*x))))/d - (2*a^2*sqrt(a + b*tan(c + d*x)))/(d*sqrt(tan(c + d*x)))]
    @test_int [tan(c + d*x)^(-5/2)*(a + b*tan(c + d*x))^(5/2), x, 9, -((im*(im*a - b)^(5/2)*atan((sqrt(im*a - b)*sqrt(tan(c + d*x)))/sqrt(a + b*tan(c + d*x))))/d) - (im*(im*a + b)^(5/2)*atanh((sqrt(im*a + b)*sqrt(tan(c + d*x)))/sqrt(a + b*tan(c + d*x))))/d - (2*a^2*sqrt(a + b*tan(c + d*x)))/(3*d*tan(c + d*x)^(3/2)) - (14*a*b*sqrt(a + b*tan(c + d*x)))/(3*d*sqrt(tan(c + d*x)))]
    @test_int [tan(c + d*x)^(-7/2)*(a + b*tan(c + d*x))^(5/2), x, 10, -(((im*a - b)^(5/2)*atan((sqrt(im*a - b)*sqrt(tan(c + d*x)))/sqrt(a + b*tan(c + d*x))))/d) + ((im*a + b)^(5/2)*atanh((sqrt(im*a + b)*sqrt(tan(c + d*x)))/sqrt(a + b*tan(c + d*x))))/d - (2*a^2*sqrt(a + b*tan(c + d*x)))/(5*d*tan(c + d*x)^(5/2)) - (22*a*b*sqrt(a + b*tan(c + d*x)))/(15*d*tan(c + d*x)^(3/2)) + (2*(15*a^2 - 23*b^2)*sqrt(a + b*tan(c + d*x)))/(15*d*sqrt(tan(c + d*x)))]
    @test_int [tan(c + d*x)^(-9/2)*(a + b*tan(c + d*x))^(5/2), x, 11, (im*(im*a - b)^(5/2)*atan((sqrt(im*a - b)*sqrt(tan(c + d*x)))/sqrt(a + b*tan(c + d*x))))/d + (im*(im*a + b)^(5/2)*atanh((sqrt(im*a + b)*sqrt(tan(c + d*x)))/sqrt(a + b*tan(c + d*x))))/d - (2*a^2*sqrt(a + b*tan(c + d*x)))/(7*d*tan(c + d*x)^(7/2)) - (6*a*b*sqrt(a + b*tan(c + d*x)))/(7*d*tan(c + d*x)^(5/2)) + (2*(7*a^2 - 9*b^2)*sqrt(a + b*tan(c + d*x)))/(21*d*tan(c + d*x)^(3/2)) + (2*b*(49*a^2 - 3*b^2)*sqrt(a + b*tan(c + d*x)))/(21*a*d*sqrt(tan(c + d*x)))]
    @test_int [tan(c + d*x)^(-11/2)*(a + b*tan(c + d*x))^(5/2), x, 12, ((im*a - b)^(5/2)*atan((sqrt(im*a - b)*sqrt(tan(c + d*x)))/sqrt(a + b*tan(c + d*x))))/d - ((im*a + b)^(5/2)*atanh((sqrt(im*a + b)*sqrt(tan(c + d*x)))/sqrt(a + b*tan(c + d*x))))/d - (2*a^2*sqrt(a + b*tan(c + d*x)))/(9*d*tan(c + d*x)^(9/2)) - (38*a*b*sqrt(a + b*tan(c + d*x)))/(63*d*tan(c + d*x)^(7/2)) + (2*(21*a^2 - 25*b^2)*sqrt(a + b*tan(c + d*x)))/(105*d*tan(c + d*x)^(5/2)) + (2*b*(231*a^2 - 5*b^2)*sqrt(a + b*tan(c + d*x)))/(315*a*d*tan(c + d*x)^(3/2)) - (2*(315*a^4 - 483*a^2*b^2 - 10*b^4)*sqrt(a + b*tan(c + d*x)))/(315*a^2*d*sqrt(tan(c + d*x)))]


    #= ::Subsubsection::Closed:: =#
    #=m<0=#


    @test_int [tan(c + d*x)^(7/2)/(a + b*tan(c + d*x))^(1/2), x, 14, atan((sqrt(im*a - b)*sqrt(tan(c + d*x)))/sqrt(a + b*tan(c + d*x)))/(sqrt(im*a - b)*d) + ((3*a^2 - 8*b^2)*atanh((sqrt(b)*sqrt(tan(c + d*x)))/sqrt(a + b*tan(c + d*x))))/(4*b^(5/2)*d) + atanh((sqrt(im*a + b)*sqrt(tan(c + d*x)))/sqrt(a + b*tan(c + d*x)))/(sqrt(im*a + b)*d) - (3*a*sqrt(tan(c + d*x))*sqrt(a + b*tan(c + d*x)))/(4*b^2*d) + (tan(c + d*x)^(3/2)*sqrt(a + b*tan(c + d*x)))/(2*b*d)]
    @test_int [tan(c + d*x)^(5/2)/(a + b*tan(c + d*x))^(1/2), x, 13, -((im*atan((sqrt(im*a - b)*sqrt(tan(c + d*x)))/sqrt(a + b*tan(c + d*x))))/(sqrt(im*a - b)*d)) - (a*atanh((sqrt(b)*sqrt(tan(c + d*x)))/sqrt(a + b*tan(c + d*x))))/(b^(3/2)*d) + (im*atanh((sqrt(im*a + b)*sqrt(tan(c + d*x)))/sqrt(a + b*tan(c + d*x))))/(sqrt(im*a + b)*d) + (sqrt(tan(c + d*x))*sqrt(a + b*tan(c + d*x)))/(b*d)]
    @test_int [tan(c + d*x)^(3/2)/(a + b*tan(c + d*x))^(1/2), x, 12, -(atan((sqrt(im*a - b)*sqrt(tan(c + d*x)))/sqrt(a + b*tan(c + d*x)))/(sqrt(im*a - b)*d)) + (2*atanh((sqrt(b)*sqrt(tan(c + d*x)))/sqrt(a + b*tan(c + d*x))))/(sqrt(b)*d) - atanh((sqrt(im*a + b)*sqrt(tan(c + d*x)))/sqrt(a + b*tan(c + d*x)))/(sqrt(im*a + b)*d)]
    @test_int [tan(c + d*x)^(1/2)/(a + b*tan(c + d*x))^(1/2), x, 7, (im*atan((sqrt(im*a - b)*sqrt(tan(c + d*x)))/sqrt(a + b*tan(c + d*x))))/(sqrt(im*a - b)*d) - (im*atanh((sqrt(im*a + b)*sqrt(tan(c + d*x)))/sqrt(a + b*tan(c + d*x))))/(sqrt(im*a + b)*d)]
    @test_int [tan(c + d*x)^(-1/2)/(a + b*tan(c + d*x))^(1/2), x, 7, atan((sqrt(im*a - b)*sqrt(tan(c + d*x)))/sqrt(a + b*tan(c + d*x)))/(sqrt(im*a - b)*d) + atanh((sqrt(im*a + b)*sqrt(tan(c + d*x)))/sqrt(a + b*tan(c + d*x)))/(sqrt(im*a + b)*d)]
    @test_int [tan(c + d*x)^(-3/2)/(a + b*tan(c + d*x))^(1/2), x, 9, -((im*atan((sqrt(im*a - b)*sqrt(tan(c + d*x)))/sqrt(a + b*tan(c + d*x))))/(sqrt(im*a - b)*d)) + (im*atanh((sqrt(im*a + b)*sqrt(tan(c + d*x)))/sqrt(a + b*tan(c + d*x))))/(sqrt(im*a + b)*d) - (2*sqrt(a + b*tan(c + d*x)))/(a*d*sqrt(tan(c + d*x)))]
    @test_int [tan(c + d*x)^(-5/2)/(a + b*tan(c + d*x))^(1/2), x, 10, -(atan((sqrt(im*a - b)*sqrt(tan(c + d*x)))/sqrt(a + b*tan(c + d*x)))/(sqrt(im*a - b)*d)) - atanh((sqrt(im*a + b)*sqrt(tan(c + d*x)))/sqrt(a + b*tan(c + d*x)))/(sqrt(im*a + b)*d) - (2*sqrt(a + b*tan(c + d*x)))/(3*a*d*tan(c + d*x)^(3/2)) + (4*b*sqrt(a + b*tan(c + d*x)))/(3*a^2*d*sqrt(tan(c + d*x)))]
    @test_int [tan(c + d*x)^(-7/2)/(a + b*tan(c + d*x))^(1/2), x, 11, (im*atan((sqrt(im*a - b)*sqrt(tan(c + d*x)))/sqrt(a + b*tan(c + d*x))))/(sqrt(im*a - b)*d) - (im*atanh((sqrt(im*a + b)*sqrt(tan(c + d*x)))/sqrt(a + b*tan(c + d*x))))/(sqrt(im*a + b)*d) - (2*sqrt(a + b*tan(c + d*x)))/(5*a*d*tan(c + d*x)^(5/2)) + (8*b*sqrt(a + b*tan(c + d*x)))/(15*a^2*d*tan(c + d*x)^(3/2)) + (2*(15*a^2 - 8*b^2)*sqrt(a + b*tan(c + d*x)))/(15*a^3*d*sqrt(tan(c + d*x)))]


    @test_int [tan(c + d*x)^(7/2)/(a + b*tan(c + d*x))^(3/2), x, 14, (im*atan((sqrt(im*a - b)*sqrt(tan(c + d*x)))/sqrt(a + b*tan(c + d*x))))/((im*a - b)^(3/2)*d) - (3*a*atanh((sqrt(b)*sqrt(tan(c + d*x)))/sqrt(a + b*tan(c + d*x))))/(b^(5/2)*d) + (im*atanh((sqrt(im*a + b)*sqrt(tan(c + d*x)))/sqrt(a + b*tan(c + d*x))))/((im*a + b)^(3/2)*d) - (2*a^2*tan(c + d*x)^(3/2))/(b*(a^2 + b^2)*d*sqrt(a + b*tan(c + d*x))) + ((3*a^2 + b^2)*sqrt(tan(c + d*x))*sqrt(a + b*tan(c + d*x)))/(b^2*(a^2 + b^2)*d)]
    @test_int [tan(c + d*x)^(5/2)/(a + b*tan(c + d*x))^(3/2), x, 13, atan((sqrt(im*a - b)*sqrt(tan(c + d*x)))/sqrt(a + b*tan(c + d*x)))/((im*a - b)^(3/2)*d) + (2*atanh((sqrt(b)*sqrt(tan(c + d*x)))/sqrt(a + b*tan(c + d*x))))/(b^(3/2)*d) - atanh((sqrt(im*a + b)*sqrt(tan(c + d*x)))/sqrt(a + b*tan(c + d*x)))/((im*a + b)^(3/2)*d) - (2*a^2*sqrt(tan(c + d*x)))/(b*(a^2 + b^2)*d*sqrt(a + b*tan(c + d*x)))]
    @test_int [tan(c + d*x)^(3/2)/(a + b*tan(c + d*x))^(3/2), x, 8, -((im*atan((sqrt(im*a - b)*sqrt(tan(c + d*x)))/sqrt(a + b*tan(c + d*x))))/((im*a - b)^(3/2)*d)) - (im*atanh((sqrt(im*a + b)*sqrt(tan(c + d*x)))/sqrt(a + b*tan(c + d*x))))/((im*a + b)^(3/2)*d) + (2*a*sqrt(tan(c + d*x)))/((a^2 + b^2)*d*sqrt(a + b*tan(c + d*x)))]
    @test_int [tan(c + d*x)^(1/2)/(a + b*tan(c + d*x))^(3/2), x, 8, -(atan((sqrt(im*a - b)*sqrt(tan(c + d*x)))/sqrt(a + b*tan(c + d*x)))/((im*a - b)^(3/2)*d)) + atanh((sqrt(im*a + b)*sqrt(tan(c + d*x)))/sqrt(a + b*tan(c + d*x)))/((im*a + b)^(3/2)*d) - (2*b*sqrt(tan(c + d*x)))/((a^2 + b^2)*d*sqrt(a + b*tan(c + d*x)))]
    @test_int [tan(c + d*x)^(-1/2)/(a + b*tan(c + d*x))^(3/2), x, 8, (im*atan((sqrt(im*a - b)*sqrt(tan(c + d*x)))/sqrt(a + b*tan(c + d*x))))/((im*a - b)^(3/2)*d) + (im*atanh((sqrt(im*a + b)*sqrt(tan(c + d*x)))/sqrt(a + b*tan(c + d*x))))/((im*a + b)^(3/2)*d) + (2*b^2*sqrt(tan(c + d*x)))/(a*(a^2 + b^2)*d*sqrt(a + b*tan(c + d*x)))]
    @test_int [tan(c + d*x)^(-3/2)/(a + b*tan(c + d*x))^(3/2), x, 9, atan((sqrt(im*a - b)*sqrt(tan(c + d*x)))/sqrt(a + b*tan(c + d*x)))/((im*a - b)^(3/2)*d) - atanh((sqrt(im*a + b)*sqrt(tan(c + d*x)))/sqrt(a + b*tan(c + d*x)))/((im*a + b)^(3/2)*d) - 2/(a*d*sqrt(tan(c + d*x))*sqrt(a + b*tan(c + d*x))) - (2*b*(a^2 + 2*b^2)*sqrt(tan(c + d*x)))/(a^2*(a^2 + b^2)*d*sqrt(a + b*tan(c + d*x)))]
    @test_int [tan(c + d*x)^(-5/2)/(a + b*tan(c + d*x))^(3/2), x, 10, -((im*atan((sqrt(im*a - b)*sqrt(tan(c + d*x)))/sqrt(a + b*tan(c + d*x))))/((im*a - b)^(3/2)*d)) - (im*atanh((sqrt(im*a + b)*sqrt(tan(c + d*x)))/sqrt(a + b*tan(c + d*x))))/((im*a + b)^(3/2)*d) - 2/(3*a*d*tan(c + d*x)^(3/2)*sqrt(a + b*tan(c + d*x))) + (8*b)/(3*a^2*d*sqrt(tan(c + d*x))*sqrt(a + b*tan(c + d*x))) + (2*b^2*(5*a^2 + 8*b^2)*sqrt(tan(c + d*x)))/(3*a^3*(a^2 + b^2)*d*sqrt(a + b*tan(c + d*x)))]


    @test_int [tan(c + d*x)^(9/2)/(a + b*tan(c + d*x))^(5/2), x, 15, -((im*atan((sqrt(im*a - b)*sqrt(tan(c + d*x)))/sqrt(a + b*tan(c + d*x))))/((im*a - b)^(5/2)*d)) - (5*a*atanh((sqrt(b)*sqrt(tan(c + d*x)))/sqrt(a + b*tan(c + d*x))))/(b^(7/2)*d) + (im*atanh((sqrt(im*a + b)*sqrt(tan(c + d*x)))/sqrt(a + b*tan(c + d*x))))/((im*a + b)^(5/2)*d) - (2*a^2*tan(c + d*x)^(5/2))/(3*b*(a^2 + b^2)*d*(a + b*tan(c + d*x))^(3/2)) - (2*a^2*(5*a^2 + 11*b^2)*tan(c + d*x)^(3/2))/(3*b^2*(a^2 + b^2)^2*d*sqrt(a + b*tan(c + d*x))) + ((5*a^4 + 10*a^2*b^2 + b^4)*sqrt(tan(c + d*x))*sqrt(a + b*tan(c + d*x)))/(b^3*(a^2 + b^2)^2*d)]
    @test_int [tan(c + d*x)^(7/2)/(a + b*tan(c + d*x))^(5/2), x, 14, -(atan((sqrt(im*a - b)*sqrt(tan(c + d*x)))/sqrt(a + b*tan(c + d*x)))/((im*a - b)^(5/2)*d)) + (2*atanh((sqrt(b)*sqrt(tan(c + d*x)))/sqrt(a + b*tan(c + d*x))))/(b^(5/2)*d) - atanh((sqrt(im*a + b)*sqrt(tan(c + d*x)))/sqrt(a + b*tan(c + d*x)))/((im*a + b)^(5/2)*d) - (2*a^2*tan(c + d*x)^(3/2))/(3*b*(a^2 + b^2)*d*(a + b*tan(c + d*x))^(3/2)) - (2*a^2*(a^2 + 3*b^2)*sqrt(tan(c + d*x)))/(b^2*(a^2 + b^2)^2*d*sqrt(a + b*tan(c + d*x)))]
    @test_int [tan(c + d*x)^(5/2)/(a + b*tan(c + d*x))^(5/2), x, 9, (im*atan((sqrt(im*a - b)*sqrt(tan(c + d*x)))/sqrt(a + b*tan(c + d*x))))/((im*a - b)^(5/2)*d) - (im*atanh((sqrt(im*a + b)*sqrt(tan(c + d*x)))/sqrt(a + b*tan(c + d*x))))/((im*a + b)^(5/2)*d) - (2*a^2*sqrt(tan(c + d*x)))/(3*b*(a^2 + b^2)*d*(a + b*tan(c + d*x))^(3/2)) + (2*a*(a^2 + 7*b^2)*sqrt(tan(c + d*x)))/(3*b*(a^2 + b^2)^2*d*sqrt(a + b*tan(c + d*x)))]
    @test_int [tan(c + d*x)^(3/2)/(a + b*tan(c + d*x))^(5/2), x, 9, atan((sqrt(im*a - b)*sqrt(tan(c + d*x)))/sqrt(a + b*tan(c + d*x)))/((im*a - b)^(5/2)*d) + atanh((sqrt(im*a + b)*sqrt(tan(c + d*x)))/sqrt(a + b*tan(c + d*x)))/((im*a + b)^(5/2)*d) + (2*a*sqrt(tan(c + d*x)))/(3*(a^2 + b^2)*d*(a + b*tan(c + d*x))^(3/2)) + (4*(a^2 - 2*b^2)*sqrt(tan(c + d*x)))/(3*(a^2 + b^2)^2*d*sqrt(a + b*tan(c + d*x)))]
    @test_int [tan(c + d*x)^(1/2)/(a + b*tan(c + d*x))^(5/2), x, 9, -((im*atan((sqrt(im*a - b)*sqrt(tan(c + d*x)))/sqrt(a + b*tan(c + d*x))))/((im*a - b)^(5/2)*d)) + (im*atanh((sqrt(im*a + b)*sqrt(tan(c + d*x)))/sqrt(a + b*tan(c + d*x))))/((im*a + b)^(5/2)*d) - (2*b*sqrt(tan(c + d*x)))/(3*(a^2 + b^2)*d*(a + b*tan(c + d*x))^(3/2)) - (2*b*(5*a^2 - b^2)*sqrt(tan(c + d*x)))/(3*a*(a^2 + b^2)^2*d*sqrt(a + b*tan(c + d*x)))]
    @test_int [tan(c + d*x)^(-1/2)/(a + b*tan(c + d*x))^(5/2), x, 9, -(atan((sqrt(im*a - b)*sqrt(tan(c + d*x)))/sqrt(a + b*tan(c + d*x)))/((im*a - b)^(5/2)*d)) - atanh((sqrt(im*a + b)*sqrt(tan(c + d*x)))/sqrt(a + b*tan(c + d*x)))/((im*a + b)^(5/2)*d) + (2*b^2*sqrt(tan(c + d*x)))/(3*a*(a^2 + b^2)*d*(a + b*tan(c + d*x))^(3/2)) + (4*b^2*(4*a^2 + b^2)*sqrt(tan(c + d*x)))/(3*a^2*(a^2 + b^2)^2*d*sqrt(a + b*tan(c + d*x)))]
    @test_int [tan(c + d*x)^(-3/2)/(a + b*tan(c + d*x))^(5/2), x, 10, (im*atan((sqrt(im*a - b)*sqrt(tan(c + d*x)))/sqrt(a + b*tan(c + d*x))))/((im*a - b)^(5/2)*d) - (im*atanh((sqrt(im*a + b)*sqrt(tan(c + d*x)))/sqrt(a + b*tan(c + d*x))))/((im*a + b)^(5/2)*d) - 2/(a*d*sqrt(tan(c + d*x))*(a + b*tan(c + d*x))^(3/2)) - (2*b*(3*a^2 + 4*b^2)*sqrt(tan(c + d*x)))/(3*a^2*(a^2 + b^2)*d*(a + b*tan(c + d*x))^(3/2)) - (2*b*(3*a^4 + 17*a^2*b^2 + 8*b^4)*sqrt(tan(c + d*x)))/(3*a^3*(a^2 + b^2)^2*d*sqrt(a + b*tan(c + d*x)))]
    @test_int [tan(c + d*x)^(-5/2)/(a + b*tan(c + d*x))^(5/2), x, 11, atan((sqrt(im*a - b)*sqrt(tan(c + d*x)))/sqrt(a + b*tan(c + d*x)))/((im*a - b)^(5/2)*d) + atanh((sqrt(im*a + b)*sqrt(tan(c + d*x)))/sqrt(a + b*tan(c + d*x)))/((im*a + b)^(5/2)*d) - 2/(3*a*d*tan(c + d*x)^(3/2)*(a + b*tan(c + d*x))^(3/2)) + (4*b)/(a^2*d*sqrt(tan(c + d*x))*(a + b*tan(c + d*x))^(3/2)) + (2*b^2*(7*a^2 + 8*b^2)*sqrt(tan(c + d*x)))/(3*a^3*(a^2 + b^2)*d*(a + b*tan(c + d*x))^(3/2)) + (4*b^2*(4*a^4 + 15*a^2*b^2 + 8*b^4)*sqrt(tan(c + d*x)))/(3*a^4*(a^2 + b^2)^2*d*sqrt(a + b*tan(c + d*x)))]


    @test_int [1/(sqrt(tan(c + d*x))*sqrt(2 + 3*tan(c + d*x))), x, 7, atanh((sqrt(3 - 2*im)*sqrt(tan(c + d*x)))/sqrt(2 + 3*tan(c + d*x)))/(sqrt(3 - 2*im)*d) + atanh((sqrt(3 + 2*im)*sqrt(tan(c + d*x)))/sqrt(2 + 3*tan(c + d*x)))/(sqrt(3 + 2*im)*d)]
    @test_int [1/(sqrt(tan(c + d*x))*sqrt(-2 + 3*tan(c + d*x))), x, 7, atanh((sqrt(3 - 2*im)*sqrt(tan(c + d*x)))/sqrt(-2 + 3*tan(c + d*x)))/(sqrt(3 - 2*im)*d) + atanh((sqrt(3 + 2*im)*sqrt(tan(c + d*x)))/sqrt(-2 + 3*tan(c + d*x)))/(sqrt(3 + 2*im)*d)]

    @test_int [1/(sqrt(tan(c + d*x))*sqrt(2 - 3*tan(c + d*x))), x, 7, atan((sqrt(3 - 2*im)*sqrt(tan(c + d*x)))/sqrt(2 - 3*tan(c + d*x)))/(sqrt(3 - 2*im)*d) + atan((sqrt(3 + 2*im)*sqrt(tan(c + d*x)))/sqrt(2 - 3*tan(c + d*x)))/(sqrt(3 + 2*im)*d)]
    @test_int [1/(sqrt(tan(c + d*x))*sqrt(-2 - 3*tan(c + d*x))), x, 7, atan((sqrt(3 - 2*im)*sqrt(tan(c + d*x)))/sqrt(-2 - 3*tan(c + d*x)))/(sqrt(3 - 2*im)*d) + atan((sqrt(3 + 2*im)*sqrt(tan(c + d*x)))/sqrt(-2 - 3*tan(c + d*x)))/(sqrt(3 + 2*im)*d)]

    @test_int [1/(sqrt(tan(c + d*x))*sqrt(3 + 2*tan(c + d*x))), x, 7, atanh((sqrt(2 - 3*im)*sqrt(tan(c + d*x)))/sqrt(3 + 2*tan(c + d*x)))/(sqrt(2 - 3*im)*d) + atanh((sqrt(2 + 3*im)*sqrt(tan(c + d*x)))/sqrt(3 + 2*tan(c + d*x)))/(sqrt(2 + 3*im)*d)]
    @test_int [1/(sqrt(tan(c + d*x))*sqrt(3 - 2*tan(c + d*x))), x, 7, atan((sqrt(2 - 3*im)*sqrt(tan(c + d*x)))/sqrt(3 - 2*tan(c + d*x)))/(sqrt(2 - 3*im)*d) + atan((sqrt(2 + 3*im)*sqrt(tan(c + d*x)))/sqrt(3 - 2*tan(c + d*x)))/(sqrt(2 + 3*im)*d)]

    @test_int [1/(sqrt(tan(c + d*x))*sqrt(-3 + 2*tan(c + d*x))), x, 7, atanh((sqrt(2 - 3*im)*sqrt(tan(c + d*x)))/sqrt(-3 + 2*tan(c + d*x)))/(sqrt(2 - 3*im)*d) + atanh((sqrt(2 + 3*im)*sqrt(tan(c + d*x)))/sqrt(-3 + 2*tan(c + d*x)))/(sqrt(2 + 3*im)*d)]
    @test_int [1/(sqrt(tan(c + d*x))*sqrt(-3 - 2*tan(c + d*x))), x, 7, atan((sqrt(2 - 3*im)*sqrt(tan(c + d*x)))/sqrt(-3 - 2*tan(c + d*x)))/(sqrt(2 - 3*im)*d) + atan((sqrt(2 + 3*im)*sqrt(tan(c + d*x)))/sqrt(-3 - 2*tan(c + d*x)))/(sqrt(2 + 3*im)*d)]


    @test_int [sqrt(tan(c + d*x))/sqrt(2 + 3*tan(c + d*x)), x, 7, (im*atanh((sqrt(3 - 2*im)*sqrt(tan(c + d*x)))/sqrt(2 + 3*tan(c + d*x))))/(sqrt(3 - 2*im)*d) - (im*atanh((sqrt(3 + 2*im)*sqrt(tan(c + d*x)))/sqrt(2 + 3*tan(c + d*x))))/(sqrt(3 + 2*im)*d)]
    @test_int [sqrt(tan(c + d*x))/sqrt(-2 + 3*tan(c + d*x)), x, 7, -((im*atanh((sqrt(3 - 2*im)*sqrt(tan(c + d*x)))/sqrt(-2 + 3*tan(c + d*x))))/(sqrt(3 - 2*im)*d)) + (im*atanh((sqrt(3 + 2*im)*sqrt(tan(c + d*x)))/sqrt(-2 + 3*tan(c + d*x))))/(sqrt(3 + 2*im)*d)]

    @test_int [sqrt(tan(c + d*x))/sqrt(2 - 3*tan(c + d*x)), x, 7, -((im*atan((sqrt(3 - 2*im)*sqrt(tan(c + d*x)))/sqrt(2 - 3*tan(c + d*x))))/(sqrt(3 - 2*im)*d)) + (im*atan((sqrt(3 + 2*im)*sqrt(tan(c + d*x)))/sqrt(2 - 3*tan(c + d*x))))/(sqrt(3 + 2*im)*d)]
    @test_int [sqrt(tan(c + d*x))/sqrt(-2 - 3*tan(c + d*x)), x, 7, (im*atan((sqrt(3 - 2*im)*sqrt(tan(c + d*x)))/sqrt(-2 - 3*tan(c + d*x))))/(sqrt(3 - 2*im)*d) - (im*atan((sqrt(3 + 2*im)*sqrt(tan(c + d*x)))/sqrt(-2 - 3*tan(c + d*x))))/(sqrt(3 + 2*im)*d)]

    @test_int [sqrt(tan(c + d*x))/sqrt(3 + 2*tan(c + d*x)), x, 7, (im*atanh((sqrt(2 - 3*im)*sqrt(tan(c + d*x)))/sqrt(3 + 2*tan(c + d*x))))/(sqrt(2 - 3*im)*d) - (im*atanh((sqrt(2 + 3*im)*sqrt(tan(c + d*x)))/sqrt(3 + 2*tan(c + d*x))))/(sqrt(2 + 3*im)*d)]
    @test_int [sqrt(tan(c + d*x))/sqrt(3 - 2*tan(c + d*x)), x, 7, -((im*atan((sqrt(2 - 3*im)*sqrt(tan(c + d*x)))/sqrt(3 - 2*tan(c + d*x))))/(sqrt(2 - 3*im)*d)) + (im*atan((sqrt(2 + 3*im)*sqrt(tan(c + d*x)))/sqrt(3 - 2*tan(c + d*x))))/(sqrt(2 + 3*im)*d)]

    @test_int [sqrt(tan(c + d*x))/sqrt(-3 + 2*tan(c + d*x)), x, 7, -((im*atanh((sqrt(2 - 3*im)*sqrt(tan(c + d*x)))/sqrt(-3 + 2*tan(c + d*x))))/(sqrt(2 - 3*im)*d)) + (im*atanh((sqrt(2 + 3*im)*sqrt(tan(c + d*x)))/sqrt(-3 + 2*tan(c + d*x))))/(sqrt(2 + 3*im)*d)]
    @test_int [sqrt(tan(c + d*x))/sqrt(-3 - 2*tan(c + d*x)), x, 7, (im*atan((sqrt(2 - 3*im)*sqrt(tan(c + d*x)))/sqrt(-3 - 2*tan(c + d*x))))/(sqrt(2 - 3*im)*d) - (im*atan((sqrt(2 + 3*im)*sqrt(tan(c + d*x)))/sqrt(-3 - 2*tan(c + d*x))))/(sqrt(2 + 3*im)*d)]


    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*(a+b*tan(e+f*x))^m*(d*tan(e+f*x))^(n/3)=#


    #= ::Subsubsection:: =#
    #=m>0=#


    #= ::Subsubsection::Closed:: =#
    #=m<0=#


    @test_int [tan(c + d*x)^(5/3)/(a + b*tan(c + d*x)), x, 32, -((b*atan(sqrt(3) - 2*tan(c + d*x)^(1/3)))/(2*(a^2 + b^2)*d)) + (b*atan(sqrt(3) + 2*tan(c + d*x)^(1/3)))/(2*(a^2 + b^2)*d) - (sqrt(3)*a^(5/3)*atan((a^(1/3) - 2*b^(1/3)*tan(c + d*x)^(1/3))/(sqrt(3)*a^(1/3))))/(b^(2/3)*(a^2 + b^2)*d) + (sqrt(3)*a*atan((1 - 2*tan(c + d*x)^(2/3))/sqrt(3)))/(2*(a^2 + b^2)*d) + (b*atan(tan(c + d*x)^(1/3)))/((a^2 + b^2)*d) - (3*a^(5/3)*log(a^(1/3) + b^(1/3)*tan(c + d*x)^(1/3)))/(2*b^(2/3)*(a^2 + b^2)*d) - (a*log(1 + tan(c + d*x)^(2/3)))/(2*(a^2 + b^2)*d) + (sqrt(3)*b*log(1 - sqrt(3)*tan(c + d*x)^(1/3) + tan(c + d*x)^(2/3)))/(4*(a^2 + b^2)*d) - (sqrt(3)*b*log(1 + sqrt(3)*tan(c + d*x)^(1/3) + tan(c + d*x)^(2/3)))/(4*(a^2 + b^2)*d) + (a^(5/3)*log(a + b*tan(c + d*x)))/(2*b^(2/3)*(a^2 + b^2)*d) + (a*log(1 - tan(c + d*x)^(2/3) + tan(c + d*x)^(4/3)))/(4*(a^2 + b^2)*d)]
    @test_int [tan(c + d*x)^(1/3)/(a + b*tan(c + d*x)), x, 30, -((b*atan(sqrt(3) - 2*tan(c + d*x)^(1/3)))/(2*(a^2 + b^2)*d)) + (b*atan(sqrt(3) + 2*tan(c + d*x)^(1/3)))/(2*(a^2 + b^2)*d) + (sqrt(3)*a^(1/3)*b^(2/3)*atan((a^(1/3) - 2*b^(1/3)*tan(c + d*x)^(1/3))/(sqrt(3)*a^(1/3))))/((a^2 + b^2)*d) - (sqrt(3)*a*atan((1 - 2*tan(c + d*x)^(2/3))/sqrt(3)))/(2*(a^2 + b^2)*d) + (b*atan(tan(c + d*x)^(1/3)))/((a^2 + b^2)*d) - (3*a^(1/3)*b^(2/3)*log(a^(1/3) + b^(1/3)*tan(c + d*x)^(1/3)))/(2*(a^2 + b^2)*d) - (a*log(1 + tan(c + d*x)^(2/3)))/(2*(a^2 + b^2)*d) - (sqrt(3)*b*log(1 - sqrt(3)*tan(c + d*x)^(1/3) + tan(c + d*x)^(2/3)))/(4*(a^2 + b^2)*d) + (sqrt(3)*b*log(1 + sqrt(3)*tan(c + d*x)^(1/3) + tan(c + d*x)^(2/3)))/(4*(a^2 + b^2)*d) + (a^(1/3)*b^(2/3)*log(a + b*tan(c + d*x)))/(2*(a^2 + b^2)*d) + (a*log(1 - tan(c + d*x)^(2/3) + tan(c + d*x)^(4/3)))/(4*(a^2 + b^2)*d)]
    @test_int [tan(c + d*x)^(-1/3)/(a + b*tan(c + d*x)), x, 28, (b*atan(sqrt(3) - 2*tan(c + d*x)^(1/3)))/(2*(a^2 + b^2)*d) - (b*atan(sqrt(3) + 2*tan(c + d*x)^(1/3)))/(2*(a^2 + b^2)*d) - (sqrt(3)*b^(4/3)*atan((a^(1/3) - 2*b^(1/3)*tan(c + d*x)^(1/3))/(sqrt(3)*a^(1/3))))/(a^(1/3)*(a^2 + b^2)*d) - (sqrt(3)*a*atan((1 - 2*tan(c + d*x)^(2/3))/sqrt(3)))/(2*(a^2 + b^2)*d) - (b*atan(tan(c + d*x)^(1/3)))/((a^2 + b^2)*d) - (3*b^(4/3)*log(a^(1/3) + b^(1/3)*tan(c + d*x)^(1/3)))/(2*a^(1/3)*(a^2 + b^2)*d) + (a*log(1 + tan(c + d*x)^(2/3)))/(2*(a^2 + b^2)*d) - (sqrt(3)*b*log(1 - sqrt(3)*tan(c + d*x)^(1/3) + tan(c + d*x)^(2/3)))/(4*(a^2 + b^2)*d) + (sqrt(3)*b*log(1 + sqrt(3)*tan(c + d*x)^(1/3) + tan(c + d*x)^(2/3)))/(4*(a^2 + b^2)*d) + (b^(4/3)*log(a + b*tan(c + d*x)))/(2*a^(1/3)*(a^2 + b^2)*d) - (a*log(1 - tan(c + d*x)^(2/3) + tan(c + d*x)^(4/3)))/(4*(a^2 + b^2)*d)]
    @test_int [tan(c + d*x)^(-5/3)/(a + b*tan(c + d*x)), x, 30, (b*atan(sqrt(3) - 2*tan(c + d*x)^(1/3)))/(2*(a^2 + b^2)*d) - (b*atan(sqrt(3) + 2*tan(c + d*x)^(1/3)))/(2*(a^2 + b^2)*d) + (sqrt(3)*b^(8/3)*atan((a^(1/3) - 2*b^(1/3)*tan(c + d*x)^(1/3))/(sqrt(3)*a^(1/3))))/(a^(5/3)*(a^2 + b^2)*d) + (sqrt(3)*a*atan((1 - 2*tan(c + d*x)^(2/3))/sqrt(3)))/(2*(a^2 + b^2)*d) - (b*atan(tan(c + d*x)^(1/3)))/((a^2 + b^2)*d) - (3*b^(8/3)*log(a^(1/3) + b^(1/3)*tan(c + d*x)^(1/3)))/(2*a^(5/3)*(a^2 + b^2)*d) + (a*log(1 + tan(c + d*x)^(2/3)))/(2*(a^2 + b^2)*d) + (sqrt(3)*b*log(1 - sqrt(3)*tan(c + d*x)^(1/3) + tan(c + d*x)^(2/3)))/(4*(a^2 + b^2)*d) - (sqrt(3)*b*log(1 + sqrt(3)*tan(c + d*x)^(1/3) + tan(c + d*x)^(2/3)))/(4*(a^2 + b^2)*d) + (b^(8/3)*log(a + b*tan(c + d*x)))/(2*a^(5/3)*(a^2 + b^2)*d) - (a*log(1 - tan(c + d*x)^(2/3) + tan(c + d*x)^(4/3)))/(4*(a^2 + b^2)*d) - (3*a)/(2*(a^2 + b^2)*d*tan(c + d*x)^(2/3)) - (3*b^2)/(2*a*(a^2 + b^2)*d*tan(c + d*x)^(2/3))]


    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*(a+b*tan(e+f*x))^(m/2)*(d*tan(e+f*x))^(n/3)=#


    #= ::Subsubsection:: =#
    #=m>0=#


    #= ::Subsubsection::Closed:: =#
    #=m<0=#


    @test_int [tan(c + d*x)^(4/3)/sqrt(a + b*tan(c + d*x)), x, 9, (3*AppellF1(7/3, 1, 1/2, 10/3, (-im)*tan(c + d*x), -((b*tan(c + d*x))/a))*tan(c + d*x)^(7/3)*sqrt(1 + (b*tan(c + d*x))/a))/(14*d*sqrt(a + b*tan(c + d*x))) + (3*AppellF1(7/3, 1, 1/2, 10/3, im*tan(c + d*x), -((b*tan(c + d*x))/a))*tan(c + d*x)^(7/3)*sqrt(1 + (b*tan(c + d*x))/a))/(14*d*sqrt(a + b*tan(c + d*x)))]
    @test_int [tan(c + d*x)^(2/3)/sqrt(a + b*tan(c + d*x)), x, 9, (3*AppellF1(5/3, 1, 1/2, 8/3, (-im)*tan(c + d*x), -((b*tan(c + d*x))/a))*tan(c + d*x)^(5/3)*sqrt(1 + (b*tan(c + d*x))/a))/(10*d*sqrt(a + b*tan(c + d*x))) + (3*AppellF1(5/3, 1, 1/2, 8/3, im*tan(c + d*x), -((b*tan(c + d*x))/a))*tan(c + d*x)^(5/3)*sqrt(1 + (b*tan(c + d*x))/a))/(10*d*sqrt(a + b*tan(c + d*x)))]
    @test_int [tan(c + d*x)^(1/3)/sqrt(a + b*tan(c + d*x)), x, 9, (3*AppellF1(4/3, 1, 1/2, 7/3, (-im)*tan(c + d*x), -((b*tan(c + d*x))/a))*tan(c + d*x)^(4/3)*sqrt(1 + (b*tan(c + d*x))/a))/(8*d*sqrt(a + b*tan(c + d*x))) + (3*AppellF1(4/3, 1, 1/2, 7/3, im*tan(c + d*x), -((b*tan(c + d*x))/a))*tan(c + d*x)^(4/3)*sqrt(1 + (b*tan(c + d*x))/a))/(8*d*sqrt(a + b*tan(c + d*x)))]
    @test_int [tan(c + d*x)^(-1/3)/sqrt(a + b*tan(c + d*x)), x, 9, (3*AppellF1(2/3, 1, 1/2, 5/3, (-im)*tan(c + d*x), -((b*tan(c + d*x))/a))*tan(c + d*x)^(2/3)*sqrt(1 + (b*tan(c + d*x))/a))/(4*d*sqrt(a + b*tan(c + d*x))) + (3*AppellF1(2/3, 1, 1/2, 5/3, im*tan(c + d*x), -((b*tan(c + d*x))/a))*tan(c + d*x)^(2/3)*sqrt(1 + (b*tan(c + d*x))/a))/(4*d*sqrt(a + b*tan(c + d*x)))]
    @test_int [tan(c + d*x)^(-2/3)/sqrt(a + b*tan(c + d*x)), x, 9, (3*AppellF1(1/3, 1, 1/2, 4/3, (-im)*tan(c + d*x), -((b*tan(c + d*x))/a))*tan(c + d*x)^(1/3)*sqrt(1 + (b*tan(c + d*x))/a))/(2*d*sqrt(a + b*tan(c + d*x))) + (3*AppellF1(1/3, 1, 1/2, 4/3, im*tan(c + d*x), -((b*tan(c + d*x))/a))*tan(c + d*x)^(1/3)*sqrt(1 + (b*tan(c + d*x))/a))/(2*d*sqrt(a + b*tan(c + d*x)))]
    @test_int [tan(c + d*x)^(-4/3)/sqrt(a + b*tan(c + d*x)), x, 9, -((3*AppellF1(-(1/3), 1, 1/2, 2/3, (-im)*tan(c + d*x), -((b*tan(c + d*x))/a))*sqrt(1 + (b*tan(c + d*x))/a))/(2*d*tan(c + d*x)^(1/3)*sqrt(a + b*tan(c + d*x)))) - (3*AppellF1(-(1/3), 1, 1/2, 2/3, im*tan(c + d*x), -((b*tan(c + d*x))/a))*sqrt(1 + (b*tan(c + d*x))/a))/(2*d*tan(c + d*x)^(1/3)*sqrt(a + b*tan(c + d*x)))]


    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*(a+b*tan(e+f*x))^(m/3)*(d*tan(e+f*x))^n=#


    #= ::Subsubsection::Closed:: =#
    #=m>0=#


    @test_int [tan(e + f*x)^4*(c + d*tan(e + f*x))^(1/3), x, 16, (-(1/4))*(c - sqrt(-d^2))^(1/3)*x - (1/4)*(c + sqrt(-d^2))^(1/3)*x - (sqrt(3)*sqrt(-d^2)*(c - sqrt(-d^2))^(1/3)*atan((1 + (2*(c + d*tan(e + f*x))^(1/3))/(c - sqrt(-d^2))^(1/3))/sqrt(3)))/(2*d*f) + (sqrt(3)*sqrt(-d^2)*(c + sqrt(-d^2))^(1/3)*atan((1 + (2*(c + d*tan(e + f*x))^(1/3))/(c + sqrt(-d^2))^(1/3))/sqrt(3)))/(2*d*f) + (sqrt(-d^2)*(c - sqrt(-d^2))^(1/3)*log(cos(e + f*x)))/(4*d*f) - (sqrt(-d^2)*(c + sqrt(-d^2))^(1/3)*log(cos(e + f*x)))/(4*d*f) + (3*sqrt(-d^2)*(c - sqrt(-d^2))^(1/3)*log((c - sqrt(-d^2))^(1/3) - (c + d*tan(e + f*x))^(1/3)))/(4*d*f) - (3*sqrt(-d^2)*(c + sqrt(-d^2))^(1/3)*log((c + sqrt(-d^2))^(1/3) - (c + d*tan(e + f*x))^(1/3)))/(4*d*f) + (3*(9*c^2 - 35*d^2)*(c + d*tan(e + f*x))^(4/3))/(140*d^3*f) - (9*c*tan(e + f*x)*(c + d*tan(e + f*x))^(4/3))/(35*d^2*f) + (3*tan(e + f*x)^2*(c + d*tan(e + f*x))^(4/3))/(10*d*f)]
    @test_int [tan(e + f*x)^3*(c + d*tan(e + f*x))^(1/3), x, 15, (-(1/4))*im*(c - im*d)^(1/3)*x + (1/4)*im*(c + im*d)^(1/3)*x + (sqrt(3)*(c - im*d)^(1/3)*atan((1 + (2*(c + d*tan(e + f*x))^(1/3))/(c - im*d)^(1/3))/sqrt(3)))/(2*f) + (sqrt(3)*(c + im*d)^(1/3)*atan((1 + (2*(c + d*tan(e + f*x))^(1/3))/(c + im*d)^(1/3))/sqrt(3)))/(2*f) - ((c - im*d)^(1/3)*log(cos(e + f*x)))/(4*f) - ((c + im*d)^(1/3)*log(cos(e + f*x)))/(4*f) - (3*(c - im*d)^(1/3)*log((c - im*d)^(1/3) - (c + d*tan(e + f*x))^(1/3)))/(4*f) - (3*(c + im*d)^(1/3)*log((c + im*d)^(1/3) - (c + d*tan(e + f*x))^(1/3)))/(4*f) - (3*(c + d*tan(e + f*x))^(1/3))/f - (9*c*(c + d*tan(e + f*x))^(4/3))/(28*d^2*f) + (3*tan(e + f*x)*(c + d*tan(e + f*x))^(4/3))/(7*d*f)]
    @test_int [tan(e + f*x)^2*(c + d*tan(e + f*x))^(1/3), x, 14, (1/4)*(c - sqrt(-d^2))^(1/3)*x + (1/4)*(c + sqrt(-d^2))^(1/3)*x - (sqrt(3)*d*(c - sqrt(-d^2))^(1/3)*atan((1 + (2*(c + d*tan(e + f*x))^(1/3))/(c - sqrt(-d^2))^(1/3))/sqrt(3)))/(2*sqrt(-d^2)*f) + (sqrt(3)*d*(c + sqrt(-d^2))^(1/3)*atan((1 + (2*(c + d*tan(e + f*x))^(1/3))/(c + sqrt(-d^2))^(1/3))/sqrt(3)))/(2*sqrt(-d^2)*f) + (d*(c - sqrt(-d^2))^(1/3)*log(cos(e + f*x)))/(4*sqrt(-d^2)*f) - (d*(c + sqrt(-d^2))^(1/3)*log(cos(e + f*x)))/(4*sqrt(-d^2)*f) + (3*d*(c - sqrt(-d^2))^(1/3)*log((c - sqrt(-d^2))^(1/3) - (c + d*tan(e + f*x))^(1/3)))/(4*sqrt(-d^2)*f) - (3*d*(c + sqrt(-d^2))^(1/3)*log((c + sqrt(-d^2))^(1/3) - (c + d*tan(e + f*x))^(1/3)))/(4*sqrt(-d^2)*f) + (3*(c + d*tan(e + f*x))^(4/3))/(4*d*f)]
    @test_int [tan(e + f*x)^1*(c + d*tan(e + f*x))^(1/3), x, 12, (1/4)*im*(c - im*d)^(1/3)*x - (1/4)*im*(c + im*d)^(1/3)*x - (sqrt(3)*(c - im*d)^(1/3)*atan((1 + (2*(c + d*tan(e + f*x))^(1/3))/(c - im*d)^(1/3))/sqrt(3)))/(2*f) - (sqrt(3)*(c + im*d)^(1/3)*atan((1 + (2*(c + d*tan(e + f*x))^(1/3))/(c + im*d)^(1/3))/sqrt(3)))/(2*f) + ((c - im*d)^(1/3)*log(cos(e + f*x)))/(4*f) + ((c + im*d)^(1/3)*log(cos(e + f*x)))/(4*f) + (3*(c - im*d)^(1/3)*log((c - im*d)^(1/3) - (c + d*tan(e + f*x))^(1/3)))/(4*f) + (3*(c + im*d)^(1/3)*log((c + im*d)^(1/3) - (c + d*tan(e + f*x))^(1/3)))/(4*f) + (3*(c + d*tan(e + f*x))^(1/3))/f]
    @test_int [tan(e + f*x)^0*(c + d*tan(e + f*x))^(1/3), x, 13, (-(1/4))*(c - sqrt(-d^2))^(1/3)*x - (1/4)*(c + sqrt(-d^2))^(1/3)*x + (sqrt(3)*d*(c - sqrt(-d^2))^(1/3)*atan((1 + (2*(c + d*tan(e + f*x))^(1/3))/(c - sqrt(-d^2))^(1/3))/sqrt(3)))/(2*sqrt(-d^2)*f) - (sqrt(3)*d*(c + sqrt(-d^2))^(1/3)*atan((1 + (2*(c + d*tan(e + f*x))^(1/3))/(c + sqrt(-d^2))^(1/3))/sqrt(3)))/(2*sqrt(-d^2)*f) - (d*(c - sqrt(-d^2))^(1/3)*log(cos(e + f*x)))/(4*sqrt(-d^2)*f) + (d*(c + sqrt(-d^2))^(1/3)*log(cos(e + f*x)))/(4*sqrt(-d^2)*f) - (3*d*(c - sqrt(-d^2))^(1/3)*log((c - sqrt(-d^2))^(1/3) - (c + d*tan(e + f*x))^(1/3)))/(4*sqrt(-d^2)*f) + (3*d*(c + sqrt(-d^2))^(1/3)*log((c + sqrt(-d^2))^(1/3) - (c + d*tan(e + f*x))^(1/3)))/(4*sqrt(-d^2)*f)]
    @test_int [cot(e + f*x)^1*(c + d*tan(e + f*x))^(1/3), x, 19, (-(1/4))*im*(c - im*d)^(1/3)*x + (1/4)*im*(c + im*d)^(1/3)*x - (sqrt(3)*c^(1/3)*atan((c^(1/3) + 2*(c + d*tan(e + f*x))^(1/3))/(sqrt(3)*c^(1/3))))/f + (sqrt(3)*(c - im*d)^(1/3)*atan((1 + (2*(c + d*tan(e + f*x))^(1/3))/(c - im*d)^(1/3))/sqrt(3)))/(2*f) + (sqrt(3)*(c + im*d)^(1/3)*atan((1 + (2*(c + d*tan(e + f*x))^(1/3))/(c + im*d)^(1/3))/sqrt(3)))/(2*f) - ((c - im*d)^(1/3)*log(cos(e + f*x)))/(4*f) - ((c + im*d)^(1/3)*log(cos(e + f*x)))/(4*f) - (c^(1/3)*log(tan(e + f*x)))/(2*f) + (3*c^(1/3)*log(c^(1/3) - (c + d*tan(e + f*x))^(1/3)))/(2*f) - (3*(c - im*d)^(1/3)*log((c - im*d)^(1/3) - (c + d*tan(e + f*x))^(1/3)))/(4*f) - (3*(c + im*d)^(1/3)*log((c + im*d)^(1/3) - (c + d*tan(e + f*x))^(1/3)))/(4*f)]
    @test_int [cot(e + f*x)^2*(c + d*tan(e + f*x))^(1/3), x, 20, (1/4)*(c - sqrt(-d^2))^(1/3)*x + (1/4)*(c + sqrt(-d^2))^(1/3)*x - (d*atan((c^(1/3) + 2*(c + d*tan(e + f*x))^(1/3))/(sqrt(3)*c^(1/3))))/(sqrt(3)*c^(2/3)*f) - (sqrt(3)*d*(c - sqrt(-d^2))^(1/3)*atan((1 + (2*(c + d*tan(e + f*x))^(1/3))/(c - sqrt(-d^2))^(1/3))/sqrt(3)))/(2*sqrt(-d^2)*f) + (sqrt(3)*d*(c + sqrt(-d^2))^(1/3)*atan((1 + (2*(c + d*tan(e + f*x))^(1/3))/(c + sqrt(-d^2))^(1/3))/sqrt(3)))/(2*sqrt(-d^2)*f) + (d*(c - sqrt(-d^2))^(1/3)*log(cos(e + f*x)))/(4*sqrt(-d^2)*f) - (d*(c + sqrt(-d^2))^(1/3)*log(cos(e + f*x)))/(4*sqrt(-d^2)*f) - (d*log(tan(e + f*x)))/(6*c^(2/3)*f) + (d*log(c^(1/3) - (c + d*tan(e + f*x))^(1/3)))/(2*c^(2/3)*f) + (3*d*(c - sqrt(-d^2))^(1/3)*log((c - sqrt(-d^2))^(1/3) - (c + d*tan(e + f*x))^(1/3)))/(4*sqrt(-d^2)*f) - (3*d*(c + sqrt(-d^2))^(1/3)*log((c + sqrt(-d^2))^(1/3) - (c + d*tan(e + f*x))^(1/3)))/(4*sqrt(-d^2)*f) - (cot(e + f*x)*(c + d*tan(e + f*x))^(1/3))/f]


    @test_int [(a + b*tan(c + d*x))^(5/3), x, 12, (-(1/4))*(a - im*b)^(5/3)*x - (1/4)*(a + im*b)^(5/3)*x + (im*sqrt(3)*(a - im*b)^(5/3)*atan((1 + (2*(a + b*tan(c + d*x))^(1/3))/(a - im*b)^(1/3))/sqrt(3)))/(2*d) - (im*sqrt(3)*(a + im*b)^(5/3)*atan((1 + (2*(a + b*tan(c + d*x))^(1/3))/(a + im*b)^(1/3))/sqrt(3)))/(2*d) + (im*(a - im*b)^(5/3)*log(cos(c + d*x)))/(4*d) - (im*(a + im*b)^(5/3)*log(cos(c + d*x)))/(4*d) + (3*im*(a - im*b)^(5/3)*log((a - im*b)^(1/3) - (a + b*tan(c + d*x))^(1/3)))/(4*d) - (3*im*(a + im*b)^(5/3)*log((a + im*b)^(1/3) - (a + b*tan(c + d*x))^(1/3)))/(4*d) + (3*b*(a + b*tan(c + d*x))^(2/3))/(2*d)]
    @test_int [(a + b*tan(c + d*x))^(4/3), x, 12, (-(1/4))*(a - im*b)^(4/3)*x - (1/4)*(a + im*b)^(4/3)*x - (im*sqrt(3)*(a - im*b)^(4/3)*atan((1 + (2*(a + b*tan(c + d*x))^(1/3))/(a - im*b)^(1/3))/sqrt(3)))/(2*d) + (im*sqrt(3)*(a + im*b)^(4/3)*atan((1 + (2*(a + b*tan(c + d*x))^(1/3))/(a + im*b)^(1/3))/sqrt(3)))/(2*d) + (im*(a - im*b)^(4/3)*log(cos(c + d*x)))/(4*d) - (im*(a + im*b)^(4/3)*log(cos(c + d*x)))/(4*d) + (3*im*(a - im*b)^(4/3)*log((a - im*b)^(1/3) - (a + b*tan(c + d*x))^(1/3)))/(4*d) - (3*im*(a + im*b)^(4/3)*log((a + im*b)^(1/3) - (a + b*tan(c + d*x))^(1/3)))/(4*d) + (3*b*(a + b*tan(c + d*x))^(1/3))/d]
    @test_int [(a + b*tan(c + d*x))^(2/3), x, 13, (-(1/4))*(a - sqrt(-b^2))^(2/3)*x - (1/4)*(a + sqrt(-b^2))^(2/3)*x - (sqrt(3)*b*(a - sqrt(-b^2))^(2/3)*atan((1 + (2*(a + b*tan(c + d*x))^(1/3))/(a - sqrt(-b^2))^(1/3))/sqrt(3)))/(2*sqrt(-b^2)*d) + (sqrt(3)*b*(a + sqrt(-b^2))^(2/3)*atan((1 + (2*(a + b*tan(c + d*x))^(1/3))/(a + sqrt(-b^2))^(1/3))/sqrt(3)))/(2*sqrt(-b^2)*d) - (b*(a - sqrt(-b^2))^(2/3)*log(cos(c + d*x)))/(4*sqrt(-b^2)*d) + (b*(a + sqrt(-b^2))^(2/3)*log(cos(c + d*x)))/(4*sqrt(-b^2)*d) - (3*b*(a - sqrt(-b^2))^(2/3)*log((a - sqrt(-b^2))^(1/3) - (a + b*tan(c + d*x))^(1/3)))/(4*sqrt(-b^2)*d) + (3*b*(a + sqrt(-b^2))^(2/3)*log((a + sqrt(-b^2))^(1/3) - (a + b*tan(c + d*x))^(1/3)))/(4*sqrt(-b^2)*d)]
    @test_int [(a + b*tan(c + d*x))^(1/3), x, 13, (-(1/4))*(a - sqrt(-b^2))^(1/3)*x - (1/4)*(a + sqrt(-b^2))^(1/3)*x + (sqrt(3)*b*(a - sqrt(-b^2))^(1/3)*atan((1 + (2*(a + b*tan(c + d*x))^(1/3))/(a - sqrt(-b^2))^(1/3))/sqrt(3)))/(2*sqrt(-b^2)*d) - (sqrt(3)*b*(a + sqrt(-b^2))^(1/3)*atan((1 + (2*(a + b*tan(c + d*x))^(1/3))/(a + sqrt(-b^2))^(1/3))/sqrt(3)))/(2*sqrt(-b^2)*d) - (b*(a - sqrt(-b^2))^(1/3)*log(cos(c + d*x)))/(4*sqrt(-b^2)*d) + (b*(a + sqrt(-b^2))^(1/3)*log(cos(c + d*x)))/(4*sqrt(-b^2)*d) - (3*b*(a - sqrt(-b^2))^(1/3)*log((a - sqrt(-b^2))^(1/3) - (a + b*tan(c + d*x))^(1/3)))/(4*sqrt(-b^2)*d) + (3*b*(a + sqrt(-b^2))^(1/3)*log((a + sqrt(-b^2))^(1/3) - (a + b*tan(c + d*x))^(1/3)))/(4*sqrt(-b^2)*d)]


    #= ::Subsubsection::Closed:: =#
    #=m<0=#


    @test_int [1/(a + b*tan(c + d*x))^(1/3), x, 11, -(x/(4*(a - sqrt(-b^2))^(1/3))) - x/(4*(a + sqrt(-b^2))^(1/3)) - (sqrt(3)*b*atan((1 + (2*(a + b*tan(c + d*x))^(1/3))/(a - sqrt(-b^2))^(1/3))/sqrt(3)))/(2*sqrt(-b^2)*(a - sqrt(-b^2))^(1/3)*d) + (sqrt(3)*b*atan((1 + (2*(a + b*tan(c + d*x))^(1/3))/(a + sqrt(-b^2))^(1/3))/sqrt(3)))/(2*sqrt(-b^2)*(a + sqrt(-b^2))^(1/3)*d) - (b*log(cos(c + d*x)))/(4*sqrt(-b^2)*(a - sqrt(-b^2))^(1/3)*d) + (b*log(cos(c + d*x)))/(4*sqrt(-b^2)*(a + sqrt(-b^2))^(1/3)*d) - (3*b*log((a - sqrt(-b^2))^(1/3) - (a + b*tan(c + d*x))^(1/3)))/(4*sqrt(-b^2)*(a - sqrt(-b^2))^(1/3)*d) + (3*b*log((a + sqrt(-b^2))^(1/3) - (a + b*tan(c + d*x))^(1/3)))/(4*sqrt(-b^2)*(a + sqrt(-b^2))^(1/3)*d)]
    @test_int [1/(a + b*tan(c + d*x))^(2/3), x, 11, -(x/(4*(a - sqrt(-b^2))^(2/3))) - x/(4*(a + sqrt(-b^2))^(2/3)) + (sqrt(3)*b*atan((1 + (2*(a + b*tan(c + d*x))^(1/3))/(a - sqrt(-b^2))^(1/3))/sqrt(3)))/(2*sqrt(-b^2)*(a - sqrt(-b^2))^(2/3)*d) - (sqrt(3)*b*atan((1 + (2*(a + b*tan(c + d*x))^(1/3))/(a + sqrt(-b^2))^(1/3))/sqrt(3)))/(2*sqrt(-b^2)*(a + sqrt(-b^2))^(2/3)*d) - (b*log(cos(c + d*x)))/(4*sqrt(-b^2)*(a - sqrt(-b^2))^(2/3)*d) + (b*log(cos(c + d*x)))/(4*sqrt(-b^2)*(a + sqrt(-b^2))^(2/3)*d) - (3*b*log((a - sqrt(-b^2))^(1/3) - (a + b*tan(c + d*x))^(1/3)))/(4*sqrt(-b^2)*(a - sqrt(-b^2))^(2/3)*d) + (3*b*log((a + sqrt(-b^2))^(1/3) - (a + b*tan(c + d*x))^(1/3)))/(4*sqrt(-b^2)*(a + sqrt(-b^2))^(2/3)*d)]
    @test_int [1/(a + b*tan(c + d*x))^(4/3), x, 12, -(x/(4*(a - im*b)^(4/3))) - x/(4*(a + im*b)^(4/3)) + (im*sqrt(3)*atan((1 + (2*(a + b*tan(c + d*x))^(1/3))/(a - im*b)^(1/3))/sqrt(3)))/(2*(a - im*b)^(4/3)*d) - (im*sqrt(3)*atan((1 + (2*(a + b*tan(c + d*x))^(1/3))/(a + im*b)^(1/3))/sqrt(3)))/(2*(a + im*b)^(4/3)*d) + (im*log(cos(c + d*x)))/(4*(a - im*b)^(4/3)*d) - (im*log(cos(c + d*x)))/(4*(a + im*b)^(4/3)*d) + (3*im*log((a - im*b)^(1/3) - (a + b*tan(c + d*x))^(1/3)))/(4*(a - im*b)^(4/3)*d) - (3*im*log((a + im*b)^(1/3) - (a + b*tan(c + d*x))^(1/3)))/(4*(a + im*b)^(4/3)*d) - (3*b)/((a^2 + b^2)*d*(a + b*tan(c + d*x))^(1/3))]
    @test_int [1/(a + b*tan(c + d*x))^(5/3), x, 12, -(x/(4*(a - im*b)^(5/3))) - x/(4*(a + im*b)^(5/3)) - (im*sqrt(3)*atan((1 + (2*(a + b*tan(c + d*x))^(1/3))/(a - im*b)^(1/3))/sqrt(3)))/(2*(a - im*b)^(5/3)*d) + (im*sqrt(3)*atan((1 + (2*(a + b*tan(c + d*x))^(1/3))/(a + im*b)^(1/3))/sqrt(3)))/(2*(a + im*b)^(5/3)*d) + (im*log(cos(c + d*x)))/(4*(a - im*b)^(5/3)*d) - (im*log(cos(c + d*x)))/(4*(a + im*b)^(5/3)*d) + (3*im*log((a - im*b)^(1/3) - (a + b*tan(c + d*x))^(1/3)))/(4*(a - im*b)^(5/3)*d) - (3*im*log((a + im*b)^(1/3) - (a + b*tan(c + d*x))^(1/3)))/(4*(a + im*b)^(5/3)*d) - (3*b)/(2*(a^2 + b^2)*d*(a + b*tan(c + d*x))^(2/3))]


    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*(a+b*tan(e+f*x))^m*(d*tan(e+f*x))^n*with*n*symbolic=#


    @test_int [(d*tan(e + f*x))^n*(a + b*tan(e + f*x))^4, x, 8, -((b^2*(b^2*(3 + n) - a^2*(17 + 5*n))*(d*tan(e + f*x))^(1 + n))/(d*f*(1 + n)*(3 + n))) + ((a^4 - 6*a^2*b^2 + b^4)*HypergeometricFunctions._₂F₁(1, (1 + n)/2, (3 + n)/2, -tan(e + f*x)^2)*(d*tan(e + f*x))^(1 + n))/(d*f*(1 + n)) + (2*a*b^3*(4 + n)*tan(e + f*x)*(d*tan(e + f*x))^(1 + n))/(d*f*(2 + n)*(3 + n)) + (4*a*b*(a^2 - b^2)*HypergeometricFunctions._₂F₁(1, (2 + n)/2, (4 + n)/2, -tan(e + f*x)^2)*(d*tan(e + f*x))^(2 + n))/(d^2*f*(2 + n)) + (b^2*(d*tan(e + f*x))^(1 + n)*(a + b*tan(e + f*x))^2)/(d*f*(3 + n))]
    @test_int [(d*tan(e + f*x))^n*(a + b*tan(e + f*x))^3, x, 7, (a*b^2*(5 + 2*n)*(d*tan(e + f*x))^(1 + n))/(d*f*(1 + n)*(2 + n)) + (a*(a^2 - 3*b^2)*HypergeometricFunctions._₂F₁(1, (1 + n)/2, (3 + n)/2, -tan(e + f*x)^2)*(d*tan(e + f*x))^(1 + n))/(d*f*(1 + n)) + (b*(3*a^2 - b^2)*HypergeometricFunctions._₂F₁(1, (2 + n)/2, (4 + n)/2, -tan(e + f*x)^2)*(d*tan(e + f*x))^(2 + n))/(d^2*f*(2 + n)) + (b^2*(d*tan(e + f*x))^(1 + n)*(a + b*tan(e + f*x)))/(d*f*(2 + n))]
    @test_int [(d*tan(e + f*x))^n*(a + b*tan(e + f*x))^2, x, 6, (b^2*(d*tan(e + f*x))^(1 + n))/(d*f*(1 + n)) + ((a^2 - b^2)*HypergeometricFunctions._₂F₁(1, (1 + n)/2, (3 + n)/2, -tan(e + f*x)^2)*(d*tan(e + f*x))^(1 + n))/(d*f*(1 + n)) + (2*a*b*HypergeometricFunctions._₂F₁(1, (2 + n)/2, (4 + n)/2, -tan(e + f*x)^2)*(d*tan(e + f*x))^(2 + n))/(d^2*f*(2 + n))]
    @test_int [(d*tan(e + f*x))^n*(a + b*tan(e + f*x))^1, x, 5, (a*HypergeometricFunctions._₂F₁(1, (1 + n)/2, (3 + n)/2, -tan(e + f*x)^2)*(d*tan(e + f*x))^(1 + n))/(d*f*(1 + n)) + (b*HypergeometricFunctions._₂F₁(1, (2 + n)/2, (4 + n)/2, -tan(e + f*x)^2)*(d*tan(e + f*x))^(2 + n))/(d^2*f*(2 + n))]
    @test_int [(d*tan(e + f*x))^n/(a + b*tan(e + f*x))^1, x, 8, (a*HypergeometricFunctions._₂F₁(1, (1 + n)/2, (3 + n)/2, -tan(e + f*x)^2)*(d*tan(e + f*x))^(1 + n))/((a^2 + b^2)*d*f*(1 + n)) + (b^2*HypergeometricFunctions._₂F₁(1, 1 + n, 2 + n, -((b*tan(e + f*x))/a))*(d*tan(e + f*x))^(1 + n))/(a*(a^2 + b^2)*d*f*(1 + n)) - (b*HypergeometricFunctions._₂F₁(1, (2 + n)/2, (4 + n)/2, -tan(e + f*x)^2)*(d*tan(e + f*x))^(2 + n))/((a^2 + b^2)*d^2*f*(2 + n))]
    @test_int [(d*tan(e + f*x))^n/(a + b*tan(e + f*x))^2, x, 9, ((a^2 - b^2)*HypergeometricFunctions._₂F₁(1, (1 + n)/2, (3 + n)/2, -tan(e + f*x)^2)*(d*tan(e + f*x))^(1 + n))/((a^2 + b^2)^2*d*f*(1 + n)) + (b^2*(a^2*(2 - n) - b^2*n)*HypergeometricFunctions._₂F₁(1, 1 + n, 2 + n, -((b*tan(e + f*x))/a))*(d*tan(e + f*x))^(1 + n))/(a^2*(a^2 + b^2)^2*d*f*(1 + n)) - (2*a*b*HypergeometricFunctions._₂F₁(1, (2 + n)/2, (4 + n)/2, -tan(e + f*x)^2)*(d*tan(e + f*x))^(2 + n))/((a^2 + b^2)^2*d^2*f*(2 + n)) + (b^2*(d*tan(e + f*x))^(1 + n))/(a*(a^2 + b^2)*d*f*(a + b*tan(e + f*x)))]


    @test_int [tan(c + d*x)^m*(a + b*tan(c + d*x))^(3/2), x, 7, (a*AppellF1(1 + m, -(3/2), 1, 2 + m, -((b*tan(c + d*x))/a), (-im)*tan(c + d*x))*tan(c + d*x)^(1 + m)*sqrt(a + b*tan(c + d*x)))/(2*d*(1 + m)*sqrt(1 + (b*tan(c + d*x))/a)) + (a*AppellF1(1 + m, -(3/2), 1, 2 + m, -((b*tan(c + d*x))/a), im*tan(c + d*x))*tan(c + d*x)^(1 + m)*sqrt(a + b*tan(c + d*x)))/(2*d*(1 + m)*sqrt(1 + (b*tan(c + d*x))/a))]
    @test_int [tan(c + d*x)^m*(a + b*tan(c + d*x))^(1/2), x, 7, (AppellF1(1 + m, -(1/2), 1, 2 + m, -((b*tan(c + d*x))/a), (-im)*tan(c + d*x))*tan(c + d*x)^(1 + m)*sqrt(a + b*tan(c + d*x)))/(2*d*(1 + m)*sqrt(1 + (b*tan(c + d*x))/a)) + (AppellF1(1 + m, -(1/2), 1, 2 + m, -((b*tan(c + d*x))/a), im*tan(c + d*x))*tan(c + d*x)^(1 + m)*sqrt(a + b*tan(c + d*x)))/(2*d*(1 + m)*sqrt(1 + (b*tan(c + d*x))/a))]
    @test_int [tan(c + d*x)^m/(a + b*tan(c + d*x))^(1/2), x, 7, (AppellF1(1 + m, 1/2, 1, 2 + m, -((b*tan(c + d*x))/a), (-im)*tan(c + d*x))*tan(c + d*x)^(1 + m)*sqrt(1 + (b*tan(c + d*x))/a))/(2*d*(1 + m)*sqrt(a + b*tan(c + d*x))) + (AppellF1(1 + m, 1/2, 1, 2 + m, -((b*tan(c + d*x))/a), im*tan(c + d*x))*tan(c + d*x)^(1 + m)*sqrt(1 + (b*tan(c + d*x))/a))/(2*d*(1 + m)*sqrt(a + b*tan(c + d*x)))]
    @test_int [tan(c + d*x)^m/(a + b*tan(c + d*x))^(3/2), x, 7, (AppellF1(1 + m, 3/2, 1, 2 + m, -((b*tan(c + d*x))/a), (-im)*tan(c + d*x))*tan(c + d*x)^(1 + m)*sqrt(1 + (b*tan(c + d*x))/a))/(2*a*d*(1 + m)*sqrt(a + b*tan(c + d*x))) + (AppellF1(1 + m, 3/2, 1, 2 + m, -((b*tan(c + d*x))/a), im*tan(c + d*x))*tan(c + d*x)^(1 + m)*sqrt(1 + (b*tan(c + d*x))/a))/(2*a*d*(1 + m)*sqrt(a + b*tan(c + d*x)))]


    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*(a+b*tan(e+f*x))^m*(d*tan(e+f*x))^n*with*m*symbolic=#


    @test_int [(a + b*tan(e + f*x))^m*(d*tan(e + f*x))^n, x, 7, (AppellF1(1 + n, -m, 1, 2 + n, -((b*tan(e + f*x))/a), (-im)*tan(e + f*x))*(d*tan(e + f*x))^(1 + n)*(a + b*tan(e + f*x))^m)/((1 + (b*tan(e + f*x))/a)^m*(2*d*f*(1 + n))) + (AppellF1(1 + n, -m, 1, 2 + n, -((b*tan(e + f*x))/a), im*tan(e + f*x))*(d*tan(e + f*x))^(1 + n)*(a + b*tan(e + f*x))^m)/((1 + (b*tan(e + f*x))/a)^m*(2*d*f*(1 + n)))]


    @test_int [tan(c + d*x)^4*(a + b*tan(c + d*x))^n, x, 8, ((2*a^2 - b^2*(2 + n)*(3 + n))*(a + b*tan(c + d*x))^(1 + n))/(b^3*d*(1 + n)*(2 + n)*(3 + n)) - (sqrt(-b^2)*HypergeometricFunctions._₂F₁(1, 1 + n, 2 + n, (a + b*tan(c + d*x))/(a - sqrt(-b^2)))*(a + b*tan(c + d*x))^(1 + n))/(2*b*(a - sqrt(-b^2))*d*(1 + n)) + (sqrt(-b^2)*HypergeometricFunctions._₂F₁(1, 1 + n, 2 + n, (a + b*tan(c + d*x))/(a + sqrt(-b^2)))*(a + b*tan(c + d*x))^(1 + n))/(2*b*(a + sqrt(-b^2))*d*(1 + n)) - (2*a*tan(c + d*x)*(a + b*tan(c + d*x))^(1 + n))/(b^2*d*(2 + n)*(3 + n)) + (tan(c + d*x)^2*(a + b*tan(c + d*x))^(1 + n))/(b*d*(3 + n))]
    @test_int [tan(c + d*x)^3*(a + b*tan(c + d*x))^n, x, 8, -((a*(a + b*tan(c + d*x))^(1 + n))/(b^2*d*(1 + n)*(2 + n))) + (HypergeometricFunctions._₂F₁(1, 1 + n, 2 + n, (a + b*tan(c + d*x))/(a - im*b))*(a + b*tan(c + d*x))^(1 + n))/(2*(a - im*b)*d*(1 + n)) + (HypergeometricFunctions._₂F₁(1, 1 + n, 2 + n, (a + b*tan(c + d*x))/(a + im*b))*(a + b*tan(c + d*x))^(1 + n))/(2*(a + im*b)*d*(1 + n)) + (tan(c + d*x)*(a + b*tan(c + d*x))^(1 + n))/(b*d*(2 + n))]
    @test_int [tan(c + d*x)^2*(a + b*tan(c + d*x))^n, x, 6, (a + b*tan(c + d*x))^(1 + n)/(b*d*(1 + n)) - (b*HypergeometricFunctions._₂F₁(1, 1 + n, 2 + n, (a + b*tan(c + d*x))/(a - sqrt(-b^2)))*(a + b*tan(c + d*x))^(1 + n))/(2*sqrt(-b^2)*(a - sqrt(-b^2))*d*(1 + n)) + (b*HypergeometricFunctions._₂F₁(1, 1 + n, 2 + n, (a + b*tan(c + d*x))/(a + sqrt(-b^2)))*(a + b*tan(c + d*x))^(1 + n))/(2*sqrt(-b^2)*(a + sqrt(-b^2))*d*(1 + n))]
    @test_int [tan(c + d*x)^1*(a + b*tan(c + d*x))^n, x, 5, -((HypergeometricFunctions._₂F₁(1, 1 + n, 2 + n, (a + b*tan(c + d*x))/(a - im*b))*(a + b*tan(c + d*x))^(1 + n))/(2*(a - im*b)*d*(1 + n))) - (HypergeometricFunctions._₂F₁(1, 1 + n, 2 + n, (a + b*tan(c + d*x))/(a + im*b))*(a + b*tan(c + d*x))^(1 + n))/(2*(a + im*b)*d*(1 + n))]
    @test_int [tan(c + d*x)^0*(a + b*tan(c + d*x))^n, x, 5, (b*HypergeometricFunctions._₂F₁(1, 1 + n, 2 + n, (a + b*tan(c + d*x))/(a - sqrt(-b^2)))*(a + b*tan(c + d*x))^(1 + n))/(2*sqrt(-b^2)*(a - sqrt(-b^2))*d*(1 + n)) - (b*HypergeometricFunctions._₂F₁(1, 1 + n, 2 + n, (a + b*tan(c + d*x))/(a + sqrt(-b^2)))*(a + b*tan(c + d*x))^(1 + n))/(2*sqrt(-b^2)*(a + sqrt(-b^2))*d*(1 + n))]
    @test_int [cot(c + d*x)^1*(a + b*tan(c + d*x))^n, x, 8, (HypergeometricFunctions._₂F₁(1, 1 + n, 2 + n, (a + b*tan(c + d*x))/(a - im*b))*(a + b*tan(c + d*x))^(1 + n))/(2*(a - im*b)*d*(1 + n)) + (HypergeometricFunctions._₂F₁(1, 1 + n, 2 + n, (a + b*tan(c + d*x))/(a + im*b))*(a + b*tan(c + d*x))^(1 + n))/(2*(a + im*b)*d*(1 + n)) - (HypergeometricFunctions._₂F₁(1, 1 + n, 2 + n, 1 + (b*tan(c + d*x))/a)*(a + b*tan(c + d*x))^(1 + n))/(a*d*(1 + n))]
    @test_int [cot(c + d*x)^2*(a + b*tan(c + d*x))^n, x, 10, -((cot(c + d*x)*(a + b*tan(c + d*x))^(1 + n))/(a*d)) - (b*HypergeometricFunctions._₂F₁(1, 1 + n, 2 + n, (a + b*tan(c + d*x))/(a - sqrt(-b^2)))*(a + b*tan(c + d*x))^(1 + n))/(2*sqrt(-b^2)*(a - sqrt(-b^2))*d*(1 + n)) + (b*HypergeometricFunctions._₂F₁(1, 1 + n, 2 + n, (a + b*tan(c + d*x))/(a + sqrt(-b^2)))*(a + b*tan(c + d*x))^(1 + n))/(2*sqrt(-b^2)*(a + sqrt(-b^2))*d*(1 + n)) - (b*n*HypergeometricFunctions._₂F₁(1, 1 + n, 2 + n, 1 + (b*tan(c + d*x))/a)*(a + b*tan(c + d*x))^(1 + n))/(a^2*d*(1 + n))]
    @test_int [cot(c + d*x)^3*(a + b*tan(c + d*x))^n, x, 11, (b*(1 - n)*cot(c + d*x)*(a + b*tan(c + d*x))^(1 + n))/(2*a^2*d) - (cot(c + d*x)^2*(a + b*tan(c + d*x))^(1 + n))/(2*a*d) - (HypergeometricFunctions._₂F₁(1, 1 + n, 2 + n, (a + b*tan(c + d*x))/(a - im*b))*(a + b*tan(c + d*x))^(1 + n))/(2*(a - im*b)*d*(1 + n)) - (HypergeometricFunctions._₂F₁(1, 1 + n, 2 + n, (a + b*tan(c + d*x))/(a + im*b))*(a + b*tan(c + d*x))^(1 + n))/(2*(a + im*b)*d*(1 + n)) + ((2*a^2 + b^2*(1 - n)*n)*HypergeometricFunctions._₂F₁(1, 1 + n, 2 + n, 1 + (b*tan(c + d*x))/a)*(a + b*tan(c + d*x))^(1 + n))/(2*a^3*d*(1 + n))]


    @test_int [tan(c + d*x)^(3/2)*(a + b*tan(c + d*x))^n, x, 9, (AppellF1(5/2, 1, -n, 7/2, (-im)*tan(c + d*x), -((b*tan(c + d*x))/a))*tan(c + d*x)^(5/2)*(a + b*tan(c + d*x))^n)/((1 + (b*tan(c + d*x))/a)^n*(5*d)) + (AppellF1(5/2, 1, -n, 7/2, im*tan(c + d*x), -((b*tan(c + d*x))/a))*tan(c + d*x)^(5/2)*(a + b*tan(c + d*x))^n)/((1 + (b*tan(c + d*x))/a)^n*(5*d))]
    @test_int [tan(c + d*x)^(1/2)*(a + b*tan(c + d*x))^n, x, 9, (AppellF1(3/2, 1, -n, 5/2, (-im)*tan(c + d*x), -((b*tan(c + d*x))/a))*tan(c + d*x)^(3/2)*(a + b*tan(c + d*x))^n)/((1 + (b*tan(c + d*x))/a)^n*(3*d)) + (AppellF1(3/2, 1, -n, 5/2, im*tan(c + d*x), -((b*tan(c + d*x))/a))*tan(c + d*x)^(3/2)*(a + b*tan(c + d*x))^n)/((1 + (b*tan(c + d*x))/a)^n*(3*d))]
    @test_int [tan(c + d*x)^(-1/2)*(a + b*tan(c + d*x))^n, x, 9, (AppellF1(1/2, 1, -n, 3/2, (-im)*tan(c + d*x), -((b*tan(c + d*x))/a))*sqrt(tan(c + d*x))*(a + b*tan(c + d*x))^n)/((1 + (b*tan(c + d*x))/a)^n*d) + (AppellF1(1/2, 1, -n, 3/2, im*tan(c + d*x), -((b*tan(c + d*x))/a))*sqrt(tan(c + d*x))*(a + b*tan(c + d*x))^n)/((1 + (b*tan(c + d*x))/a)^n*d)]
    @test_int [tan(c + d*x)^(-3/2)*(a + b*tan(c + d*x))^n, x, 9, -((AppellF1(-(1/2), 1, -n, 1/2, (-im)*tan(c + d*x), -((b*tan(c + d*x))/a))*(a + b*tan(c + d*x))^n)/((1 + (b*tan(c + d*x))/a)^n*(d*sqrt(tan(c + d*x))))) - (AppellF1(-(1/2), 1, -n, 1/2, im*tan(c + d*x), -((b*tan(c + d*x))/a))*(a + b*tan(c + d*x))^n)/((1 + (b*tan(c + d*x))/a)^n*(d*sqrt(tan(c + d*x))))]


    #= ::Title::Closed:: =#
    #=Integrands*of*the*form*(a+b*tan(e+f*x))^m*(d*cot(e+f*x))^n=#


    #= ::Section::Closed:: =#
    #=Integrands*of*the*form*(a+im*a*tan(e+f*x))^m*(d*cot(e+f*x))^n=#


    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*(a+im*a*tan(e+f*x))^m*(d*cot(e+f*x))^(n/2)=#


    #= ::Subsubsection::Closed:: =#
    #=m>0=#


    @test_int [cot(c + d*x)^(5/2)*(a + im*a*tan(c + d*x)), x, 5, (-2*(-1)^(3/4)*a*atanh((-1)^(3/4)*sqrt(cot(c + d*x))))/d - ((2*im)*a*sqrt(cot(c + d*x)))/d - (2*a*cot(c + d*x)^(3/2))/(3*d)]
    @test_int [cot(c + d*x)^(3/2)*(a + im*a*tan(c + d*x)), x, 4, (-2*(-1)^(1/4)*a*atanh((-1)^(3/4)*sqrt(cot(c + d*x))))/d - (2*a*sqrt(cot(c + d*x)))/d]
    @test_int [sqrt(cot(c + d*x))*(a + im*a*tan(c + d*x)), x, 3, (2*(-1)^(3/4)*a*atanh((-1)^(3/4)*sqrt(cot(c + d*x))))/d]
    @test_int [(a + im*a*tan(c + d*x))/sqrt(cot(c + d*x)), x, 4, (2*(-1)^(1/4)*a*atanh((-1)^(3/4)*sqrt(cot(c + d*x))))/d + ((2*im)*a)/(d*sqrt(cot(c + d*x)))]
    @test_int [(a + im*a*tan(c + d*x))/cot(c + d*x)^(3/2), x, 5, (-2*(-1)^(3/4)*a*atanh((-1)^(3/4)*sqrt(cot(c + d*x))))/d + (((2*im)/3)*a)/(d*cot(c + d*x)^(3/2)) + (2*a)/(d*sqrt(cot(c + d*x)))]


    @test_int [cot(c + d*x)^(7/2)*(a + im*a*tan(c + d*x))^2, x, 6, (4*(-1)^(1/4)*a^2*atanh((-1)^(3/4)*sqrt(cot(c + d*x))))/d + (4*a^2*sqrt(cot(c + d*x)))/d - (4*im*a^2*cot(c + d*x)^(3/2))/(3*d) - (2*a^2*cot(c + d*x)^(5/2))/(5*d)]
    @test_int [cot(c + d*x)^(5/2)*(a + im*a*tan(c + d*x))^2, x, 5, -((4*(-1)^(3/4)*a^2*atanh((-1)^(3/4)*sqrt(cot(c + d*x))))/d) - (4*im*a^2*sqrt(cot(c + d*x)))/d - (2*a^2*cot(c + d*x)^(3/2))/(3*d)]
    @test_int [cot(c + d*x)^(3/2)*(a + im*a*tan(c + d*x))^2, x, 4, -((4*(-1)^(1/4)*a^2*atanh((-1)^(3/4)*sqrt(cot(c + d*x))))/d) - (2*a^2*sqrt(cot(c + d*x)))/d]
    @test_int [sqrt(cot(c + d*x))*(a + im*a*tan(c + d*x))^2, x, 4, (4*(-1)^(3/4)*a^2*atanh((-1)^(3/4)*sqrt(cot(c + d*x))))/d - (2*a^2)/(d*sqrt(cot(c + d*x)))]
    @test_int [(a + im*a*tan(c + d*x))^2/sqrt(cot(c + d*x)), x, 5, (4*(-1)^(1/4)*a^2*atanh((-1)^(3/4)*sqrt(cot(c + d*x))))/d - (2*a^2)/(3*d*cot(c + d*x)^(3/2)) + (4*im*a^2)/(d*sqrt(cot(c + d*x)))]
    @test_int [(a + im*a*tan(c + d*x))^2/cot(c + d*x)^(3/2), x, 6, -((4*(-1)^(3/4)*a^2*atanh((-1)^(3/4)*sqrt(cot(c + d*x))))/d) - (2*a^2)/(5*d*cot(c + d*x)^(5/2)) + (4*im*a^2)/(3*d*cot(c + d*x)^(3/2)) + (4*a^2)/(d*sqrt(cot(c + d*x)))]


    @test_int [cot(c + d*x)^(7/2)*(a + im*a*tan(c + d*x))^3, x, 6, (8*(-1)^(1/4)*a^3*atanh((-1)^(3/4)*sqrt(cot(c + d*x))))/d + (8*a^3*sqrt(cot(c + d*x)))/d - (8*im*a^3*cot(c + d*x)^(3/2))/(5*d) - (2*cot(c + d*x)^(3/2)*(im*a^3 + a^3*cot(c + d*x)))/(5*d)]
    @test_int [cot(c + d*x)^(5/2)*(a + im*a*tan(c + d*x))^3, x, 5, -((8*(-1)^(3/4)*a^3*atanh((-1)^(3/4)*sqrt(cot(c + d*x))))/d) - (16*im*a^3*sqrt(cot(c + d*x)))/(3*d) - (2*sqrt(cot(c + d*x))*(im*a^3 + a^3*cot(c + d*x)))/(3*d)]
    @test_int [cot(c + d*x)^(3/2)*(a + im*a*tan(c + d*x))^3, x, 5, (-8*(-1)^(1/4)*a^3*atanh((-1)^(3/4)*sqrt(cot(c + d*x))))/d - (2*(im*a^3 + a^3*cot(c + d*x)))/(d*sqrt(cot(c + d*x)))]
    @test_int [sqrt(cot(c + d*x))*(a + im*a*tan(c + d*x))^3, x, 5, (8*(-1)^(3/4)*a^3*atanh((-1)^(3/4)*sqrt(cot(c + d*x))))/d - (16*a^3)/(3*d*sqrt(cot(c + d*x))) - (2*(im*a^3 + a^3*cot(c + d*x)))/(3*d*cot(c + d*x)^(3/2))]
    @test_int [(a + im*a*tan(c + d*x))^3/sqrt(cot(c + d*x)), x, 6, (8*(-1)^(1/4)*a^3*atanh((-1)^(3/4)*sqrt(cot(c + d*x))))/d - (8*a^3)/(5*d*cot(c + d*x)^(3/2)) + (8*im*a^3)/(d*sqrt(cot(c + d*x))) - (2*(im*a^3 + a^3*cot(c + d*x)))/(5*d*cot(c + d*x)^(5/2))]


    #= ::Subsubsection::Closed:: =#
    #=m<0=#


    @test_int [cot(c + d*x)^(3/2)/(a + im*a*tan(c + d*x)), x, 13, -(((5/4 + (3*im)/4)*atan(1 - sqrt(2)*sqrt(cot(c + d*x))))/(sqrt(2)*a*d)) + ((5/4 + (3*im)/4)*atan(1 + sqrt(2)*sqrt(cot(c + d*x))))/(sqrt(2)*a*d) - (5*sqrt(cot(c + d*x)))/(2*a*d) + cot(c + d*x)^(3/2)/(2*d*(im*a + a*cot(c + d*x))) - ((5/8 - (3*im)/8)*log(1 - sqrt(2)*sqrt(cot(c + d*x)) + cot(c + d*x)))/(sqrt(2)*a*d) + ((5/8 - (3*im)/8)*log(1 + sqrt(2)*sqrt(cot(c + d*x)) + cot(c + d*x)))/(sqrt(2)*a*d)]
    @test_int [sqrt(cot(c + d*x))/(a + im*a*tan(c + d*x)), x, 12, ((3/4 - im/4)*atan(1 - sqrt(2)*sqrt(cot(c + d*x))))/(sqrt(2)*a*d) - ((3/4 - im/4)*atan(1 + sqrt(2)*sqrt(cot(c + d*x))))/(sqrt(2)*a*d) + sqrt(cot(c + d*x))/(2*d*(im*a + a*cot(c + d*x))) - ((3/8 + im/8)*log(1 - sqrt(2)*sqrt(cot(c + d*x)) + cot(c + d*x)))/(sqrt(2)*a*d) + ((3/8 + im/8)*log(1 + sqrt(2)*sqrt(cot(c + d*x)) + cot(c + d*x)))/(sqrt(2)*a*d)]
    @test_int [1/(sqrt(cot(c + d*x))*(a + im*a*tan(c + d*x))), x, 4, ((-1)^(1/4)*atanh((-1)^(3/4)*sqrt(cot(c + d*x))))/(2*a*d) + (im*sqrt(cot(c + d*x)))/(2*d*(im*a + a*cot(c + d*x)))]
    @test_int [1/(cot(c + d*x)^(3/2)*(a + im*a*tan(c + d*x))), x, 12, ((1/4 - (3*im)/4)*atan(1 - sqrt(2)*sqrt(cot(c + d*x))))/(sqrt(2)*a*d) - ((1/4 - (3*im)/4)*atan(1 + sqrt(2)*sqrt(cot(c + d*x))))/(sqrt(2)*a*d) - sqrt(cot(c + d*x))/(2*d*(im*a + a*cot(c + d*x))) - ((1/8 + (3*im)/8)*log(1 - sqrt(2)*sqrt(cot(c + d*x)) + cot(c + d*x)))/(sqrt(2)*a*d) + ((1/8 + (3*im)/8)*log(1 + sqrt(2)*sqrt(cot(c + d*x)) + cot(c + d*x)))/(sqrt(2)*a*d)]
    @test_int [1/(cot(c + d*x)^(5/2)*(a + im*a*tan(c + d*x))), x, 13, ((3/4 + (5*im)/4)*atan(1 - sqrt(2)*sqrt(cot(c + d*x))))/(sqrt(2)*a*d) - ((3/4 + (5*im)/4)*atan(1 + sqrt(2)*sqrt(cot(c + d*x))))/(sqrt(2)*a*d) - (5*im)/(2*a*d*sqrt(cot(c + d*x))) - 1/(2*d*sqrt(cot(c + d*x))*(im*a + a*cot(c + d*x))) + ((3/8 - (5*im)/8)*log(1 - sqrt(2)*sqrt(cot(c + d*x)) + cot(c + d*x)))/(sqrt(2)*a*d) - ((3/8 - (5*im)/8)*log(1 + sqrt(2)*sqrt(cot(c + d*x)) + cot(c + d*x)))/(sqrt(2)*a*d)]


    @test_int [cot(c + d*x)^(3/2)/(a + im*a*tan(c + d*x))^2, x, 14, -(((25/16 + (21*im)/16)*atan(1 - sqrt(2)*sqrt(cot(c + d*x))))/(sqrt(2)*a^2*d)) + ((25/16 + (21*im)/16)*atan(1 + sqrt(2)*sqrt(cot(c + d*x))))/(sqrt(2)*a^2*d) - (25*sqrt(cot(c + d*x)))/(8*a^2*d) + (7*cot(c + d*x)^(3/2))/(8*a^2*d*(im + cot(c + d*x))) + cot(c + d*x)^(5/2)/(4*d*(im*a + a*cot(c + d*x))^2) - ((25/32 - (21*im)/32)*log(1 - sqrt(2)*sqrt(cot(c + d*x)) + cot(c + d*x)))/(sqrt(2)*a^2*d) + ((25/32 - (21*im)/32)*log(1 + sqrt(2)*sqrt(cot(c + d*x)) + cot(c + d*x)))/(sqrt(2)*a^2*d)]
    @test_int [sqrt(cot(c + d*x))/(a + im*a*tan(c + d*x))^2, x, 13, ((9/16 - (5*im)/16)*atan(1 - sqrt(2)*sqrt(cot(c + d*x))))/(sqrt(2)*a^2*d) - ((9/16 - (5*im)/16)*atan(1 + sqrt(2)*sqrt(cot(c + d*x))))/(sqrt(2)*a^2*d) + (5*sqrt(cot(c + d*x)))/(8*a^2*d*(im + cot(c + d*x))) + cot(c + d*x)^(3/2)/(4*d*(im*a + a*cot(c + d*x))^2) - ((9/32 + (5*im)/32)*log(1 - sqrt(2)*sqrt(cot(c + d*x)) + cot(c + d*x)))/(sqrt(2)*a^2*d) + ((9/32 + (5*im)/32)*log(1 + sqrt(2)*sqrt(cot(c + d*x)) + cot(c + d*x)))/(sqrt(2)*a^2*d)]
    @test_int [1/(sqrt(cot(c + d*x))*(a + im*a*tan(c + d*x))^2), x, 13, ((1/16 - (3*im)/16)*atan(1 - sqrt(2)*sqrt(cot(c + d*x))))/(sqrt(2)*a^2*d) - ((1/16 - (3*im)/16)*atan(1 + sqrt(2)*sqrt(cot(c + d*x))))/(sqrt(2)*a^2*d) + (3*im*sqrt(cot(c + d*x)))/(8*a^2*d*(im + cot(c + d*x))) + sqrt(cot(c + d*x))/(4*d*(im*a + a*cot(c + d*x))^2) + ((1/32 + (3*im)/32)*log(1 - sqrt(2)*sqrt(cot(c + d*x)) + cot(c + d*x)))/(sqrt(2)*a^2*d) - ((1/32 + (3*im)/32)*log(1 + sqrt(2)*sqrt(cot(c + d*x)) + cot(c + d*x)))/(sqrt(2)*a^2*d)]
    @test_int [1/(cot(c + d*x)^(3/2)*(a + im*a*tan(c + d*x))^2), x, 13, -(((1/16 + (3*im)/16)*atan(1 - sqrt(2)*sqrt(cot(c + d*x))))/(sqrt(2)*a^2*d)) + ((1/16 + (3*im)/16)*atan(1 + sqrt(2)*sqrt(cot(c + d*x))))/(sqrt(2)*a^2*d) + sqrt(cot(c + d*x))/(8*a^2*d*(im + cot(c + d*x))) + (im*sqrt(cot(c + d*x)))/(4*d*(im*a + a*cot(c + d*x))^2) + ((1/32 - (3*im)/32)*log(1 - sqrt(2)*sqrt(cot(c + d*x)) + cot(c + d*x)))/(sqrt(2)*a^2*d) - ((1/32 - (3*im)/32)*log(1 + sqrt(2)*sqrt(cot(c + d*x)) + cot(c + d*x)))/(sqrt(2)*a^2*d)]
    @test_int [1/(cot(c + d*x)^(5/2)*(a + im*a*tan(c + d*x))^2), x, 13, -(((9/16 + (5*im)/16)*atan(1 - sqrt(2)*sqrt(cot(c + d*x))))/(sqrt(2)*a^2*d)) + ((9/16 + (5*im)/16)*atan(1 + sqrt(2)*sqrt(cot(c + d*x))))/(sqrt(2)*a^2*d) + (5*im*sqrt(cot(c + d*x)))/(8*a^2*d*(im + cot(c + d*x))) - sqrt(cot(c + d*x))/(4*d*(im*a + a*cot(c + d*x))^2) - ((9/32 - (5*im)/32)*log(1 - sqrt(2)*sqrt(cot(c + d*x)) + cot(c + d*x)))/(sqrt(2)*a^2*d) + ((9/32 - (5*im)/32)*log(1 + sqrt(2)*sqrt(cot(c + d*x)) + cot(c + d*x)))/(sqrt(2)*a^2*d)]
    @test_int [1/(cot(c + d*x)^(7/2)*(a + im*a*tan(c + d*x))^2), x, 14, ((25/16 - (21*im)/16)*atan(1 - sqrt(2)*sqrt(cot(c + d*x))))/(sqrt(2)*a^2*d) - ((25/16 - (21*im)/16)*atan(1 + sqrt(2)*sqrt(cot(c + d*x))))/(sqrt(2)*a^2*d) - 25/(8*a^2*d*sqrt(cot(c + d*x))) + (7*im)/(8*a^2*d*sqrt(cot(c + d*x))*(im + cot(c + d*x))) - 1/(4*d*sqrt(cot(c + d*x))*(im*a + a*cot(c + d*x))^2) - ((25/32 + (21*im)/32)*log(1 - sqrt(2)*sqrt(cot(c + d*x)) + cot(c + d*x)))/(sqrt(2)*a^2*d) + ((25/32 + (21*im)/32)*log(1 + sqrt(2)*sqrt(cot(c + d*x)) + cot(c + d*x)))/(sqrt(2)*a^2*d)]


    @test_int [sqrt(cot(c + d*x))/(a + im*a*tan(c + d*x))^3, x, 14, ((7/16 - (5*im)/16)*atan(1 - sqrt(2)*sqrt(cot(c + d*x))))/(sqrt(2)*a^3*d) - ((7/16 - (5*im)/16)*atan(1 + sqrt(2)*sqrt(cot(c + d*x))))/(sqrt(2)*a^3*d) + cot(c + d*x)^(5/2)/(6*d*(im*a + a*cot(c + d*x))^3) + cot(c + d*x)^(3/2)/(3*a*d*(im*a + a*cot(c + d*x))^2) + (5*sqrt(cot(c + d*x)))/(8*d*(im*a^3 + a^3*cot(c + d*x))) - ((7/32 + (5*im)/32)*log(1 - sqrt(2)*sqrt(cot(c + d*x)) + cot(c + d*x)))/(sqrt(2)*a^3*d) + ((7/32 + (5*im)/32)*log(1 + sqrt(2)*sqrt(cot(c + d*x)) + cot(c + d*x)))/(sqrt(2)*a^3*d)]
    @test_int [1/(sqrt(cot(c + d*x))*(a + im*a*tan(c + d*x))^3), x, 16, -((im*atan(1 - sqrt(2)*sqrt(cot(c + d*x))))/(8*sqrt(2)*a^3*d)) + (im*atan(1 + sqrt(2)*sqrt(cot(c + d*x))))/(8*sqrt(2)*a^3*d) + cot(c + d*x)^(3/2)/(6*d*(im*a + a*cot(c + d*x))^3) + sqrt(cot(c + d*x))/(4*a*d*(im*a + a*cot(c + d*x))^2) + (im*sqrt(cot(c + d*x)))/(4*d*(im*a^3 + a^3*cot(c + d*x))) + (im*log(1 - sqrt(2)*sqrt(cot(c + d*x)) + cot(c + d*x)))/(16*sqrt(2)*a^3*d) - (im*log(1 + sqrt(2)*sqrt(cot(c + d*x)) + cot(c + d*x)))/(16*sqrt(2)*a^3*d)]
    @test_int [1/(cot(c + d*x)^(3/2)*(a + im*a*tan(c + d*x))^3), x, 7, -(((-1)^(3/4)*atanh((-1)^(3/4)*sqrt(cot(c + d*x))))/(8*a^3*d)) + sqrt(cot(c + d*x))/(6*d*(im*a + a*cot(c + d*x))^3) + (im*sqrt(cot(c + d*x)))/(6*a*d*(im*a + a*cot(c + d*x))^2) + sqrt(cot(c + d*x))/(8*d*(im*a^3 + a^3*cot(c + d*x)))]
    @test_int [1/(cot(c + d*x)^(5/2)*(a + im*a*tan(c + d*x))^3), x, 15, -(atan(1 - sqrt(2)*sqrt(cot(c + d*x)))/(8*sqrt(2)*a^3*d)) + atan(1 + sqrt(2)*sqrt(cot(c + d*x)))/(8*sqrt(2)*a^3*d) + (im*sqrt(cot(c + d*x)))/(6*d*(im*a + a*cot(c + d*x))^3) + sqrt(cot(c + d*x))/(12*a*d*(im*a + a*cot(c + d*x))^2) - log(1 - sqrt(2)*sqrt(cot(c + d*x)) + cot(c + d*x))/(16*sqrt(2)*a^3*d) + log(1 + sqrt(2)*sqrt(cot(c + d*x)) + cot(c + d*x))/(16*sqrt(2)*a^3*d)]
    @test_int [1/(cot(c + d*x)^(7/2)*(a + im*a*tan(c + d*x))^3), x, 14, -(((5/16 - (7*im)/16)*atan(1 - sqrt(2)*sqrt(cot(c + d*x))))/(sqrt(2)*a^3*d)) + ((5/16 - (7*im)/16)*atan(1 + sqrt(2)*sqrt(cot(c + d*x))))/(sqrt(2)*a^3*d) - sqrt(cot(c + d*x))/(6*d*(im*a + a*cot(c + d*x))^3) + (im*sqrt(cot(c + d*x)))/(3*a*d*(im*a + a*cot(c + d*x))^2) + (5*sqrt(cot(c + d*x)))/(8*d*(im*a^3 + a^3*cot(c + d*x))) + ((5/32 + (7*im)/32)*log(1 - sqrt(2)*sqrt(cot(c + d*x)) + cot(c + d*x)))/(sqrt(2)*a^3*d) - ((5/32 + (7*im)/32)*log(1 + sqrt(2)*sqrt(cot(c + d*x)) + cot(c + d*x)))/(sqrt(2)*a^3*d)]


    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*(a+im*a*tan(e+f*x))^(m/2)*(d*cot(e+f*x))^(n/2)=#


    #= ::Subsubsection::Closed:: =#
    #=m>0=#


    @test_int [cot(c + d*x)^(7/2)*sqrt(a + im*a*tan(c + d*x)), x, 7, ((-1 - im)*sqrt(a)*atanh(((1 + im)*sqrt(a)*sqrt(tan(c + d*x)))/sqrt(a + im*a*tan(c + d*x)))*sqrt(cot(c + d*x))*sqrt(tan(c + d*x)))/d + (26*sqrt(cot(c + d*x))*sqrt(a + im*a*tan(c + d*x)))/(15*d) - (((2*im)/15)*cot(c + d*x)^(3/2)*sqrt(a + im*a*tan(c + d*x)))/d - (2*cot(c + d*x)^(5/2)*sqrt(a + im*a*tan(c + d*x)))/(5*d)]
    @test_int [cot(c + d*x)^(5/2)*sqrt(a + im*a*tan(c + d*x)), x, 6, ((-1 + im)*sqrt(a)*atanh(((1 + im)*sqrt(a)*sqrt(tan(c + d*x)))/sqrt(a + im*a*tan(c + d*x)))*sqrt(cot(c + d*x))*sqrt(tan(c + d*x)))/d - (((2*im)/3)*sqrt(cot(c + d*x))*sqrt(a + im*a*tan(c + d*x)))/d - (2*cot(c + d*x)^(3/2)*sqrt(a + im*a*tan(c + d*x)))/(3*d)]
    @test_int [cot(c + d*x)^(3/2)*sqrt(a + im*a*tan(c + d*x)), x, 4, ((1 + im)*sqrt(a)*atanh(((1 + im)*sqrt(a)*sqrt(tan(c + d*x)))/sqrt(a + im*a*tan(c + d*x)))*sqrt(cot(c + d*x))*sqrt(tan(c + d*x)))/d - (2*sqrt(cot(c + d*x))*sqrt(a + im*a*tan(c + d*x)))/d]
    @test_int [sqrt(cot(c + d*x))*sqrt(a + im*a*tan(c + d*x)), x, 3, ((1 - im)*sqrt(a)*atanh(((1 + im)*sqrt(a)*sqrt(tan(c + d*x)))/sqrt(a + im*a*tan(c + d*x)))*sqrt(cot(c + d*x))*sqrt(tan(c + d*x)))/d]
    @test_int [sqrt(a + im*a*tan(c + d*x))/sqrt(cot(c + d*x)), x, 8, -((2*(-1)^(3/4)*sqrt(a)*atan(((-1)^(3/4)*sqrt(a)*sqrt(tan(c + d*x)))/sqrt(a + im*a*tan(c + d*x)))*sqrt(cot(c + d*x))*sqrt(tan(c + d*x)))/d) - ((1 + im)*sqrt(a)*atanh(((1 + im)*sqrt(a)*sqrt(tan(c + d*x)))/sqrt(a + im*a*tan(c + d*x)))*sqrt(cot(c + d*x))*sqrt(tan(c + d*x)))/d]
    @test_int [sqrt(a + im*a*tan(c + d*x))/cot(c + d*x)^(3/2), x, 10, -(((-1)^(1/4)*sqrt(a)*atan(((-1)^(3/4)*sqrt(a)*sqrt(tan(c + d*x)))/sqrt(a + im*a*tan(c + d*x)))*sqrt(cot(c + d*x))*sqrt(tan(c + d*x)))/d) - ((1 - im)*sqrt(a)*atanh(((1 + im)*sqrt(a)*sqrt(tan(c + d*x)))/sqrt(a + im*a*tan(c + d*x)))*sqrt(cot(c + d*x))*sqrt(tan(c + d*x)))/d + sqrt(a + im*a*tan(c + d*x))/(d*sqrt(cot(c + d*x)))]


    @test_int [cot(c + d*x)^(7/2)*(a + im*a*tan(c + d*x))^(3/2), x, 8, ((-2 - 2*im)*a^(3/2)*atanh(((1 + im)*sqrt(a)*sqrt(tan(c + d*x)))/sqrt(a + im*a*tan(c + d*x)))*sqrt(cot(c + d*x))*sqrt(tan(c + d*x)))/d - (((2*im)/5)*a^2*cot(c + d*x)^(3/2))/(d*sqrt(a + im*a*tan(c + d*x))) - (2*a^2*cot(c + d*x)^(5/2))/(5*d*sqrt(a + im*a*tan(c + d*x))) + (12*a*sqrt(cot(c + d*x))*sqrt(a + im*a*tan(c + d*x)))/(5*d) - (((4*im)/5)*a*cot(c + d*x)^(3/2)*sqrt(a + im*a*tan(c + d*x)))/d]
    @test_int [cot(c + d*x)^(5/2)*(a + im*a*tan(c + d*x))^(3/2), x, 5, ((-2 + 2*im)*a^(3/2)*atanh(((1 + im)*sqrt(a)*sqrt(tan(c + d*x)))/sqrt(a + im*a*tan(c + d*x)))*sqrt(cot(c + d*x))*sqrt(tan(c + d*x)))/d - ((2*im)*a*sqrt(cot(c + d*x))*sqrt(a + im*a*tan(c + d*x)))/d - (2*cot(c + d*x)^(3/2)*(a + im*a*tan(c + d*x))^(3/2))/(3*d)]
    @test_int [cot(c + d*x)^(3/2)*(a + im*a*tan(c + d*x))^(3/2), x, 4, ((2 + 2*im)*a^(3/2)*atanh(((1 + im)*sqrt(a)*sqrt(tan(c + d*x)))/sqrt(a + im*a*tan(c + d*x)))*sqrt(cot(c + d*x))*sqrt(tan(c + d*x)))/d - (2*a*sqrt(cot(c + d*x))*sqrt(a + im*a*tan(c + d*x)))/d]
    @test_int [sqrt(cot(c + d*x))*(a + im*a*tan(c + d*x))^(3/2), x, 8, (2*(-1)^(1/4)*a^(3/2)*atan(((-1)^(3/4)*sqrt(a)*sqrt(tan(c + d*x)))/sqrt(a + im*a*tan(c + d*x)))*sqrt(cot(c + d*x))*sqrt(tan(c + d*x)))/d + ((2 - 2*im)*a^(3/2)*atanh(((1 + im)*sqrt(a)*sqrt(tan(c + d*x)))/sqrt(a + im*a*tan(c + d*x)))*sqrt(cot(c + d*x))*sqrt(tan(c + d*x)))/d]
    @test_int [(a + im*a*tan(c + d*x))^(3/2)/sqrt(cot(c + d*x)), x, 10, -((3*(-1)^(3/4)*a^(3/2)*atan(((-1)^(3/4)*sqrt(a)*sqrt(tan(c + d*x)))/sqrt(a + im*a*tan(c + d*x)))*sqrt(cot(c + d*x))*sqrt(tan(c + d*x)))/d) - ((2 + 2*im)*a^(3/2)*atanh(((1 + im)*sqrt(a)*sqrt(tan(c + d*x)))/sqrt(a + im*a*tan(c + d*x)))*sqrt(cot(c + d*x))*sqrt(tan(c + d*x)))/d - a^2/(d*cot(c + d*x)^(3/2)*sqrt(a + im*a*tan(c + d*x))) + (im*a^2)/(d*sqrt(cot(c + d*x))*sqrt(a + im*a*tan(c + d*x)))]


    @test_int [cot(c + d*x)^(9/2)*(a + im*a*tan(c + d*x))^(5/2), x, 8, ((4 - 4*im)*a^(5/2)*atanh(((1 + im)*sqrt(a)*sqrt(tan(c + d*x)))/sqrt(a + im*a*tan(c + d*x)))*sqrt(cot(c + d*x))*sqrt(tan(c + d*x)))/d + (((104*im)/21)*a^2*sqrt(cot(c + d*x))*sqrt(a + im*a*tan(c + d*x)))/d + (32*a^2*cot(c + d*x)^(3/2)*sqrt(a + im*a*tan(c + d*x)))/(21*d) - (((6*im)/7)*a^2*cot(c + d*x)^(5/2)*sqrt(a + im*a*tan(c + d*x)))/d - (2*a^2*cot(c + d*x)^(7/2)*sqrt(a + im*a*tan(c + d*x)))/(7*d)]
    @test_int [cot(c + d*x)^(7/2)*(a + im*a*tan(c + d*x))^(5/2), x, 6, ((-4 - 4*im)*a^(5/2)*atanh(((1 + im)*sqrt(a)*sqrt(tan(c + d*x)))/sqrt(a + im*a*tan(c + d*x)))*sqrt(cot(c + d*x))*sqrt(tan(c + d*x)))/d + (4*a^2*sqrt(cot(c + d*x))*sqrt(a + im*a*tan(c + d*x)))/d - (((2*im)/3)*a*cot(c + d*x)^(3/2)*(a + im*a*tan(c + d*x))^(3/2))/d - (2*cot(c + d*x)^(5/2)*(a + im*a*tan(c + d*x))^(5/2))/(5*d)]
    @test_int [cot(c + d*x)^(5/2)*(a + im*a*tan(c + d*x))^(5/2), x, 5, ((-4 + 4*im)*a^(5/2)*atanh(((1 + im)*sqrt(a)*sqrt(tan(c + d*x)))/sqrt(a + im*a*tan(c + d*x)))*sqrt(cot(c + d*x))*sqrt(tan(c + d*x)))/d - ((4*im)*a^2*sqrt(cot(c + d*x))*sqrt(a + im*a*tan(c + d*x)))/d - (2*a*cot(c + d*x)^(3/2)*(a + im*a*tan(c + d*x))^(3/2))/(3*d)]
    @test_int [cot(c + d*x)^(3/2)*(a + im*a*tan(c + d*x))^(5/2), x, 9, (2*(-1)^(3/4)*a^(5/2)*atan(((-1)^(3/4)*sqrt(a)*sqrt(tan(c + d*x)))/sqrt(a + im*a*tan(c + d*x)))*sqrt(cot(c + d*x))*sqrt(tan(c + d*x)))/d + ((4 + 4*im)*a^(5/2)*atanh(((1 + im)*sqrt(a)*sqrt(tan(c + d*x)))/sqrt(a + im*a*tan(c + d*x)))*sqrt(cot(c + d*x))*sqrt(tan(c + d*x)))/d - (2*a^2*sqrt(cot(c + d*x))*sqrt(a + im*a*tan(c + d*x)))/d]
    @test_int [sqrt(cot(c + d*x))*(a + im*a*tan(c + d*x))^(5/2), x, 9, (5*(-1)^(1/4)*a^(5/2)*atan(((-1)^(3/4)*sqrt(a)*sqrt(tan(c + d*x)))/sqrt(a + im*a*tan(c + d*x)))*sqrt(cot(c + d*x))*sqrt(tan(c + d*x)))/d + ((4 - 4*im)*a^(5/2)*atanh(((1 + im)*sqrt(a)*sqrt(tan(c + d*x)))/sqrt(a + im*a*tan(c + d*x)))*sqrt(cot(c + d*x))*sqrt(tan(c + d*x)))/d - (a^2*sqrt(a + im*a*tan(c + d*x)))/(d*sqrt(cot(c + d*x)))]
    @test_int [(a + im*a*tan(c + d*x))^(5/2)/sqrt(cot(c + d*x)), x, 10, -((23*(-1)^(3/4)*a^(5/2)*atan(((-1)^(3/4)*sqrt(a)*sqrt(tan(c + d*x)))/sqrt(a + im*a*tan(c + d*x)))*sqrt(cot(c + d*x))*sqrt(tan(c + d*x)))/(4*d)) - ((4 + 4*im)*a^(5/2)*atanh(((1 + im)*sqrt(a)*sqrt(tan(c + d*x)))/sqrt(a + im*a*tan(c + d*x)))*sqrt(cot(c + d*x))*sqrt(tan(c + d*x)))/d - (a^2*sqrt(a + im*a*tan(c + d*x)))/(2*d*cot(c + d*x)^(3/2)) + (9*im*a^2*sqrt(a + im*a*tan(c + d*x)))/(4*d*sqrt(cot(c + d*x)))]


    #= ::Subsubsection::Closed:: =#
    #=m<0=#


    @test_int [cot(c + d*x)^(5/2)/sqrt(a + im*a*tan(c + d*x)), x, 7, -(((1/2 - im/2)*atanh(((1 + im)*sqrt(a)*sqrt(tan(c + d*x)))/sqrt(a + im*a*tan(c + d*x)))*sqrt(cot(c + d*x))*sqrt(tan(c + d*x)))/(sqrt(a)*d)) + cot(c + d*x)^(3/2)/(d*sqrt(a + im*a*tan(c + d*x))) + (7*im*sqrt(cot(c + d*x))*sqrt(a + im*a*tan(c + d*x)))/(3*a*d) - (5*cot(c + d*x)^(3/2)*sqrt(a + im*a*tan(c + d*x)))/(3*a*d)]
    @test_int [cot(c + d*x)^(3/2)/sqrt(a + im*a*tan(c + d*x)), x, 6, ((1/2 + im/2)*atanh(((1 + im)*sqrt(a)*sqrt(tan(c + d*x)))/sqrt(a + im*a*tan(c + d*x)))*sqrt(cot(c + d*x))*sqrt(tan(c + d*x)))/(sqrt(a)*d) + sqrt(cot(c + d*x))/(d*sqrt(a + im*a*tan(c + d*x))) - (3*sqrt(cot(c + d*x))*sqrt(a + im*a*tan(c + d*x)))/(a*d)]
    @test_int [sqrt(cot(c + d*x))/sqrt(a + im*a*tan(c + d*x)), x, 5, ((1/2 - im/2)*atanh(((1 + im)*sqrt(a)*sqrt(tan(c + d*x)))/sqrt(a + im*a*tan(c + d*x)))*sqrt(cot(c + d*x))*sqrt(tan(c + d*x)))/(sqrt(a)*d) + 1/(d*sqrt(cot(c + d*x))*sqrt(a + im*a*tan(c + d*x)))]
    @test_int [1/(sqrt(cot(c + d*x))*sqrt(a + im*a*tan(c + d*x))), x, 4, ((-1/2 - im/2)*atanh(((1 + im)*sqrt(a)*sqrt(tan(c + d*x)))/sqrt(a + im*a*tan(c + d*x)))*sqrt(cot(c + d*x))*sqrt(tan(c + d*x)))/(sqrt(a)*d) + im/(d*sqrt(cot(c + d*x))*sqrt(a + im*a*tan(c + d*x)))]
    @test_int [1/(cot(c + d*x)^(3/2)*sqrt(a + im*a*tan(c + d*x))), x, 9, -((2*(-1)^(1/4)*atan(((-1)^(3/4)*sqrt(a)*sqrt(tan(c + d*x)))/sqrt(a + im*a*tan(c + d*x)))*sqrt(cot(c + d*x))*sqrt(tan(c + d*x)))/(sqrt(a)*d)) - ((1/2 - im/2)*atanh(((1 + im)*sqrt(a)*sqrt(tan(c + d*x)))/sqrt(a + im*a*tan(c + d*x)))*sqrt(cot(c + d*x))*sqrt(tan(c + d*x)))/(sqrt(a)*d) - 1/(d*sqrt(cot(c + d*x))*sqrt(a + im*a*tan(c + d*x)))]
    @test_int [1/(cot(c + d*x)^(5/2)*sqrt(a + im*a*tan(c + d*x))), x, 10, -(((-1)^(3/4)*atan(((-1)^(3/4)*sqrt(a)*sqrt(tan(c + d*x)))/sqrt(a + im*a*tan(c + d*x)))*sqrt(cot(c + d*x))*sqrt(tan(c + d*x)))/(sqrt(a)*d)) + ((1/2 + im/2)*atanh(((1 + im)*sqrt(a)*sqrt(tan(c + d*x)))/sqrt(a + im*a*tan(c + d*x)))*sqrt(cot(c + d*x))*sqrt(tan(c + d*x)))/(sqrt(a)*d) - 1/(d*cot(c + d*x)^(3/2)*sqrt(a + im*a*tan(c + d*x))) - (2*im*sqrt(a + im*a*tan(c + d*x)))/(a*d*sqrt(cot(c + d*x)))]


    @test_int [cot(c + d*x)^(5/2)/(a + im*a*tan(c + d*x))^(3/2), x, 8, ((-1/4 + im/4)*atanh(((1 + im)*sqrt(a)*sqrt(tan(c + d*x)))/sqrt(a + im*a*tan(c + d*x)))*sqrt(cot(c + d*x))*sqrt(tan(c + d*x)))/(a^(3/2)*d) + cot(c + d*x)^(3/2)/(3*d*(a + im*a*tan(c + d*x))^(3/2)) + (5*cot(c + d*x)^(3/2))/(2*a*d*sqrt(a + im*a*tan(c + d*x))) + (((13*im)/2)*sqrt(cot(c + d*x))*sqrt(a + im*a*tan(c + d*x)))/(a^2*d) - (7*cot(c + d*x)^(3/2)*sqrt(a + im*a*tan(c + d*x)))/(2*a^2*d)]
    @test_int [cot(c + d*x)^(3/2)/(a + im*a*tan(c + d*x))^(3/2), x, 7, ((1/4 + im/4)*atanh(((1 + im)*sqrt(a)*sqrt(tan(c + d*x)))/sqrt(a + im*a*tan(c + d*x)))*sqrt(cot(c + d*x))*sqrt(tan(c + d*x)))/(a^(3/2)*d) + sqrt(cot(c + d*x))/(3*d*(a + im*a*tan(c + d*x))^(3/2)) + (11*sqrt(cot(c + d*x)))/(6*a*d*sqrt(a + im*a*tan(c + d*x))) - (25*sqrt(cot(c + d*x))*sqrt(a + im*a*tan(c + d*x)))/(6*a^2*d)]
    @test_int [sqrt(cot(c + d*x))/(a + im*a*tan(c + d*x))^(3/2), x, 6, ((1/4 - im/4)*atanh(((1 + im)*sqrt(a)*sqrt(tan(c + d*x)))/sqrt(a + im*a*tan(c + d*x)))*sqrt(cot(c + d*x))*sqrt(tan(c + d*x)))/(a^(3/2)*d) + 1/(3*d*sqrt(cot(c + d*x))*(a + im*a*tan(c + d*x))^(3/2)) + 7/(6*a*d*sqrt(cot(c + d*x))*sqrt(a + im*a*tan(c + d*x)))]
    @test_int [1/(sqrt(cot(c + d*x))*(a + im*a*tan(c + d*x))^(3/2)), x, 5, -(((1/4 + im/4)*atanh(((1 + im)*sqrt(a)*sqrt(tan(c + d*x)))/sqrt(a + im*a*tan(c + d*x)))*sqrt(cot(c + d*x))*sqrt(tan(c + d*x)))/(a^(3/2)*d)) + 1/(3*d*cot(c + d*x)^(3/2)*(a + im*a*tan(c + d*x))^(3/2)) + im/(2*a*d*sqrt(cot(c + d*x))*sqrt(a + im*a*tan(c + d*x)))]
    @test_int [1/(cot(c + d*x)^(3/2)*(a + im*a*tan(c + d*x))^(3/2)), x, 5, ((-1/4 + im/4)*atanh(((1 + im)*sqrt(a)*sqrt(tan(c + d*x)))/sqrt(a + im*a*tan(c + d*x)))*sqrt(cot(c + d*x))*sqrt(tan(c + d*x)))/(a^(3/2)*d) + (im/3)/(d*cot(c + d*x)^(3/2)*(a + im*a*tan(c + d*x))^(3/2)) + 1/(2*a*d*sqrt(cot(c + d*x))*sqrt(a + im*a*tan(c + d*x)))]
    @test_int [1/(cot(c + d*x)^(5/2)*(a + im*a*tan(c + d*x))^(3/2)), x, 10, (2*(-1)^(3/4)*atan(((-1)^(3/4)*sqrt(a)*sqrt(tan(c + d*x)))/sqrt(a + im*a*tan(c + d*x)))*sqrt(cot(c + d*x))*sqrt(tan(c + d*x)))/(a^(3/2)*d) + ((1/4 + im/4)*atanh(((1 + im)*sqrt(a)*sqrt(tan(c + d*x)))/sqrt(a + im*a*tan(c + d*x)))*sqrt(cot(c + d*x))*sqrt(tan(c + d*x)))/(a^(3/2)*d) - 1/(3*d*cot(c + d*x)^(3/2)*(a + im*a*tan(c + d*x))^(3/2)) + (3*im)/(2*a*d*sqrt(cot(c + d*x))*sqrt(a + im*a*tan(c + d*x)))]
    @test_int [1/(cot(c + d*x)^(7/2)*(a + im*a*tan(c + d*x))^(3/2)), x, 11, -((3*(-1)^(1/4)*atan(((-1)^(3/4)*sqrt(a)*sqrt(tan(c + d*x)))/sqrt(a + im*a*tan(c + d*x)))*sqrt(cot(c + d*x))*sqrt(tan(c + d*x)))/(a^(3/2)*d)) + ((1/4 - im/4)*atanh(((1 + im)*sqrt(a)*sqrt(tan(c + d*x)))/sqrt(a + im*a*tan(c + d*x)))*sqrt(cot(c + d*x))*sqrt(tan(c + d*x)))/(a^(3/2)*d) - 1/(3*d*cot(c + d*x)^(5/2)*(a + im*a*tan(c + d*x))^(3/2)) + (13*im)/(6*a*d*cot(c + d*x)^(3/2)*sqrt(a + im*a*tan(c + d*x))) - (7*sqrt(a + im*a*tan(c + d*x)))/(2*a^2*d*sqrt(cot(c + d*x)))]


    @test_int [cot(c + d*x)^(5/2)/(a + im*a*tan(c + d*x))^(5/2), x, 9, ((-1/8 + im/8)*atanh(((1 + im)*sqrt(a)*sqrt(tan(c + d*x)))/sqrt(a + im*a*tan(c + d*x)))*sqrt(cot(c + d*x))*sqrt(tan(c + d*x)))/(a^(5/2)*d) + cot(c + d*x)^(3/2)/(5*d*(a + im*a*tan(c + d*x))^(5/2)) + (7*cot(c + d*x)^(3/2))/(10*a*d*(a + im*a*tan(c + d*x))^(3/2)) + (89*cot(c + d*x)^(3/2))/(20*a^2*d*sqrt(a + im*a*tan(c + d*x))) + (((707*im)/60)*sqrt(cot(c + d*x))*sqrt(a + im*a*tan(c + d*x)))/(a^3*d) - (361*cot(c + d*x)^(3/2)*sqrt(a + im*a*tan(c + d*x)))/(60*a^3*d)]
    @test_int [cot(c + d*x)^(3/2)/(a + im*a*tan(c + d*x))^(5/2), x, 8, ((1/8 + im/8)*atanh(((1 + im)*sqrt(a)*sqrt(tan(c + d*x)))/sqrt(a + im*a*tan(c + d*x)))*sqrt(cot(c + d*x))*sqrt(tan(c + d*x)))/(a^(5/2)*d) + sqrt(cot(c + d*x))/(5*d*(a + im*a*tan(c + d*x))^(5/2)) + (17*sqrt(cot(c + d*x)))/(30*a*d*(a + im*a*tan(c + d*x))^(3/2)) + (151*sqrt(cot(c + d*x)))/(60*a^2*d*sqrt(a + im*a*tan(c + d*x))) - (317*sqrt(cot(c + d*x))*sqrt(a + im*a*tan(c + d*x)))/(60*a^3*d)]
    @test_int [sqrt(cot(c + d*x))/(a + im*a*tan(c + d*x))^(5/2), x, 7, ((1/8 - im/8)*atanh(((1 + im)*sqrt(a)*sqrt(tan(c + d*x)))/sqrt(a + im*a*tan(c + d*x)))*sqrt(cot(c + d*x))*sqrt(tan(c + d*x)))/(a^(5/2)*d) + 1/(5*d*sqrt(cot(c + d*x))*(a + im*a*tan(c + d*x))^(5/2)) + 13/(30*a*d*sqrt(cot(c + d*x))*(a + im*a*tan(c + d*x))^(3/2)) + 67/(60*a^2*d*sqrt(cot(c + d*x))*sqrt(a + im*a*tan(c + d*x)))]
    @test_int [1/(sqrt(cot(c + d*x))*(a + im*a*tan(c + d*x))^(5/2)), x, 7, ((-1/8 - im/8)*atanh(((1 + im)*sqrt(a)*sqrt(tan(c + d*x)))/sqrt(a + im*a*tan(c + d*x)))*sqrt(cot(c + d*x))*sqrt(tan(c + d*x)))/(a^(5/2)*d) + (im/5)/(d*sqrt(cot(c + d*x))*(a + im*a*tan(c + d*x))^(5/2)) + (im/10)/(a*d*sqrt(cot(c + d*x))*(a + im*a*tan(c + d*x))^(3/2)) - (im/20)/(a^2*d*sqrt(cot(c + d*x))*sqrt(a + im*a*tan(c + d*x)))]
    @test_int [1/(cot(c + d*x)^(3/2)*(a + im*a*tan(c + d*x))^(5/2)), x, 6, ((-1/8 + im/8)*atanh(((1 + im)*sqrt(a)*sqrt(tan(c + d*x)))/sqrt(a + im*a*tan(c + d*x)))*sqrt(cot(c + d*x))*sqrt(tan(c + d*x)))/(a^(5/2)*d) + 1/(5*d*cot(c + d*x)^(5/2)*(a + im*a*tan(c + d*x))^(5/2)) + (im/6)/(a*d*cot(c + d*x)^(3/2)*(a + im*a*tan(c + d*x))^(3/2)) + 1/(4*a^2*d*sqrt(cot(c + d*x))*sqrt(a + im*a*tan(c + d*x)))]
    @test_int [1/(cot(c + d*x)^(5/2)*(a + im*a*tan(c + d*x))^(5/2)), x, 6, ((1/8 + im/8)*atanh(((1 + im)*sqrt(a)*sqrt(tan(c + d*x)))/sqrt(a + im*a*tan(c + d*x)))*sqrt(cot(c + d*x))*sqrt(tan(c + d*x)))/(a^(5/2)*d) + (im/5)/(d*cot(c + d*x)^(5/2)*(a + im*a*tan(c + d*x))^(5/2)) + 1/(6*a*d*cot(c + d*x)^(3/2)*(a + im*a*tan(c + d*x))^(3/2)) - (im/4)/(a^2*d*sqrt(cot(c + d*x))*sqrt(a + im*a*tan(c + d*x)))]
    @test_int [1/(cot(c + d*x)^(7/2)*(a + im*a*tan(c + d*x))^(5/2)), x, 11, (2*(-1)^(1/4)*atan(((-1)^(3/4)*sqrt(a)*sqrt(tan(c + d*x)))/sqrt(a + im*a*tan(c + d*x)))*sqrt(cot(c + d*x))*sqrt(tan(c + d*x)))/(a^(5/2)*d) + ((1/8 - im/8)*atanh(((1 + im)*sqrt(a)*sqrt(tan(c + d*x)))/sqrt(a + im*a*tan(c + d*x)))*sqrt(cot(c + d*x))*sqrt(tan(c + d*x)))/(a^(5/2)*d) - 1/(5*d*cot(c + d*x)^(5/2)*(a + im*a*tan(c + d*x))^(5/2)) + im/(2*a*d*cot(c + d*x)^(3/2)*(a + im*a*tan(c + d*x))^(3/2)) + 7/(4*a^2*d*sqrt(cot(c + d*x))*sqrt(a + im*a*tan(c + d*x)))]


    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*(a+im*a*tan(e+f*x))^m*(d*cot(e+f*x))^n*with*n*symbolic=#


    @test_int [(d*cot(e + f*x))^n*(a + im*a*tan(e + f*x))^3, x, 6, If(13>=8, (im*a^3*d^2*(1 - 2*n)*(d*cot(e + f*x))^(-2 + n))/(f*(1 - n)*(2 - n)) + (d^2*(d*cot(e + f*x))^(-2 + n)*(im*a^3 + a^3*cot(e + f*x)))/(f*(1 - n)) - (4*im*a^3*d^2*(d*cot(e + f*x))^(-2 + n)*HypergeometricFunctions._₂F₁(1, -2 + n, -1 + n, (-im)*cot(e + f*x)))/(f*(2 - n)), (im*a^3*d^2*(1 - 2*n)*(d*cot(e + f*x))^(-2 + n))/(f*(2 - 3*n + n^2)) + (d^2*(d*cot(e + f*x))^(-2 + n)*(im*a^3 + a^3*cot(e + f*x)))/(f*(1 - n)) - (4*im*a^3*d^2*(d*cot(e + f*x))^(-2 + n)*HypergeometricFunctions._₂F₁(1, -2 + n, -1 + n, (-im)*cot(e + f*x)))/(f*(2 - n)))]
    @test_int [(d*cot(e + f*x))^n*(a + im*a*tan(e + f*x))^2, x, 5, (a^2*d*(d*cot(e + f*x))^(-1 + n))/(f*(1 - n)) - (2*a^2*d*(d*cot(e + f*x))^(-1 + n)*HypergeometricFunctions._₂F₁(1, -1 + n, n, (-im)*cot(e + f*x)))/(f*(1 - n))]
    @test_int [(d*cot(e + f*x))^n*(a + im*a*tan(e + f*x))^1, x, 3, -((im*a*(d*cot(e + f*x))^n*HypergeometricFunctions._₂F₁(1, n, 1 + n, (-im)*cot(e + f*x)))/(f*n))]
    @test_int [(d*cot(e + f*x))^n/(a + im*a*tan(e + f*x))^1, x, 7, -((d*cot(e + f*x))^(2 + n)/(2*d^2*f*(im*a + a*cot(e + f*x)))) - (im*n*(d*cot(e + f*x))^(2 + n)*HypergeometricFunctions._₂F₁(1, (2 + n)/2, (4 + n)/2, -cot(e + f*x)^2))/(2*a*d^2*f*(2 + n)) + ((1 + n)*(d*cot(e + f*x))^(3 + n)*HypergeometricFunctions._₂F₁(1, (3 + n)/2, (5 + n)/2, -cot(e + f*x)^2))/(2*a*d^3*f*(3 + n))]
    @test_int [(d*cot(e + f*x))^n/(a + im*a*tan(e + f*x))^2, x, 8, -((im*n*(d*cot(e + f*x))^(3 + n))/(4*a^2*d^3*f*(im + cot(e + f*x)))) - (d*cot(e + f*x))^(3 + n)/(4*d^3*f*(im*a + a*cot(e + f*x))^2) + ((1 + n)^2*(d*cot(e + f*x))^(3 + n)*HypergeometricFunctions._₂F₁(1, (3 + n)/2, (5 + n)/2, -cot(e + f*x)^2))/(4*a^2*d^3*f*(3 + n)) + (im*n*(2 + n)*(d*cot(e + f*x))^(4 + n)*HypergeometricFunctions._₂F₁(1, (4 + n)/2, (6 + n)/2, -cot(e + f*x)^2))/(4*a^2*d^4*f*(4 + n))]


    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*(a+im*a*tan(e+f*x))^m*(d*cot(e+f*x))^n*with*m*symbolic=#


    @test_int [(a + im*a*tan(e + f*x))^m*(d*cot(e + f*x))^n, x, 4, (AppellF1(1 - n, 1 - m, 1, 2 - n, (-im)*tan(e + f*x), im*tan(e + f*x))*(d*cot(e + f*x))^n*tan(e + f*x)*(a + im*a*tan(e + f*x))^m)/((1 + im*tan(e + f*x))^m*(f*(1 - n)))]


    @test_int [cot(c + d*x)^(3/2)*(a + im*a*tan(c + d*x))^n, x, 5, -((2*AppellF1(-(1/2), 1 - n, 1, 1/2, (-im)*tan(c + d*x), im*tan(c + d*x))*sqrt(cot(c + d*x))*(a + im*a*tan(c + d*x))^n)/((1 + im*tan(c + d*x))^n*d))]
    @test_int [cot(c + d*x)^(1/2)*(a + im*a*tan(c + d*x))^n, x, 5, (2*AppellF1(1/2, 1 - n, 1, 3/2, (-im)*tan(c + d*x), im*tan(c + d*x))*(a + im*a*tan(c + d*x))^n)/((1 + im*tan(c + d*x))^n*(d*sqrt(cot(c + d*x))))]
    @test_int [(a + im*a*tan(c + d*x))^n/cot(c + d*x)^(1/2), x, 5, (2*AppellF1(3/2, 1 - n, 1, 5/2, (-im)*tan(c + d*x), im*tan(c + d*x))*(a + im*a*tan(c + d*x))^n)/((1 + im*tan(c + d*x))^n*(3*d*cot(c + d*x)^(3/2)))]
    @test_int [(a + im*a*tan(c + d*x))^n/cot(c + d*x)^(3/2), x, 5, (2*AppellF1(5/2, 1 - n, 1, 7/2, (-im)*tan(c + d*x), im*tan(c + d*x))*(a + im*a*tan(c + d*x))^n)/((1 + im*tan(c + d*x))^n*(5*d*cot(c + d*x)^(5/2)))]


    #= ::Section:: =#
    #=Integrands*of*the*form*(a+a*tan(e+f*x))^m*(d*cot(e+f*x))^n=#


    #= ::Section::Closed:: =#
    #=Integrands*of*the*form*(a+b*tan(e+f*x))^m*(d*cot(e+f*x))^n=#


    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*(a+b*tan(e+f*x))^m*(d*cot(e+f*x))^(n/2)=#


    #= ::Subsubsection::Closed:: =#
    #=m>0=#


    @test_int [cot(c + d*x)^(7/2)*(a + b*tan(c + d*x)), x, 14, ((a - b)*atan(1 - sqrt(2)*sqrt(cot(c + d*x))))/(sqrt(2)*d) - ((a - b)*atan(1 + sqrt(2)*sqrt(cot(c + d*x))))/(sqrt(2)*d) + (2*a*sqrt(cot(c + d*x)))/d - (2*b*cot(c + d*x)^(3/2))/(3*d) - (2*a*cot(c + d*x)^(5/2))/(5*d) + ((a + b)*log(1 - sqrt(2)*sqrt(cot(c + d*x)) + cot(c + d*x)))/(2*sqrt(2)*d) - ((a + b)*log(1 + sqrt(2)*sqrt(cot(c + d*x)) + cot(c + d*x)))/(2*sqrt(2)*d)]
    @test_int [cot(c + d*x)^(5/2)*(a + b*tan(c + d*x)), x, 13, -(((a + b)*atan(1 - sqrt(2)*sqrt(cot(c + d*x))))/(sqrt(2)*d)) + ((a + b)*atan(1 + sqrt(2)*sqrt(cot(c + d*x))))/(sqrt(2)*d) - (2*b*sqrt(cot(c + d*x)))/d - (2*a*cot(c + d*x)^(3/2))/(3*d) + ((a - b)*log(1 - sqrt(2)*sqrt(cot(c + d*x)) + cot(c + d*x)))/(2*sqrt(2)*d) - ((a - b)*log(1 + sqrt(2)*sqrt(cot(c + d*x)) + cot(c + d*x)))/(2*sqrt(2)*d)]
    @test_int [cot(c + d*x)^(3/2)*(a + b*tan(c + d*x)), x, 12, -(((a - b)*atan(1 - sqrt(2)*sqrt(cot(c + d*x))))/(sqrt(2)*d)) + ((a - b)*atan(1 + sqrt(2)*sqrt(cot(c + d*x))))/(sqrt(2)*d) - (2*a*sqrt(cot(c + d*x)))/d - ((a + b)*log(1 - sqrt(2)*sqrt(cot(c + d*x)) + cot(c + d*x)))/(2*sqrt(2)*d) + ((a + b)*log(1 + sqrt(2)*sqrt(cot(c + d*x)) + cot(c + d*x)))/(2*sqrt(2)*d)]
    @test_int [sqrt(cot(c + d*x))*(a + b*tan(c + d*x)), x, 11, ((a + b)*atan(1 - sqrt(2)*sqrt(cot(c + d*x))))/(sqrt(2)*d) - ((a + b)*atan(1 + sqrt(2)*sqrt(cot(c + d*x))))/(sqrt(2)*d) - ((a - b)*log(1 - sqrt(2)*sqrt(cot(c + d*x)) + cot(c + d*x)))/(2*sqrt(2)*d) + ((a - b)*log(1 + sqrt(2)*sqrt(cot(c + d*x)) + cot(c + d*x)))/(2*sqrt(2)*d)]
    @test_int [(a + b*tan(c + d*x))/sqrt(cot(c + d*x)), x, 12, ((a - b)*atan(1 - sqrt(2)*sqrt(cot(c + d*x))))/(sqrt(2)*d) - ((a - b)*atan(1 + sqrt(2)*sqrt(cot(c + d*x))))/(sqrt(2)*d) + (2*b)/(d*sqrt(cot(c + d*x))) + ((a + b)*log(1 - sqrt(2)*sqrt(cot(c + d*x)) + cot(c + d*x)))/(2*sqrt(2)*d) - ((a + b)*log(1 + sqrt(2)*sqrt(cot(c + d*x)) + cot(c + d*x)))/(2*sqrt(2)*d)]
    @test_int [(a + b*tan(c + d*x))/cot(c + d*x)^(3/2), x, 13, -(((a + b)*atan(1 - sqrt(2)*sqrt(cot(c + d*x))))/(sqrt(2)*d)) + ((a + b)*atan(1 + sqrt(2)*sqrt(cot(c + d*x))))/(sqrt(2)*d) + (2*b)/(3*d*cot(c + d*x)^(3/2)) + (2*a)/(d*sqrt(cot(c + d*x))) + ((a - b)*log(1 - sqrt(2)*sqrt(cot(c + d*x)) + cot(c + d*x)))/(2*sqrt(2)*d) - ((a - b)*log(1 + sqrt(2)*sqrt(cot(c + d*x)) + cot(c + d*x)))/(2*sqrt(2)*d)]
    @test_int [(a + b*tan(c + d*x))/cot(c + d*x)^(5/2), x, 14, -(((a - b)*atan(1 - sqrt(2)*sqrt(cot(c + d*x))))/(sqrt(2)*d)) + ((a - b)*atan(1 + sqrt(2)*sqrt(cot(c + d*x))))/(sqrt(2)*d) + (2*b)/(5*d*cot(c + d*x)^(5/2)) + (2*a)/(3*d*cot(c + d*x)^(3/2)) - (2*b)/(d*sqrt(cot(c + d*x))) - ((a + b)*log(1 - sqrt(2)*sqrt(cot(c + d*x)) + cot(c + d*x)))/(2*sqrt(2)*d) + ((a + b)*log(1 + sqrt(2)*sqrt(cot(c + d*x)) + cot(c + d*x)))/(2*sqrt(2)*d)]


    @test_int [cot(c + d*x)^(9/2)*(a + b*tan(c + d*x))^2, x, 15, ((a^2 + 2*a*b - b^2)*atan(1 - sqrt(2)*sqrt(cot(c + d*x))))/(sqrt(2)*d) - ((a^2 + 2*a*b - b^2)*atan(1 + sqrt(2)*sqrt(cot(c + d*x))))/(sqrt(2)*d) + (4*a*b*sqrt(cot(c + d*x)))/d + (2*(a^2 - b^2)*cot(c + d*x)^(3/2))/(3*d) - (4*a*b*cot(c + d*x)^(5/2))/(5*d) - (2*a^2*cot(c + d*x)^(7/2))/(7*d) - ((a^2 - 2*a*b - b^2)*log(1 - sqrt(2)*sqrt(cot(c + d*x)) + cot(c + d*x)))/(2*sqrt(2)*d) + ((a^2 - 2*a*b - b^2)*log(1 + sqrt(2)*sqrt(cot(c + d*x)) + cot(c + d*x)))/(2*sqrt(2)*d)]
    @test_int [cot(c + d*x)^(7/2)*(a + b*tan(c + d*x))^2, x, 14, ((a^2 - 2*a*b - b^2)*atan(1 - sqrt(2)*sqrt(cot(c + d*x))))/(sqrt(2)*d) - ((a^2 - 2*a*b - b^2)*atan(1 + sqrt(2)*sqrt(cot(c + d*x))))/(sqrt(2)*d) + (2*(a^2 - b^2)*sqrt(cot(c + d*x)))/d - (4*a*b*cot(c + d*x)^(3/2))/(3*d) - (2*a^2*cot(c + d*x)^(5/2))/(5*d) + ((a^2 + 2*a*b - b^2)*log(1 - sqrt(2)*sqrt(cot(c + d*x)) + cot(c + d*x)))/(2*sqrt(2)*d) - ((a^2 + 2*a*b - b^2)*log(1 + sqrt(2)*sqrt(cot(c + d*x)) + cot(c + d*x)))/(2*sqrt(2)*d)]
    @test_int [cot(c + d*x)^(5/2)*(a + b*tan(c + d*x))^2, x, 13, -(((a^2 + 2*a*b - b^2)*atan(1 - sqrt(2)*sqrt(cot(c + d*x))))/(sqrt(2)*d)) + ((a^2 + 2*a*b - b^2)*atan(1 + sqrt(2)*sqrt(cot(c + d*x))))/(sqrt(2)*d) - (4*a*b*sqrt(cot(c + d*x)))/d - (2*a^2*cot(c + d*x)^(3/2))/(3*d) + ((a^2 - 2*a*b - b^2)*log(1 - sqrt(2)*sqrt(cot(c + d*x)) + cot(c + d*x)))/(2*sqrt(2)*d) - ((a^2 - 2*a*b - b^2)*log(1 + sqrt(2)*sqrt(cot(c + d*x)) + cot(c + d*x)))/(2*sqrt(2)*d)]
    @test_int [cot(c + d*x)^(3/2)*(a + b*tan(c + d*x))^2, x, 12, -(((a^2 - 2*a*b - b^2)*atan(1 - sqrt(2)*sqrt(cot(c + d*x))))/(sqrt(2)*d)) + ((a^2 - 2*a*b - b^2)*atan(1 + sqrt(2)*sqrt(cot(c + d*x))))/(sqrt(2)*d) - (2*a^2*sqrt(cot(c + d*x)))/d - ((a^2 + 2*a*b - b^2)*log(1 - sqrt(2)*sqrt(cot(c + d*x)) + cot(c + d*x)))/(2*sqrt(2)*d) + ((a^2 + 2*a*b - b^2)*log(1 + sqrt(2)*sqrt(cot(c + d*x)) + cot(c + d*x)))/(2*sqrt(2)*d)]
    @test_int [sqrt(cot(c + d*x))*(a + b*tan(c + d*x))^2, x, 12, ((a^2 + 2*a*b - b^2)*atan(1 - sqrt(2)*sqrt(cot(c + d*x))))/(sqrt(2)*d) - ((a^2 + 2*a*b - b^2)*atan(1 + sqrt(2)*sqrt(cot(c + d*x))))/(sqrt(2)*d) + (2*b^2)/(d*sqrt(cot(c + d*x))) - ((a^2 - 2*a*b - b^2)*log(1 - sqrt(2)*sqrt(cot(c + d*x)) + cot(c + d*x)))/(2*sqrt(2)*d) + ((a^2 - 2*a*b - b^2)*log(1 + sqrt(2)*sqrt(cot(c + d*x)) + cot(c + d*x)))/(2*sqrt(2)*d)]
    @test_int [(a + b*tan(c + d*x))^2/sqrt(cot(c + d*x)), x, 13, ((a^2 - 2*a*b - b^2)*atan(1 - sqrt(2)*sqrt(cot(c + d*x))))/(sqrt(2)*d) - ((a^2 - 2*a*b - b^2)*atan(1 + sqrt(2)*sqrt(cot(c + d*x))))/(sqrt(2)*d) + (2*b^2)/(3*d*cot(c + d*x)^(3/2)) + (4*a*b)/(d*sqrt(cot(c + d*x))) + ((a^2 + 2*a*b - b^2)*log(1 - sqrt(2)*sqrt(cot(c + d*x)) + cot(c + d*x)))/(2*sqrt(2)*d) - ((a^2 + 2*a*b - b^2)*log(1 + sqrt(2)*sqrt(cot(c + d*x)) + cot(c + d*x)))/(2*sqrt(2)*d)]
    @test_int [(a + b*tan(c + d*x))^2/cot(c + d*x)^(3/2), x, 14, -(((a^2 + 2*a*b - b^2)*atan(1 - sqrt(2)*sqrt(cot(c + d*x))))/(sqrt(2)*d)) + ((a^2 + 2*a*b - b^2)*atan(1 + sqrt(2)*sqrt(cot(c + d*x))))/(sqrt(2)*d) + (2*b^2)/(5*d*cot(c + d*x)^(5/2)) + (4*a*b)/(3*d*cot(c + d*x)^(3/2)) + (2*(a^2 - b^2))/(d*sqrt(cot(c + d*x))) + ((a^2 - 2*a*b - b^2)*log(1 - sqrt(2)*sqrt(cot(c + d*x)) + cot(c + d*x)))/(2*sqrt(2)*d) - ((a^2 - 2*a*b - b^2)*log(1 + sqrt(2)*sqrt(cot(c + d*x)) + cot(c + d*x)))/(2*sqrt(2)*d)]
    @test_int [(a + b*tan(c + d*x))^2/cot(c + d*x)^(5/2), x, 15, -(((a^2 - 2*a*b - b^2)*atan(1 - sqrt(2)*sqrt(cot(c + d*x))))/(sqrt(2)*d)) + ((a^2 - 2*a*b - b^2)*atan(1 + sqrt(2)*sqrt(cot(c + d*x))))/(sqrt(2)*d) + (2*b^2)/(7*d*cot(c + d*x)^(7/2)) + (4*a*b)/(5*d*cot(c + d*x)^(5/2)) + (2*(a^2 - b^2))/(3*d*cot(c + d*x)^(3/2)) - (4*a*b)/(d*sqrt(cot(c + d*x))) - ((a^2 + 2*a*b - b^2)*log(1 - sqrt(2)*sqrt(cot(c + d*x)) + cot(c + d*x)))/(2*sqrt(2)*d) + ((a^2 + 2*a*b - b^2)*log(1 + sqrt(2)*sqrt(cot(c + d*x)) + cot(c + d*x)))/(2*sqrt(2)*d)]


    @test_int [cot(c + d*x)^(9/2)*(a + b*tan(c + d*x))^3, x, 15, ((a - b)*(a^2 + 4*a*b + b^2)*atan(1 - sqrt(2)*sqrt(cot(c + d*x))))/(sqrt(2)*d) - ((a - b)*(a^2 + 4*a*b + b^2)*atan(1 + sqrt(2)*sqrt(cot(c + d*x))))/(sqrt(2)*d) + (2*b*(3*a^2 - b^2)*sqrt(cot(c + d*x)))/d + (2*a*(a^2 - 3*b^2)*cot(c + d*x)^(3/2))/(3*d) - (32*a^2*b*cot(c + d*x)^(5/2))/(35*d) - (2*a^2*cot(c + d*x)^(5/2)*(b + a*cot(c + d*x)))/(7*d) - ((a + b)*(a^2 - 4*a*b + b^2)*log(1 - sqrt(2)*sqrt(cot(c + d*x)) + cot(c + d*x)))/(2*sqrt(2)*d) + ((a + b)*(a^2 - 4*a*b + b^2)*log(1 + sqrt(2)*sqrt(cot(c + d*x)) + cot(c + d*x)))/(2*sqrt(2)*d)]
    @test_int [cot(c + d*x)^(7/2)*(a + b*tan(c + d*x))^3, x, 14, ((a + b)*(a^2 - 4*a*b + b^2)*atan(1 - sqrt(2)*sqrt(cot(c + d*x))))/(sqrt(2)*d) - ((a + b)*(a^2 - 4*a*b + b^2)*atan(1 + sqrt(2)*sqrt(cot(c + d*x))))/(sqrt(2)*d) + (2*a*(a^2 - 3*b^2)*sqrt(cot(c + d*x)))/d - (8*a^2*b*cot(c + d*x)^(3/2))/(5*d) - (2*a^2*cot(c + d*x)^(3/2)*(b + a*cot(c + d*x)))/(5*d) + ((a - b)*(a^2 + 4*a*b + b^2)*log(1 - sqrt(2)*sqrt(cot(c + d*x)) + cot(c + d*x)))/(2*sqrt(2)*d) - ((a - b)*(a^2 + 4*a*b + b^2)*log(1 + sqrt(2)*sqrt(cot(c + d*x)) + cot(c + d*x)))/(2*sqrt(2)*d)]
    @test_int [cot(c + d*x)^(5/2)*(a + b*tan(c + d*x))^3, x, 13, -(((a - b)*(a^2 + 4*a*b + b^2)*atan(1 - sqrt(2)*sqrt(cot(c + d*x))))/(sqrt(2)*d)) + ((a - b)*(a^2 + 4*a*b + b^2)*atan(1 + sqrt(2)*sqrt(cot(c + d*x))))/(sqrt(2)*d) - (16*a^2*b*sqrt(cot(c + d*x)))/(3*d) - (2*a^2*sqrt(cot(c + d*x))*(b + a*cot(c + d*x)))/(3*d) + ((a + b)*(a^2 - 4*a*b + b^2)*log(1 - sqrt(2)*sqrt(cot(c + d*x)) + cot(c + d*x)))/(2*sqrt(2)*d) - ((a + b)*(a^2 - 4*a*b + b^2)*log(1 + sqrt(2)*sqrt(cot(c + d*x)) + cot(c + d*x)))/(2*sqrt(2)*d)]
    @test_int [cot(c + d*x)^(3/2)*(a + b*tan(c + d*x))^3, x, 13, -(((a + b)*(a^2 - 4*a*b + b^2)*atan(1 - sqrt(2)*sqrt(cot(c + d*x))))/(sqrt(2)*d)) + ((a + b)*(a^2 - 4*a*b + b^2)*atan(1 + sqrt(2)*sqrt(cot(c + d*x))))/(sqrt(2)*d) - (2*a*(a^2 + b^2)*sqrt(cot(c + d*x)))/d + (2*b^2*(b + a*cot(c + d*x)))/(d*sqrt(cot(c + d*x))) - ((a - b)*(a^2 + 4*a*b + b^2)*log(1 - sqrt(2)*sqrt(cot(c + d*x)) + cot(c + d*x)))/(2*sqrt(2)*d) + ((a - b)*(a^2 + 4*a*b + b^2)*log(1 + sqrt(2)*sqrt(cot(c + d*x)) + cot(c + d*x)))/(2*sqrt(2)*d)]
    @test_int [sqrt(cot(c + d*x))*(a + b*tan(c + d*x))^3, x, 13, ((a - b)*(a^2 + 4*a*b + b^2)*atan(1 - sqrt(2)*sqrt(cot(c + d*x))))/(sqrt(2)*d) - ((a - b)*(a^2 + 4*a*b + b^2)*atan(1 + sqrt(2)*sqrt(cot(c + d*x))))/(sqrt(2)*d) + (16*a*b^2)/(3*d*sqrt(cot(c + d*x))) + (2*b^2*(b + a*cot(c + d*x)))/(3*d*cot(c + d*x)^(3/2)) - ((a + b)*(a^2 - 4*a*b + b^2)*log(1 - sqrt(2)*sqrt(cot(c + d*x)) + cot(c + d*x)))/(2*sqrt(2)*d) + ((a + b)*(a^2 - 4*a*b + b^2)*log(1 + sqrt(2)*sqrt(cot(c + d*x)) + cot(c + d*x)))/(2*sqrt(2)*d)]
    @test_int [(a + b*tan(c + d*x))^3/sqrt(cot(c + d*x)), x, 14, ((a + b)*(a^2 - 4*a*b + b^2)*atan(1 - sqrt(2)*sqrt(cot(c + d*x))))/(sqrt(2)*d) - ((a + b)*(a^2 - 4*a*b + b^2)*atan(1 + sqrt(2)*sqrt(cot(c + d*x))))/(sqrt(2)*d) + (8*a*b^2)/(5*d*cot(c + d*x)^(3/2)) + (2*b*(3*a^2 - b^2))/(d*sqrt(cot(c + d*x))) + (2*b^2*(b + a*cot(c + d*x)))/(5*d*cot(c + d*x)^(5/2)) + ((a - b)*(a^2 + 4*a*b + b^2)*log(1 - sqrt(2)*sqrt(cot(c + d*x)) + cot(c + d*x)))/(2*sqrt(2)*d) - ((a - b)*(a^2 + 4*a*b + b^2)*log(1 + sqrt(2)*sqrt(cot(c + d*x)) + cot(c + d*x)))/(2*sqrt(2)*d)]
    @test_int [(a + b*tan(c + d*x))^3/cot(c + d*x)^(3/2), x, 15, -(((a - b)*(a^2 + 4*a*b + b^2)*atan(1 - sqrt(2)*sqrt(cot(c + d*x))))/(sqrt(2)*d)) + ((a - b)*(a^2 + 4*a*b + b^2)*atan(1 + sqrt(2)*sqrt(cot(c + d*x))))/(sqrt(2)*d) + (32*a*b^2)/(35*d*cot(c + d*x)^(5/2)) + (2*b*(3*a^2 - b^2))/(3*d*cot(c + d*x)^(3/2)) + (2*a*(a^2 - 3*b^2))/(d*sqrt(cot(c + d*x))) + (2*b^2*(b + a*cot(c + d*x)))/(7*d*cot(c + d*x)^(7/2)) + ((a + b)*(a^2 - 4*a*b + b^2)*log(1 - sqrt(2)*sqrt(cot(c + d*x)) + cot(c + d*x)))/(2*sqrt(2)*d) - ((a + b)*(a^2 - 4*a*b + b^2)*log(1 + sqrt(2)*sqrt(cot(c + d*x)) + cot(c + d*x)))/(2*sqrt(2)*d)]


    #= ::Subsubsection::Closed:: =#
    #=m<0=#


    @test_int [cot(c + d*x)^(5/2)/(a + b*tan(c + d*x)), x, 17, -(((a - b)*atan(1 - sqrt(2)*sqrt(cot(c + d*x))))/(sqrt(2)*(a^2 + b^2)*d)) + ((a - b)*atan(1 + sqrt(2)*sqrt(cot(c + d*x))))/(sqrt(2)*(a^2 + b^2)*d) - (2*b^(7/2)*atan((sqrt(a)*sqrt(cot(c + d*x)))/sqrt(b)))/(a^(5/2)*(a^2 + b^2)*d) + (2*b*sqrt(cot(c + d*x)))/(a^2*d) - (2*cot(c + d*x)^(3/2))/(3*a*d) + ((a + b)*log(1 - sqrt(2)*sqrt(cot(c + d*x)) + cot(c + d*x)))/(2*sqrt(2)*(a^2 + b^2)*d) - ((a + b)*log(1 + sqrt(2)*sqrt(cot(c + d*x)) + cot(c + d*x)))/(2*sqrt(2)*(a^2 + b^2)*d)]
    @test_int [cot(c + d*x)^(3/2)/(a + b*tan(c + d*x)), x, 16, -(((a + b)*atan(1 - sqrt(2)*sqrt(cot(c + d*x))))/(sqrt(2)*(a^2 + b^2)*d)) + ((a + b)*atan(1 + sqrt(2)*sqrt(cot(c + d*x))))/(sqrt(2)*(a^2 + b^2)*d) + (2*b^(5/2)*atan((sqrt(a)*sqrt(cot(c + d*x)))/sqrt(b)))/(a^(3/2)*(a^2 + b^2)*d) - (2*sqrt(cot(c + d*x)))/(a*d) - ((a - b)*log(1 - sqrt(2)*sqrt(cot(c + d*x)) + cot(c + d*x)))/(2*sqrt(2)*(a^2 + b^2)*d) + ((a - b)*log(1 + sqrt(2)*sqrt(cot(c + d*x)) + cot(c + d*x)))/(2*sqrt(2)*(a^2 + b^2)*d)]
    @test_int [sqrt(cot(c + d*x))/(a + b*tan(c + d*x)), x, 15, ((a - b)*atan(1 - sqrt(2)*sqrt(cot(c + d*x))))/(sqrt(2)*(a^2 + b^2)*d) - ((a - b)*atan(1 + sqrt(2)*sqrt(cot(c + d*x))))/(sqrt(2)*(a^2 + b^2)*d) - (2*b^(3/2)*atan((sqrt(a)*sqrt(cot(c + d*x)))/sqrt(b)))/(sqrt(a)*(a^2 + b^2)*d) - ((a + b)*log(1 - sqrt(2)*sqrt(cot(c + d*x)) + cot(c + d*x)))/(2*sqrt(2)*(a^2 + b^2)*d) + ((a + b)*log(1 + sqrt(2)*sqrt(cot(c + d*x)) + cot(c + d*x)))/(2*sqrt(2)*(a^2 + b^2)*d)]
    @test_int [1/(sqrt(cot(c + d*x))*(a + b*tan(c + d*x))), x, 15, ((a + b)*atan(1 - sqrt(2)*sqrt(cot(c + d*x))))/(sqrt(2)*(a^2 + b^2)*d) - ((a + b)*atan(1 + sqrt(2)*sqrt(cot(c + d*x))))/(sqrt(2)*(a^2 + b^2)*d) + (2*sqrt(a)*sqrt(b)*atan((sqrt(a)*sqrt(cot(c + d*x)))/sqrt(b)))/((a^2 + b^2)*d) + ((a - b)*log(1 - sqrt(2)*sqrt(cot(c + d*x)) + cot(c + d*x)))/(2*sqrt(2)*(a^2 + b^2)*d) - ((a - b)*log(1 + sqrt(2)*sqrt(cot(c + d*x)) + cot(c + d*x)))/(2*sqrt(2)*(a^2 + b^2)*d)]
    @test_int [1/(cot(c + d*x)^(3/2)*(a + b*tan(c + d*x))), x, 15, -(((a - b)*atan(1 - sqrt(2)*sqrt(cot(c + d*x))))/(sqrt(2)*(a^2 + b^2)*d)) + ((a - b)*atan(1 + sqrt(2)*sqrt(cot(c + d*x))))/(sqrt(2)*(a^2 + b^2)*d) - (2*a^(3/2)*atan((sqrt(a)*sqrt(cot(c + d*x)))/sqrt(b)))/(sqrt(b)*(a^2 + b^2)*d) + ((a + b)*log(1 - sqrt(2)*sqrt(cot(c + d*x)) + cot(c + d*x)))/(2*sqrt(2)*(a^2 + b^2)*d) - ((a + b)*log(1 + sqrt(2)*sqrt(cot(c + d*x)) + cot(c + d*x)))/(2*sqrt(2)*(a^2 + b^2)*d)]
    @test_int [1/(cot(c + d*x)^(5/2)*(a + b*tan(c + d*x))), x, 16, -(((a + b)*atan(1 - sqrt(2)*sqrt(cot(c + d*x))))/(sqrt(2)*(a^2 + b^2)*d)) + ((a + b)*atan(1 + sqrt(2)*sqrt(cot(c + d*x))))/(sqrt(2)*(a^2 + b^2)*d) + (2*a^(5/2)*atan((sqrt(a)*sqrt(cot(c + d*x)))/sqrt(b)))/(b^(3/2)*(a^2 + b^2)*d) + 2/(b*d*sqrt(cot(c + d*x))) - ((a - b)*log(1 - sqrt(2)*sqrt(cot(c + d*x)) + cot(c + d*x)))/(2*sqrt(2)*(a^2 + b^2)*d) + ((a - b)*log(1 + sqrt(2)*sqrt(cot(c + d*x)) + cot(c + d*x)))/(2*sqrt(2)*(a^2 + b^2)*d)]


    @test_int [cot(c + d*x)^(5/2)/(a + b*tan(c + d*x))^2, x, 18, -(((a^2 - 2*a*b - b^2)*atan(1 - sqrt(2)*sqrt(cot(c + d*x))))/(sqrt(2)*(a^2 + b^2)^2*d)) + ((a^2 - 2*a*b - b^2)*atan(1 + sqrt(2)*sqrt(cot(c + d*x))))/(sqrt(2)*(a^2 + b^2)^2*d) - (b^(7/2)*(9*a^2 + 5*b^2)*atan((sqrt(a)*sqrt(cot(c + d*x)))/sqrt(b)))/(a^(7/2)*(a^2 + b^2)^2*d) + (b*(4*a^2 + 5*b^2)*sqrt(cot(c + d*x)))/(a^3*(a^2 + b^2)*d) - ((2*a^2 + 5*b^2)*cot(c + d*x)^(3/2))/(3*a^2*(a^2 + b^2)*d) + (b^2*cot(c + d*x)^(5/2))/(a*(a^2 + b^2)*d*(b + a*cot(c + d*x))) + ((a^2 + 2*a*b - b^2)*log(1 - sqrt(2)*sqrt(cot(c + d*x)) + cot(c + d*x)))/(2*sqrt(2)*(a^2 + b^2)^2*d) - ((a^2 + 2*a*b - b^2)*log(1 + sqrt(2)*sqrt(cot(c + d*x)) + cot(c + d*x)))/(2*sqrt(2)*(a^2 + b^2)^2*d)]
    @test_int [cot(c + d*x)^(3/2)/(a + b*tan(c + d*x))^2, x, 17, -(((a^2 + 2*a*b - b^2)*atan(1 - sqrt(2)*sqrt(cot(c + d*x))))/(sqrt(2)*(a^2 + b^2)^2*d)) + ((a^2 + 2*a*b - b^2)*atan(1 + sqrt(2)*sqrt(cot(c + d*x))))/(sqrt(2)*(a^2 + b^2)^2*d) + (b^(5/2)*(7*a^2 + 3*b^2)*atan((sqrt(a)*sqrt(cot(c + d*x)))/sqrt(b)))/(a^(5/2)*(a^2 + b^2)^2*d) - ((2*a^2 + 3*b^2)*sqrt(cot(c + d*x)))/(a^2*(a^2 + b^2)*d) + (b^2*cot(c + d*x)^(3/2))/(a*(a^2 + b^2)*d*(b + a*cot(c + d*x))) - ((a^2 - 2*a*b - b^2)*log(1 - sqrt(2)*sqrt(cot(c + d*x)) + cot(c + d*x)))/(2*sqrt(2)*(a^2 + b^2)^2*d) + ((a^2 - 2*a*b - b^2)*log(1 + sqrt(2)*sqrt(cot(c + d*x)) + cot(c + d*x)))/(2*sqrt(2)*(a^2 + b^2)^2*d)]
    @test_int [sqrt(cot(c + d*x))/(a + b*tan(c + d*x))^2, x, 16, ((a^2 - 2*a*b - b^2)*atan(1 - sqrt(2)*sqrt(cot(c + d*x))))/(sqrt(2)*(a^2 + b^2)^2*d) - ((a^2 - 2*a*b - b^2)*atan(1 + sqrt(2)*sqrt(cot(c + d*x))))/(sqrt(2)*(a^2 + b^2)^2*d) - (b^(3/2)*(5*a^2 + b^2)*atan((sqrt(a)*sqrt(cot(c + d*x)))/sqrt(b)))/(a^(3/2)*(a^2 + b^2)^2*d) + (b^2*sqrt(cot(c + d*x)))/(a*(a^2 + b^2)*d*(b + a*cot(c + d*x))) - ((a^2 + 2*a*b - b^2)*log(1 - sqrt(2)*sqrt(cot(c + d*x)) + cot(c + d*x)))/(2*sqrt(2)*(a^2 + b^2)^2*d) + ((a^2 + 2*a*b - b^2)*log(1 + sqrt(2)*sqrt(cot(c + d*x)) + cot(c + d*x)))/(2*sqrt(2)*(a^2 + b^2)^2*d)]
    @test_int [1/(sqrt(cot(c + d*x))*(a + b*tan(c + d*x))^2), x, 16, ((a^2 + 2*a*b - b^2)*atan(1 - sqrt(2)*sqrt(cot(c + d*x))))/(sqrt(2)*(a^2 + b^2)^2*d) - ((a^2 + 2*a*b - b^2)*atan(1 + sqrt(2)*sqrt(cot(c + d*x))))/(sqrt(2)*(a^2 + b^2)^2*d) + (sqrt(b)*(3*a^2 - b^2)*atan((sqrt(a)*sqrt(cot(c + d*x)))/sqrt(b)))/(sqrt(a)*(a^2 + b^2)^2*d) - (b*sqrt(cot(c + d*x)))/((a^2 + b^2)*d*(b + a*cot(c + d*x))) + ((a^2 - 2*a*b - b^2)*log(1 - sqrt(2)*sqrt(cot(c + d*x)) + cot(c + d*x)))/(2*sqrt(2)*(a^2 + b^2)^2*d) - ((a^2 - 2*a*b - b^2)*log(1 + sqrt(2)*sqrt(cot(c + d*x)) + cot(c + d*x)))/(2*sqrt(2)*(a^2 + b^2)^2*d)]
    @test_int [1/(cot(c + d*x)^(3/2)*(a + b*tan(c + d*x))^2), x, 16, -(((a^2 - 2*a*b - b^2)*atan(1 - sqrt(2)*sqrt(cot(c + d*x))))/(sqrt(2)*(a^2 + b^2)^2*d)) + ((a^2 - 2*a*b - b^2)*atan(1 + sqrt(2)*sqrt(cot(c + d*x))))/(sqrt(2)*(a^2 + b^2)^2*d) - (sqrt(a)*(a^2 - 3*b^2)*atan((sqrt(a)*sqrt(cot(c + d*x)))/sqrt(b)))/(sqrt(b)*(a^2 + b^2)^2*d) + (a*sqrt(cot(c + d*x)))/((a^2 + b^2)*d*(b + a*cot(c + d*x))) + ((a^2 + 2*a*b - b^2)*log(1 - sqrt(2)*sqrt(cot(c + d*x)) + cot(c + d*x)))/(2*sqrt(2)*(a^2 + b^2)^2*d) - ((a^2 + 2*a*b - b^2)*log(1 + sqrt(2)*sqrt(cot(c + d*x)) + cot(c + d*x)))/(2*sqrt(2)*(a^2 + b^2)^2*d)]
    @test_int [1/(cot(c + d*x)^(5/2)*(a + b*tan(c + d*x))^2), x, 16, -(((a^2 + 2*a*b - b^2)*atan(1 - sqrt(2)*sqrt(cot(c + d*x))))/(sqrt(2)*(a^2 + b^2)^2*d)) + ((a^2 + 2*a*b - b^2)*atan(1 + sqrt(2)*sqrt(cot(c + d*x))))/(sqrt(2)*(a^2 + b^2)^2*d) - (a^(3/2)*(a^2 + 5*b^2)*atan((sqrt(a)*sqrt(cot(c + d*x)))/sqrt(b)))/(b^(3/2)*(a^2 + b^2)^2*d) - (a^2*sqrt(cot(c + d*x)))/(b*(a^2 + b^2)*d*(b + a*cot(c + d*x))) - ((a^2 - 2*a*b - b^2)*log(1 - sqrt(2)*sqrt(cot(c + d*x)) + cot(c + d*x)))/(2*sqrt(2)*(a^2 + b^2)^2*d) + ((a^2 - 2*a*b - b^2)*log(1 + sqrt(2)*sqrt(cot(c + d*x)) + cot(c + d*x)))/(2*sqrt(2)*(a^2 + b^2)^2*d)]
    @test_int [1/(cot(c + d*x)^(7/2)*(a + b*tan(c + d*x))^2), x, 17, ((a^2 - 2*a*b - b^2)*atan(1 - sqrt(2)*sqrt(cot(c + d*x))))/(sqrt(2)*(a^2 + b^2)^2*d) - ((a^2 - 2*a*b - b^2)*atan(1 + sqrt(2)*sqrt(cot(c + d*x))))/(sqrt(2)*(a^2 + b^2)^2*d) + (a^(5/2)*(3*a^2 + 7*b^2)*atan((sqrt(a)*sqrt(cot(c + d*x)))/sqrt(b)))/(b^(5/2)*(a^2 + b^2)^2*d) + (3*a^2 + 2*b^2)/(b^2*(a^2 + b^2)*d*sqrt(cot(c + d*x))) - a^2/(b*(a^2 + b^2)*d*sqrt(cot(c + d*x))*(b + a*cot(c + d*x))) - ((a^2 + 2*a*b - b^2)*log(1 - sqrt(2)*sqrt(cot(c + d*x)) + cot(c + d*x)))/(2*sqrt(2)*(a^2 + b^2)^2*d) + ((a^2 + 2*a*b - b^2)*log(1 + sqrt(2)*sqrt(cot(c + d*x)) + cot(c + d*x)))/(2*sqrt(2)*(a^2 + b^2)^2*d)]


    @test_int [cot(c + d*x)^(5/2)/(a + b*tan(c + d*x))^3, x, 19, -(((a + b)*(a^2 - 4*a*b + b^2)*atan(1 - sqrt(2)*sqrt(cot(c + d*x))))/(sqrt(2)*(a^2 + b^2)^3*d)) + ((a + b)*(a^2 - 4*a*b + b^2)*atan(1 + sqrt(2)*sqrt(cot(c + d*x))))/(sqrt(2)*(a^2 + b^2)^3*d) - (b^(7/2)*(99*a^4 + 102*a^2*b^2 + 35*b^4)*atan((sqrt(a)*sqrt(cot(c + d*x)))/sqrt(b)))/(4*a^(9/2)*(a^2 + b^2)^3*d) + (b*(24*a^4 + 67*a^2*b^2 + 35*b^4)*sqrt(cot(c + d*x)))/(4*a^4*(a^2 + b^2)^2*d) - ((8*a^4 + 67*a^2*b^2 + 35*b^4)*cot(c + d*x)^(3/2))/(12*a^3*(a^2 + b^2)^2*d) + (b^2*cot(c + d*x)^(7/2))/(2*a*(a^2 + b^2)*d*(b + a*cot(c + d*x))^2) + (b^2*(15*a^2 + 7*b^2)*cot(c + d*x)^(5/2))/(4*a^2*(a^2 + b^2)^2*d*(b + a*cot(c + d*x))) + ((a - b)*(a^2 + 4*a*b + b^2)*log(1 - sqrt(2)*sqrt(cot(c + d*x)) + cot(c + d*x)))/(2*sqrt(2)*(a^2 + b^2)^3*d) - ((a - b)*(a^2 + 4*a*b + b^2)*log(1 + sqrt(2)*sqrt(cot(c + d*x)) + cot(c + d*x)))/(2*sqrt(2)*(a^2 + b^2)^3*d)]
    @test_int [cot(c + d*x)^(3/2)/(a + b*tan(c + d*x))^3, x, 18, -(((a - b)*(a^2 + 4*a*b + b^2)*atan(1 - sqrt(2)*sqrt(cot(c + d*x))))/(sqrt(2)*(a^2 + b^2)^3*d)) + ((a - b)*(a^2 + 4*a*b + b^2)*atan(1 + sqrt(2)*sqrt(cot(c + d*x))))/(sqrt(2)*(a^2 + b^2)^3*d) + (b^(5/2)*(63*a^4 + 46*a^2*b^2 + 15*b^4)*atan((sqrt(a)*sqrt(cot(c + d*x)))/sqrt(b)))/(4*a^(7/2)*(a^2 + b^2)^3*d) - ((8*a^4 + 31*a^2*b^2 + 15*b^4)*sqrt(cot(c + d*x)))/(4*a^3*(a^2 + b^2)^2*d) + (b^2*cot(c + d*x)^(5/2))/(2*a*(a^2 + b^2)*d*(b + a*cot(c + d*x))^2) + (b^2*(13*a^2 + 5*b^2)*cot(c + d*x)^(3/2))/(4*a^2*(a^2 + b^2)^2*d*(b + a*cot(c + d*x))) - ((a + b)*(a^2 - 4*a*b + b^2)*log(1 - sqrt(2)*sqrt(cot(c + d*x)) + cot(c + d*x)))/(2*sqrt(2)*(a^2 + b^2)^3*d) + ((a + b)*(a^2 - 4*a*b + b^2)*log(1 + sqrt(2)*sqrt(cot(c + d*x)) + cot(c + d*x)))/(2*sqrt(2)*(a^2 + b^2)^3*d)]
    @test_int [sqrt(cot(c + d*x))/(a + b*tan(c + d*x))^3, x, 17, ((a + b)*(a^2 - 4*a*b + b^2)*atan(1 - sqrt(2)*sqrt(cot(c + d*x))))/(sqrt(2)*(a^2 + b^2)^3*d) - ((a + b)*(a^2 - 4*a*b + b^2)*atan(1 + sqrt(2)*sqrt(cot(c + d*x))))/(sqrt(2)*(a^2 + b^2)^3*d) - (b^(3/2)*(35*a^4 + 6*a^2*b^2 + 3*b^4)*atan((sqrt(a)*sqrt(cot(c + d*x)))/sqrt(b)))/(4*a^(5/2)*(a^2 + b^2)^3*d) + (b^2*cot(c + d*x)^(3/2))/(2*a*(a^2 + b^2)*d*(b + a*cot(c + d*x))^2) + (b^2*(11*a^2 + 3*b^2)*sqrt(cot(c + d*x)))/(4*a^2*(a^2 + b^2)^2*d*(b + a*cot(c + d*x))) - ((a - b)*(a^2 + 4*a*b + b^2)*log(1 - sqrt(2)*sqrt(cot(c + d*x)) + cot(c + d*x)))/(2*sqrt(2)*(a^2 + b^2)^3*d) + ((a - b)*(a^2 + 4*a*b + b^2)*log(1 + sqrt(2)*sqrt(cot(c + d*x)) + cot(c + d*x)))/(2*sqrt(2)*(a^2 + b^2)^3*d)]
    @test_int [1/(sqrt(cot(c + d*x))*(a + b*tan(c + d*x))^3), x, 17, ((a - b)*(a^2 + 4*a*b + b^2)*atan(1 - sqrt(2)*sqrt(cot(c + d*x))))/(sqrt(2)*(a^2 + b^2)^3*d) - ((a - b)*(a^2 + 4*a*b + b^2)*atan(1 + sqrt(2)*sqrt(cot(c + d*x))))/(sqrt(2)*(a^2 + b^2)^3*d) + (sqrt(b)*(15*a^4 - 18*a^2*b^2 - b^4)*atan((sqrt(a)*sqrt(cot(c + d*x)))/sqrt(b)))/(4*a^(3/2)*(a^2 + b^2)^3*d) + (b^2*sqrt(cot(c + d*x)))/(2*a*(a^2 + b^2)*d*(b + a*cot(c + d*x))^2) - (b*(9*a^2 + b^2)*sqrt(cot(c + d*x)))/(4*a*(a^2 + b^2)^2*d*(b + a*cot(c + d*x))) + ((a + b)*(a^2 - 4*a*b + b^2)*log(1 - sqrt(2)*sqrt(cot(c + d*x)) + cot(c + d*x)))/(2*sqrt(2)*(a^2 + b^2)^3*d) - ((a + b)*(a^2 - 4*a*b + b^2)*log(1 + sqrt(2)*sqrt(cot(c + d*x)) + cot(c + d*x)))/(2*sqrt(2)*(a^2 + b^2)^3*d)]
    @test_int [1/(cot(c + d*x)^(3/2)*(a + b*tan(c + d*x))^3), x, 17, -(((a + b)*(a^2 - 4*a*b + b^2)*atan(1 - sqrt(2)*sqrt(cot(c + d*x))))/(sqrt(2)*(a^2 + b^2)^3*d)) + ((a + b)*(a^2 - 4*a*b + b^2)*atan(1 + sqrt(2)*sqrt(cot(c + d*x))))/(sqrt(2)*(a^2 + b^2)^3*d) - ((3*a^4 - 26*a^2*b^2 + 3*b^4)*atan((sqrt(a)*sqrt(cot(c + d*x)))/sqrt(b)))/(4*sqrt(a)*sqrt(b)*(a^2 + b^2)^3*d) - (b*sqrt(cot(c + d*x)))/(2*(a^2 + b^2)*d*(b + a*cot(c + d*x))^2) + ((5*a^2 - 3*b^2)*sqrt(cot(c + d*x)))/(4*(a^2 + b^2)^2*d*(b + a*cot(c + d*x))) + ((a - b)*(a^2 + 4*a*b + b^2)*log(1 - sqrt(2)*sqrt(cot(c + d*x)) + cot(c + d*x)))/(2*sqrt(2)*(a^2 + b^2)^3*d) - ((a - b)*(a^2 + 4*a*b + b^2)*log(1 + sqrt(2)*sqrt(cot(c + d*x)) + cot(c + d*x)))/(2*sqrt(2)*(a^2 + b^2)^3*d)]
    @test_int [1/(cot(c + d*x)^(5/2)*(a + b*tan(c + d*x))^3), x, 17, -(((a - b)*(a^2 + 4*a*b + b^2)*atan(1 - sqrt(2)*sqrt(cot(c + d*x))))/(sqrt(2)*(a^2 + b^2)^3*d)) + ((a - b)*(a^2 + 4*a*b + b^2)*atan(1 + sqrt(2)*sqrt(cot(c + d*x))))/(sqrt(2)*(a^2 + b^2)^3*d) - (sqrt(a)*(a^4 + 18*a^2*b^2 - 15*b^4)*atan((sqrt(a)*sqrt(cot(c + d*x)))/sqrt(b)))/(4*b^(3/2)*(a^2 + b^2)^3*d) + (a*sqrt(cot(c + d*x)))/(2*(a^2 + b^2)*d*(b + a*cot(c + d*x))^2) - (a*(a^2 - 7*b^2)*sqrt(cot(c + d*x)))/(4*b*(a^2 + b^2)^2*d*(b + a*cot(c + d*x))) - ((a + b)*(a^2 - 4*a*b + b^2)*log(1 - sqrt(2)*sqrt(cot(c + d*x)) + cot(c + d*x)))/(2*sqrt(2)*(a^2 + b^2)^3*d) + ((a + b)*(a^2 - 4*a*b + b^2)*log(1 + sqrt(2)*sqrt(cot(c + d*x)) + cot(c + d*x)))/(2*sqrt(2)*(a^2 + b^2)^3*d)]
    @test_int [1/(cot(c + d*x)^(7/2)*(a + b*tan(c + d*x))^3), x, 17, ((a + b)*(a^2 - 4*a*b + b^2)*atan(1 - sqrt(2)*sqrt(cot(c + d*x))))/(sqrt(2)*(a^2 + b^2)^3*d) - ((a + b)*(a^2 - 4*a*b + b^2)*atan(1 + sqrt(2)*sqrt(cot(c + d*x))))/(sqrt(2)*(a^2 + b^2)^3*d) - (a^(3/2)*(3*a^4 + 6*a^2*b^2 + 35*b^4)*atan((sqrt(a)*sqrt(cot(c + d*x)))/sqrt(b)))/(4*b^(5/2)*(a^2 + b^2)^3*d) - (a^2*sqrt(cot(c + d*x)))/(2*b*(a^2 + b^2)*d*(b + a*cot(c + d*x))^2) - (a^2*(3*a^2 + 11*b^2)*sqrt(cot(c + d*x)))/(4*b^2*(a^2 + b^2)^2*d*(b + a*cot(c + d*x))) - ((a - b)*(a^2 + 4*a*b + b^2)*log(1 - sqrt(2)*sqrt(cot(c + d*x)) + cot(c + d*x)))/(2*sqrt(2)*(a^2 + b^2)^3*d) + ((a - b)*(a^2 + 4*a*b + b^2)*log(1 + sqrt(2)*sqrt(cot(c + d*x)) + cot(c + d*x)))/(2*sqrt(2)*(a^2 + b^2)^3*d)]


    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*(a+b*tan(e+f*x))^(m/2)*(d*cot(e+f*x))^(n/2)=#


    #= ::Subsubsection::Closed:: =#
    #=m>0=#


    @test_int [cot(c + d*x)^(7/2)*sqrt(a + b*tan(c + d*x)), x, 11, (sqrt(im*a - b)*atan((sqrt(im*a - b)*sqrt(tan(c + d*x)))/sqrt(a + b*tan(c + d*x)))*sqrt(cot(c + d*x))*sqrt(tan(c + d*x)))/d - (sqrt(im*a + b)*atanh((sqrt(im*a + b)*sqrt(tan(c + d*x)))/sqrt(a + b*tan(c + d*x)))*sqrt(cot(c + d*x))*sqrt(tan(c + d*x)))/d + (2*(15*a^2 + 2*b^2)*sqrt(cot(c + d*x))*sqrt(a + b*tan(c + d*x)))/(15*a^2*d) - (2*b*cot(c + d*x)^(3/2)*sqrt(a + b*tan(c + d*x)))/(15*a*d) - (2*cot(c + d*x)^(5/2)*sqrt(a + b*tan(c + d*x)))/(5*d)]
    @test_int [cot(c + d*x)^(5/2)*sqrt(a + b*tan(c + d*x)), x, 11, (im*sqrt(im*a - b)*atan((sqrt(im*a - b)*sqrt(tan(c + d*x)))/sqrt(a + b*tan(c + d*x)))*sqrt(cot(c + d*x))*sqrt(tan(c + d*x)))/d + (im*sqrt(im*a + b)*atanh((sqrt(im*a + b)*sqrt(tan(c + d*x)))/sqrt(a + b*tan(c + d*x)))*sqrt(cot(c + d*x))*sqrt(tan(c + d*x)))/d - (2*b*sqrt(cot(c + d*x))*sqrt(a + b*tan(c + d*x)))/(3*a*d) - (2*cot(c + d*x)^(3/2)*sqrt(a + b*tan(c + d*x)))/(3*d)]
    @test_int [cot(c + d*x)^(3/2)*sqrt(a + b*tan(c + d*x)), x, 9, -((sqrt(im*a - b)*atan((sqrt(im*a - b)*sqrt(tan(c + d*x)))/sqrt(a + b*tan(c + d*x)))*sqrt(cot(c + d*x))*sqrt(tan(c + d*x)))/d) + (sqrt(im*a + b)*atanh((sqrt(im*a + b)*sqrt(tan(c + d*x)))/sqrt(a + b*tan(c + d*x)))*sqrt(cot(c + d*x))*sqrt(tan(c + d*x)))/d - (2*sqrt(cot(c + d*x))*sqrt(a + b*tan(c + d*x)))/d]
    @test_int [sqrt(cot(c + d*x))*sqrt(a + b*tan(c + d*x)), x, 8, ((-im)*sqrt(im*a - b)*atan((sqrt(im*a - b)*sqrt(tan(c + d*x)))/sqrt(a + b*tan(c + d*x)))*sqrt(cot(c + d*x))*sqrt(tan(c + d*x)))/d - (im*sqrt(im*a + b)*atanh((sqrt(im*a + b)*sqrt(tan(c + d*x)))/sqrt(a + b*tan(c + d*x)))*sqrt(cot(c + d*x))*sqrt(tan(c + d*x)))/d]
    @test_int [sqrt(a + b*tan(c + d*x))/sqrt(cot(c + d*x)), x, 12, (sqrt(im*a - b)*atan((sqrt(im*a - b)*sqrt(tan(c + d*x)))/sqrt(a + b*tan(c + d*x)))*sqrt(cot(c + d*x))*sqrt(tan(c + d*x)))/d + (2*sqrt(b)*atanh((sqrt(b)*sqrt(tan(c + d*x)))/sqrt(a + b*tan(c + d*x)))*sqrt(cot(c + d*x))*sqrt(tan(c + d*x)))/d - (sqrt(im*a + b)*atanh((sqrt(im*a + b)*sqrt(tan(c + d*x)))/sqrt(a + b*tan(c + d*x)))*sqrt(cot(c + d*x))*sqrt(tan(c + d*x)))/d]
    @test_int [sqrt(a + b*tan(c + d*x))/cot(c + d*x)^(3/2), x, 14, (im*sqrt(im*a - b)*atan((sqrt(im*a - b)*sqrt(tan(c + d*x)))/sqrt(a + b*tan(c + d*x)))*sqrt(cot(c + d*x))*sqrt(tan(c + d*x)))/d + (a*atanh((sqrt(b)*sqrt(tan(c + d*x)))/sqrt(a + b*tan(c + d*x)))*sqrt(cot(c + d*x))*sqrt(tan(c + d*x)))/(sqrt(b)*d) + (im*sqrt(im*a + b)*atanh((sqrt(im*a + b)*sqrt(tan(c + d*x)))/sqrt(a + b*tan(c + d*x)))*sqrt(cot(c + d*x))*sqrt(tan(c + d*x)))/d + sqrt(a + b*tan(c + d*x))/(d*sqrt(cot(c + d*x)))]


    @test_int [cot(c + d*x)^(9/2)*(a + b*tan(c + d*x))^(3/2), x, 12, -(((im*a - b)^(3/2)*atan((sqrt(im*a - b)*sqrt(tan(c + d*x)))/sqrt(a + b*tan(c + d*x)))*sqrt(cot(c + d*x))*sqrt(tan(c + d*x)))/d) - ((im*a + b)^(3/2)*atanh((sqrt(im*a + b)*sqrt(tan(c + d*x)))/sqrt(a + b*tan(c + d*x)))*sqrt(cot(c + d*x))*sqrt(tan(c + d*x)))/d + (4*b*(70*a^2 + 3*b^2)*sqrt(cot(c + d*x))*sqrt(a + b*tan(c + d*x)))/(105*a^2*d) + (2*(35*a^2 - 3*b^2)*cot(c + d*x)^(3/2)*sqrt(a + b*tan(c + d*x)))/(105*a*d) - (16*b*cot(c + d*x)^(5/2)*sqrt(a + b*tan(c + d*x)))/(35*d) - (2*a*cot(c + d*x)^(7/2)*sqrt(a + b*tan(c + d*x)))/(7*d)]
    @test_int [cot(c + d*x)^(7/2)*(a + b*tan(c + d*x))^(3/2), x, 11, -((im*(im*a - b)^(3/2)*atan((sqrt(im*a - b)*sqrt(tan(c + d*x)))/sqrt(a + b*tan(c + d*x)))*sqrt(cot(c + d*x))*sqrt(tan(c + d*x)))/d) + (im*(im*a + b)^(3/2)*atanh((sqrt(im*a + b)*sqrt(tan(c + d*x)))/sqrt(a + b*tan(c + d*x)))*sqrt(cot(c + d*x))*sqrt(tan(c + d*x)))/d + (2*(5*a^2 - b^2)*sqrt(cot(c + d*x))*sqrt(a + b*tan(c + d*x)))/(5*a*d) - (4*b*cot(c + d*x)^(3/2)*sqrt(a + b*tan(c + d*x)))/(5*d) - (2*a*cot(c + d*x)^(5/2)*sqrt(a + b*tan(c + d*x)))/(5*d)]
    @test_int [cot(c + d*x)^(5/2)*(a + b*tan(c + d*x))^(3/2), x, 10, ((im*a - b)^(3/2)*atan((sqrt(im*a - b)*sqrt(tan(c + d*x)))/sqrt(a + b*tan(c + d*x)))*sqrt(cot(c + d*x))*sqrt(tan(c + d*x)))/d + ((im*a + b)^(3/2)*atanh((sqrt(im*a + b)*sqrt(tan(c + d*x)))/sqrt(a + b*tan(c + d*x)))*sqrt(cot(c + d*x))*sqrt(tan(c + d*x)))/d - (8*b*sqrt(cot(c + d*x))*sqrt(a + b*tan(c + d*x)))/(3*d) - (2*a*cot(c + d*x)^(3/2)*sqrt(a + b*tan(c + d*x)))/(3*d)]
    @test_int [cot(c + d*x)^(3/2)*(a + b*tan(c + d*x))^(3/2), x, 9, (im*(im*a - b)^(3/2)*atan((sqrt(im*a - b)*sqrt(tan(c + d*x)))/sqrt(a + b*tan(c + d*x)))*sqrt(cot(c + d*x))*sqrt(tan(c + d*x)))/d - (im*(im*a + b)^(3/2)*atanh((sqrt(im*a + b)*sqrt(tan(c + d*x)))/sqrt(a + b*tan(c + d*x)))*sqrt(cot(c + d*x))*sqrt(tan(c + d*x)))/d - (2*a*sqrt(cot(c + d*x))*sqrt(a + b*tan(c + d*x)))/d]
    @test_int [sqrt(cot(c + d*x))*(a + b*tan(c + d*x))^(3/2), x, 13, -(((im*a - b)^(3/2)*atan((sqrt(im*a - b)*sqrt(tan(c + d*x)))/sqrt(a + b*tan(c + d*x)))*sqrt(cot(c + d*x))*sqrt(tan(c + d*x)))/d) + (2*b^(3/2)*atanh((sqrt(b)*sqrt(tan(c + d*x)))/sqrt(a + b*tan(c + d*x)))*sqrt(cot(c + d*x))*sqrt(tan(c + d*x)))/d - ((im*a + b)^(3/2)*atanh((sqrt(im*a + b)*sqrt(tan(c + d*x)))/sqrt(a + b*tan(c + d*x)))*sqrt(cot(c + d*x))*sqrt(tan(c + d*x)))/d]
    @test_int [(a + b*tan(c + d*x))^(3/2)/sqrt(cot(c + d*x)), x, 14, -((im*(im*a - b)^(3/2)*atan((sqrt(im*a - b)*sqrt(tan(c + d*x)))/sqrt(a + b*tan(c + d*x)))*sqrt(cot(c + d*x))*sqrt(tan(c + d*x)))/d) + (3*a*sqrt(b)*atanh((sqrt(b)*sqrt(tan(c + d*x)))/sqrt(a + b*tan(c + d*x)))*sqrt(cot(c + d*x))*sqrt(tan(c + d*x)))/d + (im*(im*a + b)^(3/2)*atanh((sqrt(im*a + b)*sqrt(tan(c + d*x)))/sqrt(a + b*tan(c + d*x)))*sqrt(cot(c + d*x))*sqrt(tan(c + d*x)))/d + (b*sqrt(a + b*tan(c + d*x)))/(d*sqrt(cot(c + d*x)))]
    @test_int [(a + b*tan(c + d*x))^(3/2)/cot(c + d*x)^(3/2), x, 15, ((im*a - b)^(3/2)*atan((sqrt(im*a - b)*sqrt(tan(c + d*x)))/sqrt(a + b*tan(c + d*x)))*sqrt(cot(c + d*x))*sqrt(tan(c + d*x)))/d + ((3*a^2 - 8*b^2)*atanh((sqrt(b)*sqrt(tan(c + d*x)))/sqrt(a + b*tan(c + d*x)))*sqrt(cot(c + d*x))*sqrt(tan(c + d*x)))/(4*sqrt(b)*d) + ((im*a + b)^(3/2)*atanh((sqrt(im*a + b)*sqrt(tan(c + d*x)))/sqrt(a + b*tan(c + d*x)))*sqrt(cot(c + d*x))*sqrt(tan(c + d*x)))/d + (3*a*sqrt(a + b*tan(c + d*x)))/(4*d*sqrt(cot(c + d*x))) + (a + b*tan(c + d*x))^(3/2)/(2*d*sqrt(cot(c + d*x)))]


    @test_int [cot(c + d*x)^(11/2)*(a + b*tan(c + d*x))^(5/2), x, 13, ((im*a - b)^(5/2)*atan((sqrt(im*a - b)*sqrt(tan(c + d*x)))/sqrt(a + b*tan(c + d*x)))*sqrt(cot(c + d*x))*sqrt(tan(c + d*x)))/d - ((im*a + b)^(5/2)*atanh((sqrt(im*a + b)*sqrt(tan(c + d*x)))/sqrt(a + b*tan(c + d*x)))*sqrt(cot(c + d*x))*sqrt(tan(c + d*x)))/d - (2*(315*a^4 - 483*a^2*b^2 - 10*b^4)*sqrt(cot(c + d*x))*sqrt(a + b*tan(c + d*x)))/(315*a^2*d) + (2*b*(231*a^2 - 5*b^2)*cot(c + d*x)^(3/2)*sqrt(a + b*tan(c + d*x)))/(315*a*d) + (2*(21*a^2 - 25*b^2)*cot(c + d*x)^(5/2)*sqrt(a + b*tan(c + d*x)))/(105*d) - (38*a*b*cot(c + d*x)^(7/2)*sqrt(a + b*tan(c + d*x)))/(63*d) - (2*a^2*cot(c + d*x)^(9/2)*sqrt(a + b*tan(c + d*x)))/(9*d)]
    @test_int [cot(c + d*x)^(9/2)*(a + b*tan(c + d*x))^(5/2), x, 12, (im*(im*a - b)^(5/2)*atan((sqrt(im*a - b)*sqrt(tan(c + d*x)))/sqrt(a + b*tan(c + d*x)))*sqrt(cot(c + d*x))*sqrt(tan(c + d*x)))/d + (im*(im*a + b)^(5/2)*atanh((sqrt(im*a + b)*sqrt(tan(c + d*x)))/sqrt(a + b*tan(c + d*x)))*sqrt(cot(c + d*x))*sqrt(tan(c + d*x)))/d + (2*b*(49*a^2 - 3*b^2)*sqrt(cot(c + d*x))*sqrt(a + b*tan(c + d*x)))/(21*a*d) + (2*(7*a^2 - 9*b^2)*cot(c + d*x)^(3/2)*sqrt(a + b*tan(c + d*x)))/(21*d) - (6*a*b*cot(c + d*x)^(5/2)*sqrt(a + b*tan(c + d*x)))/(7*d) - (2*a^2*cot(c + d*x)^(7/2)*sqrt(a + b*tan(c + d*x)))/(7*d)]
    @test_int [cot(c + d*x)^(7/2)*(a + b*tan(c + d*x))^(5/2), x, 11, -(((im*a - b)^(5/2)*atan((sqrt(im*a - b)*sqrt(tan(c + d*x)))/sqrt(a + b*tan(c + d*x)))*sqrt(cot(c + d*x))*sqrt(tan(c + d*x)))/d) + ((im*a + b)^(5/2)*atanh((sqrt(im*a + b)*sqrt(tan(c + d*x)))/sqrt(a + b*tan(c + d*x)))*sqrt(cot(c + d*x))*sqrt(tan(c + d*x)))/d + (2*(15*a^2 - 23*b^2)*sqrt(cot(c + d*x))*sqrt(a + b*tan(c + d*x)))/(15*d) - (22*a*b*cot(c + d*x)^(3/2)*sqrt(a + b*tan(c + d*x)))/(15*d) - (2*a^2*cot(c + d*x)^(5/2)*sqrt(a + b*tan(c + d*x)))/(5*d)]
    @test_int [cot(c + d*x)^(5/2)*(a + b*tan(c + d*x))^(5/2), x, 10, -((im*(im*a - b)^(5/2)*atan((sqrt(im*a - b)*sqrt(tan(c + d*x)))/sqrt(a + b*tan(c + d*x)))*sqrt(cot(c + d*x))*sqrt(tan(c + d*x)))/d) - (im*(im*a + b)^(5/2)*atanh((sqrt(im*a + b)*sqrt(tan(c + d*x)))/sqrt(a + b*tan(c + d*x)))*sqrt(cot(c + d*x))*sqrt(tan(c + d*x)))/d - (14*a*b*sqrt(cot(c + d*x))*sqrt(a + b*tan(c + d*x)))/(3*d) - (2*a^2*cot(c + d*x)^(3/2)*sqrt(a + b*tan(c + d*x)))/(3*d)]
    @test_int [cot(c + d*x)^(3/2)*(a + b*tan(c + d*x))^(5/2), x, 14, ((im*a - b)^(5/2)*atan((sqrt(im*a - b)*sqrt(tan(c + d*x)))/sqrt(a + b*tan(c + d*x)))*sqrt(cot(c + d*x))*sqrt(tan(c + d*x)))/d + (2*b^(5/2)*atanh((sqrt(b)*sqrt(tan(c + d*x)))/sqrt(a + b*tan(c + d*x)))*sqrt(cot(c + d*x))*sqrt(tan(c + d*x)))/d - ((im*a + b)^(5/2)*atanh((sqrt(im*a + b)*sqrt(tan(c + d*x)))/sqrt(a + b*tan(c + d*x)))*sqrt(cot(c + d*x))*sqrt(tan(c + d*x)))/d - (2*a^2*sqrt(cot(c + d*x))*sqrt(a + b*tan(c + d*x)))/d]
    @test_int [sqrt(cot(c + d*x))*(a + b*tan(c + d*x))^(5/2), x, 14, (im*(im*a - b)^(5/2)*atan((sqrt(im*a - b)*sqrt(tan(c + d*x)))/sqrt(a + b*tan(c + d*x)))*sqrt(cot(c + d*x))*sqrt(tan(c + d*x)))/d + (5*a*b^(3/2)*atanh((sqrt(b)*sqrt(tan(c + d*x)))/sqrt(a + b*tan(c + d*x)))*sqrt(cot(c + d*x))*sqrt(tan(c + d*x)))/d + (im*(im*a + b)^(5/2)*atanh((sqrt(im*a + b)*sqrt(tan(c + d*x)))/sqrt(a + b*tan(c + d*x)))*sqrt(cot(c + d*x))*sqrt(tan(c + d*x)))/d + (b^2*sqrt(a + b*tan(c + d*x)))/(d*sqrt(cot(c + d*x)))]
    @test_int [(a + b*tan(c + d*x))^(5/2)/sqrt(cot(c + d*x)), x, 15, -(((im*a - b)^(5/2)*atan((sqrt(im*a - b)*sqrt(tan(c + d*x)))/sqrt(a + b*tan(c + d*x)))*sqrt(cot(c + d*x))*sqrt(tan(c + d*x)))/d) + (sqrt(b)*(15*a^2 - 8*b^2)*atanh((sqrt(b)*sqrt(tan(c + d*x)))/sqrt(a + b*tan(c + d*x)))*sqrt(cot(c + d*x))*sqrt(tan(c + d*x)))/(4*d) + ((im*a + b)^(5/2)*atanh((sqrt(im*a + b)*sqrt(tan(c + d*x)))/sqrt(a + b*tan(c + d*x)))*sqrt(cot(c + d*x))*sqrt(tan(c + d*x)))/d + (b^2*sqrt(a + b*tan(c + d*x)))/(2*d*cot(c + d*x)^(3/2)) + (9*a*b*sqrt(a + b*tan(c + d*x)))/(4*d*sqrt(cot(c + d*x)))]
    @test_int [(a + b*tan(c + d*x))^(5/2)/cot(c + d*x)^(3/2), x, 16, -((im*(im*a - b)^(5/2)*atan((sqrt(im*a - b)*sqrt(tan(c + d*x)))/sqrt(a + b*tan(c + d*x)))*sqrt(cot(c + d*x))*sqrt(tan(c + d*x)))/d) + (5*a*(a^2 - 8*b^2)*atanh((sqrt(b)*sqrt(tan(c + d*x)))/sqrt(a + b*tan(c + d*x)))*sqrt(cot(c + d*x))*sqrt(tan(c + d*x)))/(8*sqrt(b)*d) - (im*(im*a + b)^(5/2)*atanh((sqrt(im*a + b)*sqrt(tan(c + d*x)))/sqrt(a + b*tan(c + d*x)))*sqrt(cot(c + d*x))*sqrt(tan(c + d*x)))/d + (b^2*sqrt(a + b*tan(c + d*x)))/(3*d*cot(c + d*x)^(5/2)) + (13*a*b*sqrt(a + b*tan(c + d*x)))/(12*d*cot(c + d*x)^(3/2)) + ((11*a^2 - 8*b^2)*sqrt(a + b*tan(c + d*x)))/(8*d*sqrt(cot(c + d*x)))]


    #= ::Subsubsection::Closed:: =#
    #=m<0=#


    @test_int [cot(c + d*x)^(5/2)/sqrt(a + b*tan(c + d*x)), x, 11, -((atan((sqrt(im*a - b)*sqrt(tan(c + d*x)))/sqrt(a + b*tan(c + d*x)))*sqrt(cot(c + d*x))*sqrt(tan(c + d*x)))/(sqrt(im*a - b)*d)) - (atanh((sqrt(im*a + b)*sqrt(tan(c + d*x)))/sqrt(a + b*tan(c + d*x)))*sqrt(cot(c + d*x))*sqrt(tan(c + d*x)))/(sqrt(im*a + b)*d) + (4*b*sqrt(cot(c + d*x))*sqrt(a + b*tan(c + d*x)))/(3*a^2*d) - (2*cot(c + d*x)^(3/2)*sqrt(a + b*tan(c + d*x)))/(3*a*d)]
    @test_int [cot(c + d*x)^(3/2)/sqrt(a + b*tan(c + d*x)), x, 10, ((-im)*atan((sqrt(im*a - b)*sqrt(tan(c + d*x)))/sqrt(a + b*tan(c + d*x)))*sqrt(cot(c + d*x))*sqrt(tan(c + d*x)))/(sqrt(im*a - b)*d) + (im*atanh((sqrt(im*a + b)*sqrt(tan(c + d*x)))/sqrt(a + b*tan(c + d*x)))*sqrt(cot(c + d*x))*sqrt(tan(c + d*x)))/(sqrt(im*a + b)*d) - (2*sqrt(cot(c + d*x))*sqrt(a + b*tan(c + d*x)))/(a*d)]
    @test_int [sqrt(cot(c + d*x))/sqrt(a + b*tan(c + d*x)), x, 8, (atan((sqrt(im*a - b)*sqrt(tan(c + d*x)))/sqrt(a + b*tan(c + d*x)))*sqrt(cot(c + d*x))*sqrt(tan(c + d*x)))/(sqrt(im*a - b)*d) + (atanh((sqrt(im*a + b)*sqrt(tan(c + d*x)))/sqrt(a + b*tan(c + d*x)))*sqrt(cot(c + d*x))*sqrt(tan(c + d*x)))/(sqrt(im*a + b)*d)]
    @test_int [1/(sqrt(cot(c + d*x))*sqrt(a + b*tan(c + d*x))), x, 8, (im*atan((sqrt(im*a - b)*sqrt(tan(c + d*x)))/sqrt(a + b*tan(c + d*x)))*sqrt(cot(c + d*x))*sqrt(tan(c + d*x)))/(sqrt(im*a - b)*d) - (im*atanh((sqrt(im*a + b)*sqrt(tan(c + d*x)))/sqrt(a + b*tan(c + d*x)))*sqrt(cot(c + d*x))*sqrt(tan(c + d*x)))/(sqrt(im*a + b)*d)]
    @test_int [1/(cot(c + d*x)^(3/2)*sqrt(a + b*tan(c + d*x))), x, 13, -((atan((sqrt(im*a - b)*sqrt(tan(c + d*x)))/sqrt(a + b*tan(c + d*x)))*sqrt(cot(c + d*x))*sqrt(tan(c + d*x)))/(sqrt(im*a - b)*d)) + (2*atanh((sqrt(b)*sqrt(tan(c + d*x)))/sqrt(a + b*tan(c + d*x)))*sqrt(cot(c + d*x))*sqrt(tan(c + d*x)))/(sqrt(b)*d) - (atanh((sqrt(im*a + b)*sqrt(tan(c + d*x)))/sqrt(a + b*tan(c + d*x)))*sqrt(cot(c + d*x))*sqrt(tan(c + d*x)))/(sqrt(im*a + b)*d)]
    @test_int [1/(cot(c + d*x)^(5/2)*sqrt(a + b*tan(c + d*x))), x, 14, ((-im)*atan((sqrt(im*a - b)*sqrt(tan(c + d*x)))/sqrt(a + b*tan(c + d*x)))*sqrt(cot(c + d*x))*sqrt(tan(c + d*x)))/(sqrt(im*a - b)*d) - (a*atanh((sqrt(b)*sqrt(tan(c + d*x)))/sqrt(a + b*tan(c + d*x)))*sqrt(cot(c + d*x))*sqrt(tan(c + d*x)))/(b^(3/2)*d) + (im*atanh((sqrt(im*a + b)*sqrt(tan(c + d*x)))/sqrt(a + b*tan(c + d*x)))*sqrt(cot(c + d*x))*sqrt(tan(c + d*x)))/(sqrt(im*a + b)*d) + sqrt(a + b*tan(c + d*x))/(b*d*sqrt(cot(c + d*x)))]


    @test_int [cot(c + d*x)^(5/2)/(a + b*tan(c + d*x))^(3/2), x, 11, -((im*atan((sqrt(im*a - b)*sqrt(tan(c + d*x)))/sqrt(a + b*tan(c + d*x)))*sqrt(cot(c + d*x))*sqrt(tan(c + d*x)))/((im*a - b)^(3/2)*d)) - (im*atanh((sqrt(im*a + b)*sqrt(tan(c + d*x)))/sqrt(a + b*tan(c + d*x)))*sqrt(cot(c + d*x))*sqrt(tan(c + d*x)))/((im*a + b)^(3/2)*d) + (2*b^2*(5*a^2 + 8*b^2))/(3*a^3*(a^2 + b^2)*d*sqrt(cot(c + d*x))*sqrt(a + b*tan(c + d*x))) + (8*b*sqrt(cot(c + d*x)))/(3*a^2*d*sqrt(a + b*tan(c + d*x))) - (2*cot(c + d*x)^(3/2))/(3*a*d*sqrt(a + b*tan(c + d*x)))]
    @test_int [cot(c + d*x)^(3/2)/(a + b*tan(c + d*x))^(3/2), x, 10, (atan((sqrt(im*a - b)*sqrt(tan(c + d*x)))/sqrt(a + b*tan(c + d*x)))*sqrt(cot(c + d*x))*sqrt(tan(c + d*x)))/((im*a - b)^(3/2)*d) - (atanh((sqrt(im*a + b)*sqrt(tan(c + d*x)))/sqrt(a + b*tan(c + d*x)))*sqrt(cot(c + d*x))*sqrt(tan(c + d*x)))/((im*a + b)^(3/2)*d) - (2*b*(a^2 + 2*b^2))/(a^2*(a^2 + b^2)*d*sqrt(cot(c + d*x))*sqrt(a + b*tan(c + d*x))) - (2*sqrt(cot(c + d*x)))/(a*d*sqrt(a + b*tan(c + d*x)))]
    @test_int [sqrt(cot(c + d*x))/(a + b*tan(c + d*x))^(3/2), x, 9, (im*atan((sqrt(im*a - b)*sqrt(tan(c + d*x)))/sqrt(a + b*tan(c + d*x)))*sqrt(cot(c + d*x))*sqrt(tan(c + d*x)))/((im*a - b)^(3/2)*d) + (im*atanh((sqrt(im*a + b)*sqrt(tan(c + d*x)))/sqrt(a + b*tan(c + d*x)))*sqrt(cot(c + d*x))*sqrt(tan(c + d*x)))/((im*a + b)^(3/2)*d) + (2*b^2)/(a*(a^2 + b^2)*d*sqrt(cot(c + d*x))*sqrt(a + b*tan(c + d*x)))]
    @test_int [1/(sqrt(cot(c + d*x))*(a + b*tan(c + d*x))^(3/2)), x, 9, -((atan((sqrt(im*a - b)*sqrt(tan(c + d*x)))/sqrt(a + b*tan(c + d*x)))*sqrt(cot(c + d*x))*sqrt(tan(c + d*x)))/((im*a - b)^(3/2)*d)) + (atanh((sqrt(im*a + b)*sqrt(tan(c + d*x)))/sqrt(a + b*tan(c + d*x)))*sqrt(cot(c + d*x))*sqrt(tan(c + d*x)))/((im*a + b)^(3/2)*d) - (2*b)/((a^2 + b^2)*d*sqrt(cot(c + d*x))*sqrt(a + b*tan(c + d*x)))]
    @test_int [1/(cot(c + d*x)^(3/2)*(a + b*tan(c + d*x))^(3/2)), x, 9, -((im*atan((sqrt(im*a - b)*sqrt(tan(c + d*x)))/sqrt(a + b*tan(c + d*x)))*sqrt(cot(c + d*x))*sqrt(tan(c + d*x)))/((im*a - b)^(3/2)*d)) - (im*atanh((sqrt(im*a + b)*sqrt(tan(c + d*x)))/sqrt(a + b*tan(c + d*x)))*sqrt(cot(c + d*x))*sqrt(tan(c + d*x)))/((im*a + b)^(3/2)*d) + (2*a)/((a^2 + b^2)*d*sqrt(cot(c + d*x))*sqrt(a + b*tan(c + d*x)))]
    @test_int [1/(cot(c + d*x)^(5/2)*(a + b*tan(c + d*x))^(3/2)), x, 14, (atan((sqrt(im*a - b)*sqrt(tan(c + d*x)))/sqrt(a + b*tan(c + d*x)))*sqrt(cot(c + d*x))*sqrt(tan(c + d*x)))/((im*a - b)^(3/2)*d) + (2*atanh((sqrt(b)*sqrt(tan(c + d*x)))/sqrt(a + b*tan(c + d*x)))*sqrt(cot(c + d*x))*sqrt(tan(c + d*x)))/(b^(3/2)*d) - (atanh((sqrt(im*a + b)*sqrt(tan(c + d*x)))/sqrt(a + b*tan(c + d*x)))*sqrt(cot(c + d*x))*sqrt(tan(c + d*x)))/((im*a + b)^(3/2)*d) - (2*a^2)/(b*(a^2 + b^2)*d*sqrt(cot(c + d*x))*sqrt(a + b*tan(c + d*x)))]
    @test_int [1/(cot(c + d*x)^(7/2)*(a + b*tan(c + d*x))^(3/2)), x, 15, (im*atan((sqrt(im*a - b)*sqrt(tan(c + d*x)))/sqrt(a + b*tan(c + d*x)))*sqrt(cot(c + d*x))*sqrt(tan(c + d*x)))/((im*a - b)^(3/2)*d) - (3*a*atanh((sqrt(b)*sqrt(tan(c + d*x)))/sqrt(a + b*tan(c + d*x)))*sqrt(cot(c + d*x))*sqrt(tan(c + d*x)))/(b^(5/2)*d) + (im*atanh((sqrt(im*a + b)*sqrt(tan(c + d*x)))/sqrt(a + b*tan(c + d*x)))*sqrt(cot(c + d*x))*sqrt(tan(c + d*x)))/((im*a + b)^(3/2)*d) - (2*a^2)/(b*(a^2 + b^2)*d*cot(c + d*x)^(3/2)*sqrt(a + b*tan(c + d*x))) + ((3*a^2 + b^2)*sqrt(a + b*tan(c + d*x)))/(b^2*(a^2 + b^2)*d*sqrt(cot(c + d*x)))]


    @test_int [cot(c + d*x)^(5/2)/(a + b*tan(c + d*x))^(5/2), x, 12, (atan((sqrt(im*a - b)*sqrt(tan(c + d*x)))/sqrt(a + b*tan(c + d*x)))*sqrt(cot(c + d*x))*sqrt(tan(c + d*x)))/((im*a - b)^(5/2)*d) + (atanh((sqrt(im*a + b)*sqrt(tan(c + d*x)))/sqrt(a + b*tan(c + d*x)))*sqrt(cot(c + d*x))*sqrt(tan(c + d*x)))/((im*a + b)^(5/2)*d) + (2*b^2*(7*a^2 + 8*b^2))/(3*a^3*(a^2 + b^2)*d*sqrt(cot(c + d*x))*(a + b*tan(c + d*x))^(3/2)) + (4*b*sqrt(cot(c + d*x)))/(a^2*d*(a + b*tan(c + d*x))^(3/2)) - (2*cot(c + d*x)^(3/2))/(3*a*d*(a + b*tan(c + d*x))^(3/2)) + (4*b^2*(4*a^4 + 15*a^2*b^2 + 8*b^4))/(3*a^4*(a^2 + b^2)^2*d*sqrt(cot(c + d*x))*sqrt(a + b*tan(c + d*x)))]
    @test_int [cot(c + d*x)^(3/2)/(a + b*tan(c + d*x))^(5/2), x, 11, (im*atan((sqrt(im*a - b)*sqrt(tan(c + d*x)))/sqrt(a + b*tan(c + d*x)))*sqrt(cot(c + d*x))*sqrt(tan(c + d*x)))/((im*a - b)^(5/2)*d) - (im*atanh((sqrt(im*a + b)*sqrt(tan(c + d*x)))/sqrt(a + b*tan(c + d*x)))*sqrt(cot(c + d*x))*sqrt(tan(c + d*x)))/((im*a + b)^(5/2)*d) - (2*b*(3*a^2 + 4*b^2))/(3*a^2*(a^2 + b^2)*d*sqrt(cot(c + d*x))*(a + b*tan(c + d*x))^(3/2)) - (2*sqrt(cot(c + d*x)))/(a*d*(a + b*tan(c + d*x))^(3/2)) - (2*b*(3*a^4 + 17*a^2*b^2 + 8*b^4))/(3*a^3*(a^2 + b^2)^2*d*sqrt(cot(c + d*x))*sqrt(a + b*tan(c + d*x)))]
    @test_int [sqrt(cot(c + d*x))/(a + b*tan(c + d*x))^(5/2), x, 10, -((atan((sqrt(im*a - b)*sqrt(tan(c + d*x)))/sqrt(a + b*tan(c + d*x)))*sqrt(cot(c + d*x))*sqrt(tan(c + d*x)))/((im*a - b)^(5/2)*d)) - (atanh((sqrt(im*a + b)*sqrt(tan(c + d*x)))/sqrt(a + b*tan(c + d*x)))*sqrt(cot(c + d*x))*sqrt(tan(c + d*x)))/((im*a + b)^(5/2)*d) + (2*b^2)/(3*a*(a^2 + b^2)*d*sqrt(cot(c + d*x))*(a + b*tan(c + d*x))^(3/2)) + (4*b^2*(4*a^2 + b^2))/(3*a^2*(a^2 + b^2)^2*d*sqrt(cot(c + d*x))*sqrt(a + b*tan(c + d*x)))]
    @test_int [1/(sqrt(cot(c + d*x))*(a + b*tan(c + d*x))^(5/2)), x, 10, -((im*atan((sqrt(im*a - b)*sqrt(tan(c + d*x)))/sqrt(a + b*tan(c + d*x)))*sqrt(cot(c + d*x))*sqrt(tan(c + d*x)))/((im*a - b)^(5/2)*d)) + (im*atanh((sqrt(im*a + b)*sqrt(tan(c + d*x)))/sqrt(a + b*tan(c + d*x)))*sqrt(cot(c + d*x))*sqrt(tan(c + d*x)))/((im*a + b)^(5/2)*d) - (2*b)/(3*(a^2 + b^2)*d*sqrt(cot(c + d*x))*(a + b*tan(c + d*x))^(3/2)) - (2*b*(5*a^2 - b^2))/(3*a*(a^2 + b^2)^2*d*sqrt(cot(c + d*x))*sqrt(a + b*tan(c + d*x)))]
    @test_int [1/(cot(c + d*x)^(3/2)*(a + b*tan(c + d*x))^(5/2)), x, 10, (atan((sqrt(im*a - b)*sqrt(tan(c + d*x)))/sqrt(a + b*tan(c + d*x)))*sqrt(cot(c + d*x))*sqrt(tan(c + d*x)))/((im*a - b)^(5/2)*d) + (atanh((sqrt(im*a + b)*sqrt(tan(c + d*x)))/sqrt(a + b*tan(c + d*x)))*sqrt(cot(c + d*x))*sqrt(tan(c + d*x)))/((im*a + b)^(5/2)*d) + (2*a)/(3*(a^2 + b^2)*d*sqrt(cot(c + d*x))*(a + b*tan(c + d*x))^(3/2)) + (4*(a^2 - 2*b^2))/(3*(a^2 + b^2)^2*d*sqrt(cot(c + d*x))*sqrt(a + b*tan(c + d*x)))]
    @test_int [1/(cot(c + d*x)^(5/2)*(a + b*tan(c + d*x))^(5/2)), x, 10, (im*atan((sqrt(im*a - b)*sqrt(tan(c + d*x)))/sqrt(a + b*tan(c + d*x)))*sqrt(cot(c + d*x))*sqrt(tan(c + d*x)))/((im*a - b)^(5/2)*d) - (im*atanh((sqrt(im*a + b)*sqrt(tan(c + d*x)))/sqrt(a + b*tan(c + d*x)))*sqrt(cot(c + d*x))*sqrt(tan(c + d*x)))/((im*a + b)^(5/2)*d) - (2*a^2)/(3*b*(a^2 + b^2)*d*sqrt(cot(c + d*x))*(a + b*tan(c + d*x))^(3/2)) + (2*a*(a^2 + 7*b^2))/(3*b*(a^2 + b^2)^2*d*sqrt(cot(c + d*x))*sqrt(a + b*tan(c + d*x)))]


    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*(a+b*tan(e+f*x))^m*(d*cot(e+f*x))^n*with*n*symbolic=#


    @test_int [(d*cot(e + f*x))^n*(a + b*tan(e + f*x))^3, x, 8, If(13>=8, (a^2*b*d^2*(1 - 2*n)*(d*cot(e + f*x))^(-2 + n))/(f*(1 - n)*(2 - n)) + (a^2*d^2*(d*cot(e + f*x))^(-2 + n)*(b + a*cot(e + f*x)))/(f*(1 - n)) - (b*(3*a^2 - b^2)*d^2*(d*cot(e + f*x))^(-2 + n)*HypergeometricFunctions._₂F₁(1, (1/2)*(-2 + n), n/2, -cot(e + f*x)^2))/(f*(2 - n)) - (a*(a^2 - 3*b^2)*d*(d*cot(e + f*x))^(-1 + n)*HypergeometricFunctions._₂F₁(1, (1/2)*(-1 + n), (1 + n)/2, -cot(e + f*x)^2))/(f*(1 - n)), (a^2*b*d^2*(1 - 2*n)*(d*cot(e + f*x))^(-2 + n))/(f*(2 - 3*n + n^2)) + (a^2*d^2*(d*cot(e + f*x))^(-2 + n)*(b + a*cot(e + f*x)))/(f*(1 - n)) - (b*(3*a^2 - b^2)*d^2*(d*cot(e + f*x))^(-2 + n)*HypergeometricFunctions._₂F₁(1, (1/2)*(-2 + n), n/2, -cot(e + f*x)^2))/(f*(2 - n)) - (a*(a^2 - 3*b^2)*d*(d*cot(e + f*x))^(-1 + n)*HypergeometricFunctions._₂F₁(1, (1/2)*(-1 + n), (1 + n)/2, -cot(e + f*x)^2))/(f*(1 - n)))]
    @test_int [(d*cot(e + f*x))^n*(a + b*tan(e + f*x))^2, x, 7, (a^2*d*(d*cot(e + f*x))^(-1 + n))/(f*(1 - n)) - ((a^2 - b^2)*d*(d*cot(e + f*x))^(-1 + n)*HypergeometricFunctions._₂F₁(1, (1/2)*(-1 + n), (1 + n)/2, -cot(e + f*x)^2))/(f*(1 - n)) - (2*a*b*(d*cot(e + f*x))^n*HypergeometricFunctions._₂F₁(1, n/2, (2 + n)/2, -cot(e + f*x)^2))/(f*n)]
    @test_int [(d*cot(e + f*x))^n*(a + b*tan(e + f*x))^1, x, 6, -((b*(d*cot(e + f*x))^n*HypergeometricFunctions._₂F₁(1, n/2, (2 + n)/2, -cot(e + f*x)^2))/(f*n)) - (a*(d*cot(e + f*x))^(1 + n)*HypergeometricFunctions._₂F₁(1, (1 + n)/2, (3 + n)/2, -cot(e + f*x)^2))/(d*f*(1 + n))]
    @test_int [(d*cot(e + f*x))^n/(a + b*tan(e + f*x))^1, x, 9, -((b*(d*cot(e + f*x))^(2 + n)*HypergeometricFunctions._₂F₁(1, (2 + n)/2, (4 + n)/2, -cot(e + f*x)^2))/((a^2 + b^2)*d^2*f*(2 + n))) - (a^2*(d*cot(e + f*x))^(2 + n)*HypergeometricFunctions._₂F₁(1, 2 + n, 3 + n, -((a*cot(e + f*x))/b)))/(b*(a^2 + b^2)*d^2*f*(2 + n)) + (a*(d*cot(e + f*x))^(3 + n)*HypergeometricFunctions._₂F₁(1, (3 + n)/2, (5 + n)/2, -cot(e + f*x)^2))/((a^2 + b^2)*d^3*f*(3 + n))]
    @test_int [(d*cot(e + f*x))^n/(a + b*tan(e + f*x))^2, x, 10, -((a^2*(d*cot(e + f*x))^(3 + n))/(b*(a^2 + b^2)*d^3*f*(b + a*cot(e + f*x)))) + ((a^2 - b^2)*(d*cot(e + f*x))^(3 + n)*HypergeometricFunctions._₂F₁(1, (3 + n)/2, (5 + n)/2, -cot(e + f*x)^2))/((a^2 + b^2)^2*d^3*f*(3 + n)) + (a^2*(b^2*n + a^2*(2 + n))*(d*cot(e + f*x))^(3 + n)*HypergeometricFunctions._₂F₁(1, 3 + n, 4 + n, -((a*cot(e + f*x))/b)))/(b^2*(a^2 + b^2)^2*d^3*f*(3 + n)) + (2*a*b*(d*cot(e + f*x))^(4 + n)*HypergeometricFunctions._₂F₁(1, (4 + n)/2, (6 + n)/2, -cot(e + f*x)^2))/((a^2 + b^2)^2*d^4*f*(4 + n))]


    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*(a+b*tan(e+f*x))^m*(d*cot(e+f*x))^n*with*m*symbolic=#


    @test_int [(a + b*tan(e + f*x))^m*(d*cot(e + f*x))^n, x, 8, (AppellF1(1 - n, -m, 1, 2 - n, -((b*tan(e + f*x))/a), (-im)*tan(e + f*x))*(d*cot(e + f*x))^n*tan(e + f*x)*(a + b*tan(e + f*x))^m)/((1 + (b*tan(e + f*x))/a)^m*(2*f*(1 - n))) + (AppellF1(1 - n, -m, 1, 2 - n, -((b*tan(e + f*x))/a), im*tan(e + f*x))*(d*cot(e + f*x))^n*tan(e + f*x)*(a + b*tan(e + f*x))^m)/((1 + (b*tan(e + f*x))/a)^m*(2*f*(1 - n)))]


    @test_int [cot(c + d*x)^(3/2)*(a + b*tan(c + d*x))^n, x, 10, -((AppellF1(-(1/2), 1, -n, 1/2, (-im)*tan(c + d*x), -((b*tan(c + d*x))/a))*sqrt(cot(c + d*x))*(a + b*tan(c + d*x))^n)/((1 + (b*tan(c + d*x))/a)^n*d)) - (AppellF1(-(1/2), 1, -n, 1/2, im*tan(c + d*x), -((b*tan(c + d*x))/a))*sqrt(cot(c + d*x))*(a + b*tan(c + d*x))^n)/((1 + (b*tan(c + d*x))/a)^n*d)]
    @test_int [sqrt(cot(c + d*x))*(a + b*tan(c + d*x))^n, x, 10, (AppellF1(1/2, 1, -n, 3/2, (-im)*tan(c + d*x), -((b*tan(c + d*x))/a))*(a + b*tan(c + d*x))^n)/((1 + (b*tan(c + d*x))/a)^n*(d*sqrt(cot(c + d*x)))) + (AppellF1(1/2, 1, -n, 3/2, im*tan(c + d*x), -((b*tan(c + d*x))/a))*(a + b*tan(c + d*x))^n)/((1 + (b*tan(c + d*x))/a)^n*(d*sqrt(cot(c + d*x))))]
    @test_int [(a + b*tan(c + d*x))^n/sqrt(cot(c + d*x)), x, 10, (AppellF1(3/2, 1, -n, 5/2, (-im)*tan(c + d*x), -((b*tan(c + d*x))/a))*(a + b*tan(c + d*x))^n)/((1 + (b*tan(c + d*x))/a)^n*(3*d*cot(c + d*x)^(3/2))) + (AppellF1(3/2, 1, -n, 5/2, im*tan(c + d*x), -((b*tan(c + d*x))/a))*(a + b*tan(c + d*x))^n)/((1 + (b*tan(c + d*x))/a)^n*(3*d*cot(c + d*x)^(3/2)))]
    @test_int [(a + b*tan(c + d*x))^n/cot(c + d*x)^(3/2), x, 10, (AppellF1(5/2, 1, -n, 7/2, (-im)*tan(c + d*x), -((b*tan(c + d*x))/a))*(a + b*tan(c + d*x))^n)/((1 + (b*tan(c + d*x))/a)^n*(5*d*cot(c + d*x)^(5/2))) + (AppellF1(5/2, 1, -n, 7/2, im*tan(c + d*x), -((b*tan(c + d*x))/a))*(a + b*tan(c + d*x))^n)/((1 + (b*tan(c + d*x))/a)^n*(5*d*cot(c + d*x)^(5/2)))]


    #= ::Title::Closed:: =#
    #=Integrands*of*the*form*(a+b*tan(e+f*x))^m*(c+d*tan(e+f*x))^n=#


    #= ::Section::Closed:: =#
    #=Integrands*of*the*form*(a+im*a*tan(e+f*x))^m*(c-c*im*tan(e+f*x))^n=#


    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*(a+im*a*tan(e+f*x))^m*(c-c*im*tan(e+f*x))^n=#


    #= ::Subsubsection::Closed:: =#
    #=n > 0=#


    @test_int [(a + im*a*tan(e + f*x))^3*(c - im*c*tan(e + f*x)), x, 3, -((im*c*(a + im*a*tan(e + f*x))^3)/(3*f))]
    @test_int [(a + im*a*tan(e + f*x))^2*(c - im*c*tan(e + f*x)), x, 4, -((im*c*(a + im*a*tan(e + f*x))^2)/(2*f)), (im*a^2*c*sec(e + f*x)^2)/(2*f) + (a^2*c*tan(e + f*x))/f]
    @test_int [(a + im*a*tan(e + f*x))^1*(c - im*c*tan(e + f*x)), x, 3, (a*c*tan(e + f*x))/f]
    @test_int [(c - im*c*tan(e + f*x))/(a + im*a*tan(e + f*x))^1, x, 3, (im*c)/(f*(a + im*a*tan(e + f*x)))]
    @test_int [(c - im*c*tan(e + f*x))/(a + im*a*tan(e + f*x))^2, x, 3, (im*c)/(2*f*(a + im*a*tan(e + f*x))^2)]
    @test_int [(c - im*c*tan(e + f*x))/(a + im*a*tan(e + f*x))^3, x, 3, (im*c)/(3*f*(a + im*a*tan(e + f*x))^3)]


    @test_int [(a + im*a*tan(e + f*x))^4*(c - im*c*tan(e + f*x))^2, x, 4, -((im*c^2*(a + im*a*tan(e + f*x))^4)/(2*f)) + (im*c^2*(a + im*a*tan(e + f*x))^5)/(5*a*f)]
    @test_int [(a + im*a*tan(e + f*x))^3*(c - im*c*tan(e + f*x))^2, x, 4, (im*a^3*c^2*sec(e + f*x)^4)/(4*f) + (a^3*c^2*tan(e + f*x))/f + (a^3*c^2*tan(e + f*x)^3)/(3*f)]
    @test_int [(a + im*a*tan(e + f*x))^2*(c - im*c*tan(e + f*x))^2, x, 3, (a^2*c^2*tan(e + f*x))/f + (a^2*c^2*tan(e + f*x)^3)/(3*f)]
    @test_int [(a + im*a*tan(e + f*x))^1*(c - im*c*tan(e + f*x))^2, x, 4, (im*a*(c - im*c*tan(e + f*x))^2)/(2*f), -((im*a*c^2*sec(e + f*x)^2)/(2*f)) + (a*c^2*tan(e + f*x))/f]
    @test_int [(c - im*c*tan(e + f*x))^2/(a + im*a*tan(e + f*x))^1, x, 4, -((c^2*x)/a) - (im*c^2*log(cos(e + f*x)))/(a*f) + (2*im*c^2)/(f*(a + im*a*tan(e + f*x)))]
    @test_int [(c - im*c*tan(e + f*x))^2/(a + im*a*tan(e + f*x))^2, x, 3, (c^2*tan(e + f*x))/(f*(a + im*a*tan(e + f*x))^2)]
    @test_int [(c - im*c*tan(e + f*x))^2/(a + im*a*tan(e + f*x))^3, x, 4, (2*im*c^2)/(3*f*(a + im*a*tan(e + f*x))^3) - (im*c^2)/(2*a*f*(a + im*a*tan(e + f*x))^2)]
    @test_int [(c - im*c*tan(e + f*x))^2/(a + im*a*tan(e + f*x))^4, x, 4, (im*c^2)/(2*f*(a + im*a*tan(e + f*x))^4) - (im*a^2*c^2)/(3*f*(a^2 + im*a^2*tan(e + f*x))^3)]


    @test_int [(a + im*a*tan(e + f*x))^5*(c - im*c*tan(e + f*x))^3, x, 4, -((4*im*c^3*(a + im*a*tan(e + f*x))^5)/(5*f)) + (2*im*c^3*(a + im*a*tan(e + f*x))^6)/(3*a*f) - (im*c^3*(a + im*a*tan(e + f*x))^7)/(7*a^2*f)]
    @test_int [(a + im*a*tan(e + f*x))^4*(c - im*c*tan(e + f*x))^3, x, 4, (im*a^4*c^3*sec(e + f*x)^6)/(6*f) + (a^4*c^3*tan(e + f*x))/f + (2*a^4*c^3*tan(e + f*x)^3)/(3*f) + (a^4*c^3*tan(e + f*x)^5)/(5*f)]
    @test_int [(a + im*a*tan(e + f*x))^3*(c - im*c*tan(e + f*x))^3, x, 3, (a^3*c^3*tan(e + f*x))/f + (2*a^3*c^3*tan(e + f*x)^3)/(3*f) + (a^3*c^3*tan(e + f*x)^5)/(5*f)]
    @test_int [(a + im*a*tan(e + f*x))^2*(c - im*c*tan(e + f*x))^3, x, 4, -((im*a^2*c^3*sec(e + f*x)^4)/(4*f)) + (a^2*c^3*tan(e + f*x))/f + (a^2*c^3*tan(e + f*x)^3)/(3*f)]
    @test_int [(a + im*a*tan(e + f*x))^1*(c - im*c*tan(e + f*x))^3, x, 3, (im*a*(c - im*c*tan(e + f*x))^3)/(3*f)]
    @test_int [(c - im*c*tan(e + f*x))^3/(a + im*a*tan(e + f*x))^1, x, 4, -((4*c^3*x)/a) - (4*im*c^3*log(cos(e + f*x)))/(a*f) + (c^3*tan(e + f*x))/(a*f) + (4*im*c^3)/(f*(a + im*a*tan(e + f*x)))]
    @test_int [(c - im*c*tan(e + f*x))^3/(a + im*a*tan(e + f*x))^2, x, 4, (c^3*x)/a^2 + (im*c^3*log(cos(e + f*x)))/(a^2*f) + (2*im*c^3)/(f*(a + im*a*tan(e + f*x))^2) - (4*im*c^3)/(f*(a^2 + im*a^2*tan(e + f*x)))]
    @test_int [(c - im*c*tan(e + f*x))^3/(a + im*a*tan(e + f*x))^3, x, 3, (im*c^3*(a^2 - im*a^2*tan(e + f*x))^3)/(6*f*(a^3 + im*a^3*tan(e + f*x))^3)]
    @test_int [(c - im*c*tan(e + f*x))^3/(a + im*a*tan(e + f*x))^4, x, 4, (im*c^3)/(f*(a + im*a*tan(e + f*x))^4) - (4*im*c^3)/(3*a*f*(a + im*a*tan(e + f*x))^3) + (im*c^3)/(2*f*(a^2 + im*a^2*tan(e + f*x))^2)]
    @test_int [(c - im*c*tan(e + f*x))^3/(a + im*a*tan(e + f*x))^5, x, 4, (4*im*c^3)/(5*f*(a + im*a*tan(e + f*x))^5) + (im*c^3)/(3*a^2*f*(a + im*a*tan(e + f*x))^3) - (im*a^3*c^3)/(f*(a^2 + im*a^2*tan(e + f*x))^4)]


    @test_int [(a + im*a*tan(e + f*x))^5*(c - im*c*tan(e + f*x))^4, x, 4, (im*a^5*c^4*sec(e + f*x)^8)/(8*f) + (a^5*c^4*tan(e + f*x))/f + (a^5*c^4*tan(e + f*x)^3)/f + (3*a^5*c^4*tan(e + f*x)^5)/(5*f) + (a^5*c^4*tan(e + f*x)^7)/(7*f)]
    @test_int [(a + im*a*tan(e + f*x))^4*(c - im*c*tan(e + f*x))^4, x, 3, (a^4*c^4*tan(e + f*x))/f + (a^4*c^4*tan(e + f*x)^3)/f + (3*a^4*c^4*tan(e + f*x)^5)/(5*f) + (a^4*c^4*tan(e + f*x)^7)/(7*f)]
    @test_int [(a + im*a*tan(e + f*x))^3*(c - im*c*tan(e + f*x))^4, x, 4, -((im*a^3*c^4*sec(e + f*x)^6)/(6*f)) + (a^3*c^4*tan(e + f*x))/f + (2*a^3*c^4*tan(e + f*x)^3)/(3*f) + (a^3*c^4*tan(e + f*x)^5)/(5*f)]
    @test_int [(a + im*a*tan(e + f*x))^2*(c - im*c*tan(e + f*x))^4, x, 4, (im*a^2*(c - im*c*tan(e + f*x))^4)/(2*f) - (im*a^2*(c - im*c*tan(e + f*x))^5)/(5*c*f)]
    @test_int [(a + im*a*tan(e + f*x))^1*(c - im*c*tan(e + f*x))^4, x, 3, (im*a*(c - im*c*tan(e + f*x))^4)/(4*f)]
    @test_int [(c - im*c*tan(e + f*x))^4/(a + im*a*tan(e + f*x))^1, x, 4, -((12*c^4*x)/a) - (12*im*c^4*log(cos(e + f*x)))/(a*f) + (5*c^4*tan(e + f*x))/(a*f) - (im*c^4*tan(e + f*x)^2)/(2*a*f) + (8*im*c^4)/(f*(a + im*a*tan(e + f*x)))]
    @test_int [(c - im*c*tan(e + f*x))^4/(a + im*a*tan(e + f*x))^2, x, 4, (6*c^4*x)/a^2 + (6*im*c^4*log(cos(e + f*x)))/(a^2*f) - (c^4*tan(e + f*x))/(a^2*f) + (4*im*c^4)/(f*(a + im*a*tan(e + f*x))^2) - (12*im*c^4)/(f*(a^2 + im*a^2*tan(e + f*x)))]
    @test_int [(c - im*c*tan(e + f*x))^4/(a + im*a*tan(e + f*x))^3, x, 4, -((c^4*x)/a^3) - (im*c^4*log(cos(e + f*x)))/(a^3*f) + (8*im*c^4)/(3*f*(a + im*a*tan(e + f*x))^3) - (6*im*c^4)/(a*f*(a + im*a*tan(e + f*x))^2) + (6*im*c^4)/(f*(a^3 + im*a^3*tan(e + f*x)))]
    @test_int [(c - im*c*tan(e + f*x))^4/(a + im*a*tan(e + f*x))^4, x, 3, (im*c^4*(a^2 - im*a^2*tan(e + f*x))^4)/(8*f*(a^3 + im*a^3*tan(e + f*x))^4)]
    @test_int [(c - im*c*tan(e + f*x))^4/(a + im*a*tan(e + f*x))^5, x, 4, (im*c^4*(1 - im*tan(e + f*x))^4)/(10*f*(a + im*a*tan(e + f*x))^5) + (im*c^4*(a - im*a*tan(e + f*x))^4)/(80*a^5*f*(a + im*a*tan(e + f*x))^4)]


    #= ::Subsubsection::Closed:: =#
    #=n < 0=#


    @test_int [(a + im*a*tan(e + f*x))^4/(c - im*c*tan(e + f*x)), x, 4, -((12*a^4*x)/c) + (12*im*a^4*log(cos(e + f*x)))/(c*f) + (5*a^4*tan(e + f*x))/(c*f) + (im*a^4*tan(e + f*x)^2)/(2*c*f) - (8*im*a^4)/(f*(c - im*c*tan(e + f*x)))]
    @test_int [(a + im*a*tan(e + f*x))^3/(c - im*c*tan(e + f*x)), x, 4, -((4*a^3*x)/c) + (4*im*a^3*log(cos(e + f*x)))/(c*f) + (a^3*tan(e + f*x))/(c*f) - (4*im*a^3)/(f*(c - im*c*tan(e + f*x)))]
    @test_int [(a + im*a*tan(e + f*x))^2/(c - im*c*tan(e + f*x)), x, 4, -((a^2*x)/c) + (im*a^2*log(cos(e + f*x)))/(c*f) - (2*im*a^2)/(f*(c - im*c*tan(e + f*x)))]
    @test_int [(a + im*a*tan(e + f*x))^1/(c - im*c*tan(e + f*x)), x, 3, -((im*a)/(f*(c - im*c*tan(e + f*x))))]
    @test_int [1/((a + im*a*tan(e + f*x))^1*(c - im*c*tan(e + f*x))), x, 3, x/(2*a*c) + (cos(e + f*x)*sin(e + f*x))/(2*a*c*f)]
    @test_int [1/((a + im*a*tan(e + f*x))^2*(c - im*c*tan(e + f*x))), x, 5, (3*x)/(8*a^2*c) + (im*cos(e + f*x)^4)/(4*a^2*c*f) + (3*cos(e + f*x)*sin(e + f*x))/(8*a^2*c*f) + (cos(e + f*x)^3*sin(e + f*x))/(4*a^2*c*f)]
    @test_int [1/((a + im*a*tan(e + f*x))^3*(c - im*c*tan(e + f*x))), x, 5, x/(4*a^3*c) - im/(16*a^3*f*(c - im*c*tan(e + f*x))) + (im*c^2)/(12*a^3*f*(c + im*c*tan(e + f*x))^3) + (im*c)/(8*a^3*f*(c + im*c*tan(e + f*x))^2) + (3*im)/(16*a^3*f*(c + im*c*tan(e + f*x)))]


    @test_int [(a + im*a*tan(e + f*x))^4/(c - im*c*tan(e + f*x))^2, x, 4, (6*a^4*x)/c^2 - (6*im*a^4*log(cos(e + f*x)))/(c^2*f) - (a^4*tan(e + f*x))/(c^2*f) - (4*im*a^4)/(f*(c - im*c*tan(e + f*x))^2) + (12*im*a^4)/(f*(c^2 - im*c^2*tan(e + f*x)))]
    @test_int [(a + im*a*tan(e + f*x))^3/(c - im*c*tan(e + f*x))^2, x, 4, (a^3*x)/c^2 - (im*a^3*log(cos(e + f*x)))/(c^2*f) - (2*im*a^3)/(f*(c - im*c*tan(e + f*x))^2) + (4*im*a^3)/(f*(c^2 - im*c^2*tan(e + f*x)))]
    @test_int [(a + im*a*tan(e + f*x))^2/(c - im*c*tan(e + f*x))^2, x, 3, (a^2*tan(e + f*x))/(f*(c - im*c*tan(e + f*x))^2)]
    @test_int [(a + im*a*tan(e + f*x))^1/(c - im*c*tan(e + f*x))^2, x, 3, -((im*a)/(2*f*(c - im*c*tan(e + f*x))^2))]
    @test_int [1/((a + im*a*tan(e + f*x))^1*(c - im*c*tan(e + f*x))^2), x, 5, (3*x)/(8*a*c^2) - im/(8*a*f*(c - im*c*tan(e + f*x))^2) - im/(4*a*f*(c^2 - im*c^2*tan(e + f*x))) + im/(8*a*f*(c^2 + im*c^2*tan(e + f*x)))]
    @test_int [1/((a + im*a*tan(e + f*x))^2*(c - im*c*tan(e + f*x))^2), x, 4, (3*x)/(8*a^2*c^2) + (3*cos(e + f*x)*sin(e + f*x))/(8*a^2*c^2*f) + (cos(e + f*x)^3*sin(e + f*x))/(4*a^2*c^2*f)]
    @test_int [1/((a + im*a*tan(e + f*x))^3*(c - im*c*tan(e + f*x))^2), x, 6, (5*x)/(16*a^3*c^2) + (im*cos(e + f*x)^6)/(6*a^3*c^2*f) + (5*cos(e + f*x)*sin(e + f*x))/(16*a^3*c^2*f) + (5*cos(e + f*x)^3*sin(e + f*x))/(24*a^3*c^2*f) + (cos(e + f*x)^5*sin(e + f*x))/(6*a^3*c^2*f)]


    @test_int [(a + im*a*tan(e + f*x))^6/(c - im*c*tan(e + f*x))^3, x, 4, -((40*a^6*x)/c^3) + (40*im*a^6*log(cos(e + f*x)))/(c^3*f) + (9*a^6*tan(e + f*x))/(c^3*f) + (im*a^6*tan(e + f*x)^2)/(2*c^3*f) - (32*im*a^6)/(3*f*(c - im*c*tan(e + f*x))^3) + (40*im*a^6)/(c*f*(c - im*c*tan(e + f*x))^2) - (80*im*a^6)/(f*(c^3 - im*c^3*tan(e + f*x)))]
    @test_int [(a + im*a*tan(e + f*x))^5/(c - im*c*tan(e + f*x))^3, x, 4, -((8*a^5*x)/c^3) + (8*im*a^5*log(cos(e + f*x)))/(c^3*f) + (a^5*tan(e + f*x))/(c^3*f) - (16*im*a^5)/(3*f*(c - im*c*tan(e + f*x))^3) - (24*im*a^5)/(f*(c^3 - im*c^3*tan(e + f*x))) + (16*im*a^5*c^5)/(f*(c^4 - im*c^4*tan(e + f*x))^2)]
    @test_int [(a + im*a*tan(e + f*x))^4/(c - im*c*tan(e + f*x))^3, x, 4, -((a^4*x)/c^3) + (im*a^4*log(cos(e + f*x)))/(c^3*f) - (8*im*a^4)/(3*f*(c - im*c*tan(e + f*x))^3) + (6*im*a^4)/(c*f*(c - im*c*tan(e + f*x))^2) - (6*im*a^4)/(f*(c^3 - im*c^3*tan(e + f*x)))]
    @test_int [(a + im*a*tan(e + f*x))^3/(c - im*c*tan(e + f*x))^3, x, 3, -((im*a^3*(c^2 + im*c^2*tan(e + f*x))^3)/(6*f*(c^3 - im*c^3*tan(e + f*x))^3))]
    @test_int [(a + im*a*tan(e + f*x))^2/(c - im*c*tan(e + f*x))^3, x, 4, -((2*im*a^2)/(3*f*(c - im*c*tan(e + f*x))^3)) + (im*a^2)/(2*c*f*(c - im*c*tan(e + f*x))^2)]
    @test_int [(a + im*a*tan(e + f*x))^1/(c - im*c*tan(e + f*x))^3, x, 3, -((im*a)/(3*f*(c - im*c*tan(e + f*x))^3))]
    @test_int [1/((a + im*a*tan(e + f*x))^1*(c - im*c*tan(e + f*x))^3), x, 5, x/(4*a*c^3) - im/(12*a*f*(c - im*c*tan(e + f*x))^3) - im/(8*a*c*f*(c - im*c*tan(e + f*x))^2) - (3*im)/(16*a*f*(c^3 - im*c^3*tan(e + f*x))) + im/(16*a*f*(c^3 + im*c^3*tan(e + f*x)))]
    @test_int [1/((a + im*a*tan(e + f*x))^2*(c - im*c*tan(e + f*x))^3), x, 5, (5*x)/(16*a^2*c^3) - im/(24*a^2*f*(c - im*c*tan(e + f*x))^3) - (3*im)/(32*a^2*c*f*(c - im*c*tan(e + f*x))^2) + im/(32*a^2*c*f*(c + im*c*tan(e + f*x))^2) - (3*im)/(16*a^2*f*(c^3 - im*c^3*tan(e + f*x))) + im/(8*a^2*f*(c^3 + im*c^3*tan(e + f*x)))]
    @test_int [1/((a + im*a*tan(e + f*x))^3*(c - im*c*tan(e + f*x))^3), x, 5, (5*x)/(16*a^3*c^3) + (5*cos(e + f*x)*sin(e + f*x))/(16*a^3*c^3*f) + (5*cos(e + f*x)^3*sin(e + f*x))/(24*a^3*c^3*f) + (cos(e + f*x)^5*sin(e + f*x))/(6*a^3*c^3*f)]


    @test_int [(a + im*a*tan(e + f*x))^6/(c - im*c*tan(e + f*x))^4, x, 4, (10*a^6*x)/c^4 - (10*im*a^6*log(cos(e + f*x)))/(c^4*f) - (a^6*tan(e + f*x))/(c^4*f) - (8*im*a^6)/(f*(c - im*c*tan(e + f*x))^4) + (80*im*a^6)/(3*c*f*(c - im*c*tan(e + f*x))^3) - (40*im*a^6)/(f*(c^2 - im*c^2*tan(e + f*x))^2) + (40*im*a^6)/(f*(c^4 - im*c^4*tan(e + f*x)))]
    @test_int [(a + im*a*tan(e + f*x))^5/(c - im*c*tan(e + f*x))^4, x, 4, (a^5*x)/c^4 - (im*a^5*log(cos(e + f*x)))/(c^4*f) - (4*im*a^5)/(f*(c - im*c*tan(e + f*x))^4) - (12*im*a^5)/(f*(c^2 - im*c^2*tan(e + f*x))^2) + (32*im*a^5*c^5)/(3*f*(c^3 - im*c^3*tan(e + f*x))^3) + (8*im*a^5)/(f*(c^4 - im*c^4*tan(e + f*x)))]
    @test_int [(a + im*a*tan(e + f*x))^4/(c - im*c*tan(e + f*x))^4, x, 3, -((im*a^4*(c^2 + im*c^2*tan(e + f*x))^4)/(8*f*(c^3 - im*c^3*tan(e + f*x))^4))]
    @test_int [(a + im*a*tan(e + f*x))^3/(c - im*c*tan(e + f*x))^4, x, 4, -((im*a^3)/(f*(c - im*c*tan(e + f*x))^4)) + (4*im*a^3)/(3*c*f*(c - im*c*tan(e + f*x))^3) - (im*a^3)/(2*f*(c^2 - im*c^2*tan(e + f*x))^2)]
    @test_int [(a + im*a*tan(e + f*x))^2/(c - im*c*tan(e + f*x))^4, x, 4, -((im*a^2)/(2*f*(c - im*c*tan(e + f*x))^4)) + (im*a^2*c^2)/(3*f*(c^2 - im*c^2*tan(e + f*x))^3)]
    @test_int [(a + im*a*tan(e + f*x))^1/(c - im*c*tan(e + f*x))^4, x, 3, -((im*a)/(4*f*(c - im*c*tan(e + f*x))^4))]
    @test_int [1/((a + im*a*tan(e + f*x))^1*(c - im*c*tan(e + f*x))^4), x, 5, (5*x)/(32*a*c^4) - im/(16*a*f*(c - im*c*tan(e + f*x))^4) - im/(12*a*c*f*(c - im*c*tan(e + f*x))^3) - (3*im)/(32*a*f*(c^2 - im*c^2*tan(e + f*x))^2) - im/(8*a*f*(c^4 - im*c^4*tan(e + f*x))) + im/(32*a*f*(c^4 + im*c^4*tan(e + f*x)))]
    @test_int [1/((a + im*a*tan(e + f*x))^2*(c - im*c*tan(e + f*x))^4), x, 5, (15*x)/(64*a^2*c^4) - im/(32*a^2*f*(c - im*c*tan(e + f*x))^4) - im/(16*a^2*c*f*(c - im*c*tan(e + f*x))^3) - (3*im)/(32*a^2*f*(c^2 - im*c^2*tan(e + f*x))^2) + im/(64*a^2*f*(c^2 + im*c^2*tan(e + f*x))^2) - (5*im)/(32*a^2*f*(c^4 - im*c^4*tan(e + f*x))) + (5*im)/(64*a^2*f*(c^4 + im*c^4*tan(e + f*x)))]
    @test_int [1/((a + im*a*tan(e + f*x))^3*(c - im*c*tan(e + f*x))^4), x, 5, (35*x)/(128*a^3*c^4) - im/(64*a^3*f*(c - im*c*tan(e + f*x))^4) - im/(24*a^3*c*f*(c - im*c*tan(e + f*x))^3) + im/(96*a^3*c*f*(c + im*c*tan(e + f*x))^3) - (5*im)/(64*a^3*f*(c^2 - im*c^2*tan(e + f*x))^2) + (5*im)/(128*a^3*f*(c^2 + im*c^2*tan(e + f*x))^2) - (5*im)/(32*a^3*f*(c^4 - im*c^4*tan(e + f*x))) + (15*im)/(128*a^3*f*(c^4 + im*c^4*tan(e + f*x)))]


    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*(a+im*a*tan(e+f*x))^m*(c-c*im*tan(e+f*x))^(n/2)=#


    #= ::Subsubsection::Closed:: =#
    #=n > 0=#


    @test_int [(a + im*a*tan(e + f*x))^3*sqrt(c - im*c*tan(e + f*x)), x, 4, (8*im*a^3*sqrt(c - im*c*tan(e + f*x)))/f - (8*im*a^3*(c - im*c*tan(e + f*x))^(3/2))/(3*c*f) + (2*im*a^3*(c - im*c*tan(e + f*x))^(5/2))/(5*c^2*f)]
    @test_int [(a + im*a*tan(e + f*x))^2*sqrt(c - im*c*tan(e + f*x)), x, 4, ((4*im)*a^2*sqrt(c - im*c*tan(e + f*x)))/f - (((2*im)/3)*a^2*(c - im*c*tan(e + f*x))^(3/2))/(c*f)]
    @test_int [(a + im*a*tan(e + f*x))^1*sqrt(c - im*c*tan(e + f*x)), x, 3, ((2*im)*a*sqrt(c - im*c*tan(e + f*x)))/f]
    @test_int [sqrt(c - im*c*tan(e + f*x))/(a + im*a*tan(e + f*x))^1, x, 5, (im*sqrt(c)*atanh(sqrt(c - im*c*tan(e + f*x))/(sqrt(2)*sqrt(c))))/(2*sqrt(2)*a*f) + (im*sqrt(c - im*c*tan(e + f*x)))/(2*a*f*(1 + im*tan(e + f*x)))]
    @test_int [sqrt(c - im*c*tan(e + f*x))/(a + im*a*tan(e + f*x))^2, x, 6, (3*im*sqrt(c)*atanh(sqrt(c - im*c*tan(e + f*x))/(sqrt(2)*sqrt(c))))/(16*sqrt(2)*a^2*f) + (im*sqrt(c - im*c*tan(e + f*x)))/(4*a^2*f*(1 + im*tan(e + f*x))^2) + (3*im*sqrt(c - im*c*tan(e + f*x)))/(16*a^2*f*(1 + im*tan(e + f*x)))]
    @test_int [sqrt(c - im*c*tan(e + f*x))/(a + im*a*tan(e + f*x))^3, x, 7, (5*im*sqrt(c)*atanh(sqrt(c - im*c*tan(e + f*x))/(sqrt(2)*sqrt(c))))/(64*sqrt(2)*a^3*f) + (im*sqrt(c - im*c*tan(e + f*x)))/(6*a^3*f*(1 + im*tan(e + f*x))^3) + (5*im*sqrt(c - im*c*tan(e + f*x)))/(48*a^3*f*(1 + im*tan(e + f*x))^2) + (5*im*sqrt(c - im*c*tan(e + f*x)))/(64*a^3*f*(1 + im*tan(e + f*x)))]


    @test_int [(a + im*a*tan(e + f*x))^3*(c - im*c*tan(e + f*x))^(3/2), x, 4, (8*im*a^3*(c - im*c*tan(e + f*x))^(3/2))/(3*f) - (8*im*a^3*(c - im*c*tan(e + f*x))^(5/2))/(5*c*f) + (2*im*a^3*(c - im*c*tan(e + f*x))^(7/2))/(7*c^2*f)]
    @test_int [(a + im*a*tan(e + f*x))^2*(c - im*c*tan(e + f*x))^(3/2), x, 4, (((4*im)/3)*a^2*(c - im*c*tan(e + f*x))^(3/2))/f - (((2*im)/5)*a^2*(c - im*c*tan(e + f*x))^(5/2))/(c*f)]
    @test_int [(a + im*a*tan(e + f*x))^1*(c - im*c*tan(e + f*x))^(3/2), x, 3, (((2*im)/3)*a*(c - im*c*tan(e + f*x))^(3/2))/f]
    @test_int [(c - im*c*tan(e + f*x))^(3/2)/(a + im*a*tan(e + f*x))^1, x, 5, -((im*c^(3/2)*atanh(sqrt(c - im*c*tan(e + f*x))/(sqrt(2)*sqrt(c))))/(sqrt(2)*a*f)) + (im*c^2*sqrt(c - im*c*tan(e + f*x)))/(a*f*(c + im*c*tan(e + f*x)))]
    @test_int [(c - im*c*tan(e + f*x))^(3/2)/(a + im*a*tan(e + f*x))^2, x, 6, -((im*c^(3/2)*atanh(sqrt(c - im*c*tan(e + f*x))/(sqrt(2)*sqrt(c))))/(8*sqrt(2)*a^2*f)) + (im*c^3*sqrt(c - im*c*tan(e + f*x)))/(2*a^2*f*(c + im*c*tan(e + f*x))^2) - (im*c^2*sqrt(c - im*c*tan(e + f*x)))/(8*a^2*f*(c + im*c*tan(e + f*x)))]
    @test_int [(c - im*c*tan(e + f*x))^(3/2)/(a + im*a*tan(e + f*x))^3, x, 7, -((im*c^(3/2)*atanh(sqrt(c - im*c*tan(e + f*x))/(sqrt(2)*sqrt(c))))/(32*sqrt(2)*a^3*f)) + (im*c^4*sqrt(c - im*c*tan(e + f*x)))/(3*a^3*f*(c + im*c*tan(e + f*x))^3) - (im*c^3*sqrt(c - im*c*tan(e + f*x)))/(24*a^3*f*(c + im*c*tan(e + f*x))^2) - (im*c^2*sqrt(c - im*c*tan(e + f*x)))/(32*a^3*f*(c + im*c*tan(e + f*x)))]


    @test_int [(a + im*a*tan(e + f*x))^3*(c - im*c*tan(e + f*x))^(5/2), x, 4, (8*im*a^3*(c - im*c*tan(e + f*x))^(5/2))/(5*f) - (8*im*a^3*(c - im*c*tan(e + f*x))^(7/2))/(7*c*f) + (2*im*a^3*(c - im*c*tan(e + f*x))^(9/2))/(9*c^2*f)]
    @test_int [(a + im*a*tan(e + f*x))^2*(c - im*c*tan(e + f*x))^(5/2), x, 4, (((4*im)/5)*a^2*(c - im*c*tan(e + f*x))^(5/2))/f - (((2*im)/7)*a^2*(c - im*c*tan(e + f*x))^(7/2))/(c*f)]
    @test_int [(a + im*a*tan(e + f*x))^1*(c - im*c*tan(e + f*x))^(5/2), x, 3, (((2*im)/5)*a*(c - im*c*tan(e + f*x))^(5/2))/f]
    @test_int [(c - im*c*tan(e + f*x))^(5/2)/(a + im*a*tan(e + f*x))^1, x, 6, -((3*im*sqrt(2)*c^(5/2)*atanh(sqrt(c - im*c*tan(e + f*x))/(sqrt(2)*sqrt(c))))/(a*f)) + (3*im*c^2*sqrt(c - im*c*tan(e + f*x)))/(a*f) + (im*c^2*(c - im*c*tan(e + f*x))^(3/2))/(a*f*(c + im*c*tan(e + f*x)))]
    @test_int [(c - im*c*tan(e + f*x))^(5/2)/(a + im*a*tan(e + f*x))^2, x, 6, (3*im*c^(5/2)*atanh(sqrt(c - im*c*tan(e + f*x))/(sqrt(2)*sqrt(c))))/(4*sqrt(2)*a^2*f) + (im*c^3*(c - im*c*tan(e + f*x))^(3/2))/(2*a^2*f*(c + im*c*tan(e + f*x))^2) - (3*im*c^3*sqrt(c - im*c*tan(e + f*x)))/(4*a^2*f*(c + im*c*tan(e + f*x)))]
    @test_int [(c - im*c*tan(e + f*x))^(5/2)/(a + im*a*tan(e + f*x))^3, x, 7, (im*c^(5/2)*atanh(sqrt(c - im*c*tan(e + f*x))/(sqrt(2)*sqrt(c))))/(16*sqrt(2)*a^3*f) + (im*c^4*(c - im*c*tan(e + f*x))^(3/2))/(3*a^3*f*(c + im*c*tan(e + f*x))^3) - (im*c^4*sqrt(c - im*c*tan(e + f*x)))/(4*a^3*f*(c + im*c*tan(e + f*x))^2) + (im*c^3*sqrt(c - im*c*tan(e + f*x)))/(16*a^3*f*(c + im*c*tan(e + f*x)))]


    #= ::Subsubsection::Closed:: =#
    #=n < 0=#


    @test_int [(a + im*a*tan(e + f*x))^3/sqrt(c - im*c*tan(e + f*x)), x, 4, -((8*im*a^3)/(f*sqrt(c - im*c*tan(e + f*x)))) - (8*im*a^3*sqrt(c - im*c*tan(e + f*x)))/(c*f) + (2*im*a^3*(c - im*c*tan(e + f*x))^(3/2))/(3*c^2*f)]
    @test_int [(a + im*a*tan(e + f*x))^2/sqrt(c - im*c*tan(e + f*x)), x, 4, ((-4*im)*a^2)/(f*sqrt(c - im*c*tan(e + f*x))) - ((2*im)*a^2*sqrt(c - im*c*tan(e + f*x)))/(c*f)]
    @test_int [(a + im*a*tan(e + f*x))^1/sqrt(c - im*c*tan(e + f*x)), x, 3, ((-2*im)*a)/(f*sqrt(c - im*c*tan(e + f*x)))]
    @test_int [1/((a + im*a*tan(e + f*x))^1*sqrt(c - im*c*tan(e + f*x))), x, 6, (3*im*atanh(sqrt(c - im*c*tan(e + f*x))/(sqrt(2)*sqrt(c))))/(4*sqrt(2)*a*sqrt(c)*f) - (3*im)/(4*a*f*sqrt(c - im*c*tan(e + f*x))) + im/(2*a*f*(1 + im*tan(e + f*x))*sqrt(c - im*c*tan(e + f*x)))]
    @test_int [1/((a + im*a*tan(e + f*x))^2*sqrt(c - im*c*tan(e + f*x))), x, 7, (15*im*atanh(sqrt(c - im*c*tan(e + f*x))/(sqrt(2)*sqrt(c))))/(32*sqrt(2)*a^2*sqrt(c)*f) - (15*im)/(32*a^2*f*sqrt(c - im*c*tan(e + f*x))) + im/(4*a^2*f*(1 + im*tan(e + f*x))^2*sqrt(c - im*c*tan(e + f*x))) + (5*im)/(16*a^2*f*(1 + im*tan(e + f*x))*sqrt(c - im*c*tan(e + f*x)))]
    @test_int [1/((a + im*a*tan(e + f*x))^3*sqrt(c - im*c*tan(e + f*x))), x, 8, (35*im*atanh(sqrt(c - im*c*tan(e + f*x))/(sqrt(2)*sqrt(c))))/(128*sqrt(2)*a^3*sqrt(c)*f) - (35*im)/(128*a^3*f*sqrt(c - im*c*tan(e + f*x))) + im/(6*a^3*f*(1 + im*tan(e + f*x))^3*sqrt(c - im*c*tan(e + f*x))) + (7*im)/(48*a^3*f*(1 + im*tan(e + f*x))^2*sqrt(c - im*c*tan(e + f*x))) + (35*im)/(192*a^3*f*(1 + im*tan(e + f*x))*sqrt(c - im*c*tan(e + f*x)))]


    @test_int [(a + im*a*tan(e + f*x))^3/(c - im*c*tan(e + f*x))^(3/2), x, 4, -((8*im*a^3)/(3*f*(c - im*c*tan(e + f*x))^(3/2))) + (8*im*a^3)/(c*f*sqrt(c - im*c*tan(e + f*x))) + (2*im*a^3*sqrt(c - im*c*tan(e + f*x)))/(c^2*f)]
    @test_int [(a + im*a*tan(e + f*x))^2/(c - im*c*tan(e + f*x))^(3/2), x, 4, (((-4*im)/3)*a^2)/(f*(c - im*c*tan(e + f*x))^(3/2)) + ((2*im)*a^2)/(c*f*sqrt(c - im*c*tan(e + f*x)))]
    @test_int [(a + im*a*tan(e + f*x))^1/(c - im*c*tan(e + f*x))^(3/2), x, 3, (((-2*im)/3)*a)/(f*(c - im*c*tan(e + f*x))^(3/2))]
    @test_int [1/((a + im*a*tan(e + f*x))^1*(c - im*c*tan(e + f*x))^(3/2)), x, 7, (5*im*atanh(sqrt(c - im*c*tan(e + f*x))/(sqrt(2)*sqrt(c))))/(8*sqrt(2)*a*c^(3/2)*f) - (5*im)/(12*a*f*(c - im*c*tan(e + f*x))^(3/2)) + im/(2*a*f*(1 + im*tan(e + f*x))*(c - im*c*tan(e + f*x))^(3/2)) - (5*im)/(8*a*c*f*sqrt(c - im*c*tan(e + f*x)))]
    @test_int [1/((a + im*a*tan(e + f*x))^2*(c - im*c*tan(e + f*x))^(3/2)), x, 8, (35*im*atanh(sqrt(c - im*c*tan(e + f*x))/(sqrt(2)*sqrt(c))))/(64*sqrt(2)*a^2*c^(3/2)*f) - (35*im)/(96*a^2*f*(c - im*c*tan(e + f*x))^(3/2)) + im/(4*a^2*f*(1 + im*tan(e + f*x))^2*(c - im*c*tan(e + f*x))^(3/2)) + (7*im)/(16*a^2*f*(1 + im*tan(e + f*x))*(c - im*c*tan(e + f*x))^(3/2)) - (35*im)/(64*a^2*c*f*sqrt(c - im*c*tan(e + f*x)))]
    @test_int [1/((a + im*a*tan(e + f*x))^3*(c - im*c*tan(e + f*x))^(3/2)), x, 9, (105*im*atanh(sqrt(c - im*c*tan(e + f*x))/(sqrt(2)*sqrt(c))))/(256*sqrt(2)*a^3*c^(3/2)*f) - (35*im)/(128*a^3*f*(c - im*c*tan(e + f*x))^(3/2)) + im/(6*a^3*f*(1 + im*tan(e + f*x))^3*(c - im*c*tan(e + f*x))^(3/2)) + (3*im)/(16*a^3*f*(1 + im*tan(e + f*x))^2*(c - im*c*tan(e + f*x))^(3/2)) + (21*im)/(64*a^3*f*(1 + im*tan(e + f*x))*(c - im*c*tan(e + f*x))^(3/2)) - (105*im)/(256*a^3*c*f*sqrt(c - im*c*tan(e + f*x)))]


    @test_int [(a + im*a*tan(e + f*x))^3/(c - im*c*tan(e + f*x))^(5/2), x, 4, -((8*im*a^3)/(5*f*(c - im*c*tan(e + f*x))^(5/2))) + (8*im*a^3)/(3*c*f*(c - im*c*tan(e + f*x))^(3/2)) - (2*im*a^3)/(c^2*f*sqrt(c - im*c*tan(e + f*x)))]
    @test_int [(a + im*a*tan(e + f*x))^2/(c - im*c*tan(e + f*x))^(5/2), x, 4, (((-4*im)/5)*a^2)/(f*(c - im*c*tan(e + f*x))^(5/2)) + (((2*im)/3)*a^2)/(c*f*(c - im*c*tan(e + f*x))^(3/2))]
    @test_int [(a + im*a*tan(e + f*x))^1/(c - im*c*tan(e + f*x))^(5/2), x, 3, (((-2*im)/5)*a)/(f*(c - im*c*tan(e + f*x))^(5/2))]
    @test_int [1/((a + im*a*tan(e + f*x))^1*(c - im*c*tan(e + f*x))^(5/2)), x, 8, (7*im*atanh(sqrt(c - im*c*tan(e + f*x))/(sqrt(2)*sqrt(c))))/(16*sqrt(2)*a*c^(5/2)*f) - (7*im)/(20*a*f*(c - im*c*tan(e + f*x))^(5/2)) + im/(2*a*f*(1 + im*tan(e + f*x))*(c - im*c*tan(e + f*x))^(5/2)) - (7*im)/(24*a*c*f*(c - im*c*tan(e + f*x))^(3/2)) - (7*im)/(16*a*c^2*f*sqrt(c - im*c*tan(e + f*x)))]
    @test_int [1/((a + im*a*tan(e + f*x))^2*(c - im*c*tan(e + f*x))^(5/2)), x, 9, (63*im*atanh(sqrt(c - im*c*tan(e + f*x))/(sqrt(2)*sqrt(c))))/(128*sqrt(2)*a^2*c^(5/2)*f) - (63*im)/(160*a^2*f*(c - im*c*tan(e + f*x))^(5/2)) + im/(4*a^2*f*(1 + im*tan(e + f*x))^2*(c - im*c*tan(e + f*x))^(5/2)) + (9*im)/(16*a^2*f*(1 + im*tan(e + f*x))*(c - im*c*tan(e + f*x))^(5/2)) - (21*im)/(64*a^2*c*f*(c - im*c*tan(e + f*x))^(3/2)) - (63*im)/(128*a^2*c^2*f*sqrt(c - im*c*tan(e + f*x)))]
    @test_int [1/((a + im*a*tan(e + f*x))^3*(c - im*c*tan(e + f*x))^(5/2)), x, 10, (231*im*atanh(sqrt(c - im*c*tan(e + f*x))/(sqrt(2)*sqrt(c))))/(512*sqrt(2)*a^3*c^(5/2)*f) - (231*im)/(640*a^3*f*(c - im*c*tan(e + f*x))^(5/2)) + im/(6*a^3*f*(1 + im*tan(e + f*x))^3*(c - im*c*tan(e + f*x))^(5/2)) + (11*im)/(48*a^3*f*(1 + im*tan(e + f*x))^2*(c - im*c*tan(e + f*x))^(5/2)) + (33*im)/(64*a^3*f*(1 + im*tan(e + f*x))*(c - im*c*tan(e + f*x))^(5/2)) - (77*im)/(256*a^3*c*f*(c - im*c*tan(e + f*x))^(3/2)) - (231*im)/(512*a^3*c^2*f*sqrt(c - im*c*tan(e + f*x)))]


    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*(a+im*a*tan(e+f*x))^(m/2)*(c-c*im*tan(e+f*x))^(n/2)=#


    #= ::Subsubsection::Closed:: =#
    #=n > 0=#


    @test_int [(a + im*a*tan(e + f*x))^(5/2)*sqrt(c - im*c*tan(e + f*x)), x, 6, ((-3*im)*a^(5/2)*sqrt(c)*atan((sqrt(c)*sqrt(a + im*a*tan(e + f*x)))/(sqrt(a)*sqrt(c - im*c*tan(e + f*x)))))/f + (((3*im)/2)*a^2*sqrt(a + im*a*tan(e + f*x))*sqrt(c - im*c*tan(e + f*x)))/f + ((im/2)*a*(a + im*a*tan(e + f*x))^(3/2)*sqrt(c - im*c*tan(e + f*x)))/f]
    @test_int [(a + im*a*tan(e + f*x))^(3/2)*sqrt(c - im*c*tan(e + f*x)), x, 5, ((-2*im)*a^(3/2)*sqrt(c)*atan((sqrt(c)*sqrt(a + im*a*tan(e + f*x)))/(sqrt(a)*sqrt(c - im*c*tan(e + f*x)))))/f + (im*a*sqrt(a + im*a*tan(e + f*x))*sqrt(c - im*c*tan(e + f*x)))/f]
    @test_int [(a + im*a*tan(e + f*x))^(1/2)*sqrt(c - im*c*tan(e + f*x)), x, 4, ((-2*im)*sqrt(a)*sqrt(c)*atan((sqrt(c)*sqrt(a + im*a*tan(e + f*x)))/(sqrt(a)*sqrt(c - im*c*tan(e + f*x)))))/f]
    @test_int [sqrt(c - im*c*tan(e + f*x))/(a + im*a*tan(e + f*x))^(1/2), x, 2, (im*sqrt(c - im*c*tan(e + f*x)))/(f*sqrt(a + im*a*tan(e + f*x)))]
    @test_int [sqrt(c - im*c*tan(e + f*x))/(a + im*a*tan(e + f*x))^(3/2), x, 3, ((im/3)*sqrt(c - im*c*tan(e + f*x)))/(f*(a + im*a*tan(e + f*x))^(3/2)) + ((im/3)*sqrt(c - im*c*tan(e + f*x)))/(a*f*sqrt(a + im*a*tan(e + f*x)))]
    @test_int [sqrt(c - im*c*tan(e + f*x))/(a + im*a*tan(e + f*x))^(5/2), x, 4, ((im/5)*sqrt(c - im*c*tan(e + f*x)))/(f*(a + im*a*tan(e + f*x))^(5/2)) + (((2*im)/15)*sqrt(c - im*c*tan(e + f*x)))/(a*f*(a + im*a*tan(e + f*x))^(3/2)) + (((2*im)/15)*sqrt(c - im*c*tan(e + f*x)))/(a^2*f*sqrt(a + im*a*tan(e + f*x)))]
    @test_int [sqrt(c - im*c*tan(e + f*x))/(a + im*a*tan(e + f*x))^(7/2), x, 5, (im*sqrt(c - im*c*tan(e + f*x)))/(7*f*(a + im*a*tan(e + f*x))^(7/2)) + (3*im*sqrt(c - im*c*tan(e + f*x)))/(35*a*f*(a + im*a*tan(e + f*x))^(5/2)) + (2*im*sqrt(c - im*c*tan(e + f*x)))/(35*a^2*f*(a + im*a*tan(e + f*x))^(3/2)) + (2*im*sqrt(c - im*c*tan(e + f*x)))/(35*a^3*f*sqrt(a + im*a*tan(e + f*x)))]


    @test_int [(a + im*a*tan(e + f*x))^(5/2)*(c - im*c*tan(e + f*x))^(3/2), x, 6, -((im*a^(5/2)*c^(3/2)*atan((sqrt(c)*sqrt(a + im*a*tan(e + f*x)))/(sqrt(a)*sqrt(c - im*c*tan(e + f*x)))))/f) + (a^2*c*tan(e + f*x)*sqrt(a + im*a*tan(e + f*x))*sqrt(c - im*c*tan(e + f*x)))/(2*f) + (im*a*(a + im*a*tan(e + f*x))^(3/2)*(c - im*c*tan(e + f*x))^(3/2))/(3*f)]
    @test_int [(a + im*a*tan(e + f*x))^(3/2)*(c - im*c*tan(e + f*x))^(3/2), x, 5, -((im*a^(3/2)*c^(3/2)*atan((sqrt(c)*sqrt(a + im*a*tan(e + f*x)))/(sqrt(a)*sqrt(c - im*c*tan(e + f*x)))))/f) + (a*c*tan(e + f*x)*sqrt(a + im*a*tan(e + f*x))*sqrt(c - im*c*tan(e + f*x)))/(2*f)]
    @test_int [(a + im*a*tan(e + f*x))^(1/2)*(c - im*c*tan(e + f*x))^(3/2), x, 5, ((-2*im)*sqrt(a)*c^(3/2)*atan((sqrt(c)*sqrt(a + im*a*tan(e + f*x)))/(sqrt(a)*sqrt(c - im*c*tan(e + f*x)))))/f - (im*c*sqrt(a + im*a*tan(e + f*x))*sqrt(c - im*c*tan(e + f*x)))/f]
    @test_int [(c - im*c*tan(e + f*x))^(3/2)/(a + im*a*tan(e + f*x))^(1/2), x, 5, ((2*im)*c^(3/2)*atan((sqrt(c)*sqrt(a + im*a*tan(e + f*x)))/(sqrt(a)*sqrt(c - im*c*tan(e + f*x)))))/(sqrt(a)*f) + ((2*im)*c*sqrt(c - im*c*tan(e + f*x)))/(f*sqrt(a + im*a*tan(e + f*x)))]
    @test_int [(c - im*c*tan(e + f*x))^(3/2)/(a + im*a*tan(e + f*x))^(3/2), x, 2, ((im/3)*(c - im*c*tan(e + f*x))^(3/2))/(f*(a + im*a*tan(e + f*x))^(3/2))]
    @test_int [(c - im*c*tan(e + f*x))^(3/2)/(a + im*a*tan(e + f*x))^(5/2), x, 3, ((im/5)*(c - im*c*tan(e + f*x))^(3/2))/(f*(a + im*a*tan(e + f*x))^(5/2)) + ((im/15)*(c - im*c*tan(e + f*x))^(3/2))/(a*f*(a + im*a*tan(e + f*x))^(3/2))]
    @test_int [(c - im*c*tan(e + f*x))^(3/2)/(a + im*a*tan(e + f*x))^(7/2), x, 4, (im*(c - im*c*tan(e + f*x))^(3/2))/(7*f*(a + im*a*tan(e + f*x))^(7/2)) + (2*im*(c - im*c*tan(e + f*x))^(3/2))/(35*a*f*(a + im*a*tan(e + f*x))^(5/2)) + (2*im*(c - im*c*tan(e + f*x))^(3/2))/(105*a^2*f*(a + im*a*tan(e + f*x))^(3/2))]
    @test_int [(c - im*c*tan(e + f*x))^(3/2)/(a + im*a*tan(e + f*x))^(9/2), x, 5, (im*(c - im*c*tan(e + f*x))^(3/2))/(9*f*(a + im*a*tan(e + f*x))^(9/2)) + (im*(c - im*c*tan(e + f*x))^(3/2))/(21*a*f*(a + im*a*tan(e + f*x))^(7/2)) + (2*im*(c - im*c*tan(e + f*x))^(3/2))/(105*a^2*f*(a + im*a*tan(e + f*x))^(5/2)) + (2*im*(c - im*c*tan(e + f*x))^(3/2))/(315*a^3*f*(a + im*a*tan(e + f*x))^(3/2))]


    @test_int [(a + im*a*tan(e + f*x))^(5/2)*(c - im*c*tan(e + f*x))^(5/2), x, 6, -((3*im*a^(5/2)*c^(5/2)*atan((sqrt(c)*sqrt(a + im*a*tan(e + f*x)))/(sqrt(a)*sqrt(c - im*c*tan(e + f*x)))))/(4*f)) + (3*a^2*c^2*tan(e + f*x)*sqrt(a + im*a*tan(e + f*x))*sqrt(c - im*c*tan(e + f*x)))/(8*f) + (a*c*tan(e + f*x)*(a + im*a*tan(e + f*x))^(3/2)*(c - im*c*tan(e + f*x))^(3/2))/(4*f)]
    @test_int [(a + im*a*tan(e + f*x))^(3/2)*(c - im*c*tan(e + f*x))^(5/2), x, 6, -((im*a^(3/2)*c^(5/2)*atan((sqrt(c)*sqrt(a + im*a*tan(e + f*x)))/(sqrt(a)*sqrt(c - im*c*tan(e + f*x)))))/f) + (a*c^2*tan(e + f*x)*sqrt(a + im*a*tan(e + f*x))*sqrt(c - im*c*tan(e + f*x)))/(2*f) - (im*c*(a + im*a*tan(e + f*x))^(3/2)*(c - im*c*tan(e + f*x))^(3/2))/(3*f)]
    @test_int [(a + im*a*tan(e + f*x))^(1/2)*(c - im*c*tan(e + f*x))^(5/2), x, 6, ((-3*im)*sqrt(a)*c^(5/2)*atan((sqrt(c)*sqrt(a + im*a*tan(e + f*x)))/(sqrt(a)*sqrt(c - im*c*tan(e + f*x)))))/f - (((3*im)/2)*c^2*sqrt(a + im*a*tan(e + f*x))*sqrt(c - im*c*tan(e + f*x)))/f - ((im/2)*c*sqrt(a + im*a*tan(e + f*x))*(c - im*c*tan(e + f*x))^(3/2))/f]
    @test_int [(c - im*c*tan(e + f*x))^(5/2)/(a + im*a*tan(e + f*x))^(1/2), x, 6, (6*im*c^(5/2)*atan((sqrt(c)*sqrt(a + im*a*tan(e + f*x)))/(sqrt(a)*sqrt(c - im*c*tan(e + f*x)))))/(sqrt(a)*f) + (3*im*c^2*sqrt(a + im*a*tan(e + f*x))*sqrt(c - im*c*tan(e + f*x)))/(a*f) + (2*im*c*(c - im*c*tan(e + f*x))^(3/2))/(f*sqrt(a + im*a*tan(e + f*x)))]
    @test_int [(c - im*c*tan(e + f*x))^(5/2)/(a + im*a*tan(e + f*x))^(3/2), x, 6, ((-2*im)*c^(5/2)*atan((sqrt(c)*sqrt(a + im*a*tan(e + f*x)))/(sqrt(a)*sqrt(c - im*c*tan(e + f*x)))))/(a^(3/2)*f) - ((2*im)*c^2*sqrt(c - im*c*tan(e + f*x)))/(a*f*sqrt(a + im*a*tan(e + f*x))) + (((2*im)/3)*c*(c - im*c*tan(e + f*x))^(3/2))/(f*(a + im*a*tan(e + f*x))^(3/2))]
    @test_int [(c - im*c*tan(e + f*x))^(5/2)/(a + im*a*tan(e + f*x))^(5/2), x, 2, ((im/5)*(c - im*c*tan(e + f*x))^(5/2))/(f*(a + im*a*tan(e + f*x))^(5/2))]
    @test_int [(c - im*c*tan(e + f*x))^(5/2)/(a + im*a*tan(e + f*x))^(7/2), x, 3, (im*(c - im*c*tan(e + f*x))^(5/2))/(7*f*(a + im*a*tan(e + f*x))^(7/2)) + (im*(c - im*c*tan(e + f*x))^(5/2))/(35*a*f*(a + im*a*tan(e + f*x))^(5/2))]
    @test_int [(c - im*c*tan(e + f*x))^(5/2)/(a + im*a*tan(e + f*x))^(9/2), x, 4, (im*(c - im*c*tan(e + f*x))^(5/2))/(9*f*(a + im*a*tan(e + f*x))^(9/2)) + (2*im*(c - im*c*tan(e + f*x))^(5/2))/(63*a*f*(a + im*a*tan(e + f*x))^(7/2)) + (2*im*(c - im*c*tan(e + f*x))^(5/2))/(315*a^2*f*(a + im*a*tan(e + f*x))^(5/2))]
    @test_int [(c - im*c*tan(e + f*x))^(5/2)/(a + im*a*tan(e + f*x))^(11/2), x, 5, (im*(c - im*c*tan(e + f*x))^(5/2))/(11*f*(a + im*a*tan(e + f*x))^(11/2)) + (im*(c - im*c*tan(e + f*x))^(5/2))/(33*a*f*(a + im*a*tan(e + f*x))^(9/2)) + (2*im*(c - im*c*tan(e + f*x))^(5/2))/(231*a^2*f*(a + im*a*tan(e + f*x))^(7/2)) + (2*im*(c - im*c*tan(e + f*x))^(5/2))/(1155*a^3*f*(a + im*a*tan(e + f*x))^(5/2))]


    #= ::Subsubsection::Closed:: =#
    #=n < 0=#


    @test_int [(a + im*a*tan(e + f*x))^(7/2)/sqrt(c - im*c*tan(e + f*x)), x, 7, (15*im*a^(7/2)*atan((sqrt(c)*sqrt(a + im*a*tan(e + f*x)))/(sqrt(a)*sqrt(c - im*c*tan(e + f*x)))))/(sqrt(c)*f) - (2*im*a*(a + im*a*tan(e + f*x))^(5/2))/(f*sqrt(c - im*c*tan(e + f*x))) - (15*im*a^3*sqrt(a + im*a*tan(e + f*x))*sqrt(c - im*c*tan(e + f*x)))/(2*c*f) - (5*im*a^2*(a + im*a*tan(e + f*x))^(3/2)*sqrt(c - im*c*tan(e + f*x)))/(2*c*f)]
    @test_int [(a + im*a*tan(e + f*x))^(5/2)/sqrt(c - im*c*tan(e + f*x)), x, 6, (6*im*a^(5/2)*atan((sqrt(c)*sqrt(a + im*a*tan(e + f*x)))/(sqrt(a)*sqrt(c - im*c*tan(e + f*x)))))/(sqrt(c)*f) - (2*im*a*(a + im*a*tan(e + f*x))^(3/2))/(f*sqrt(c - im*c*tan(e + f*x))) - (3*im*a^2*sqrt(a + im*a*tan(e + f*x))*sqrt(c - im*c*tan(e + f*x)))/(c*f)]
    @test_int [(a + im*a*tan(e + f*x))^(3/2)/sqrt(c - im*c*tan(e + f*x)), x, 5, ((2*im)*a^(3/2)*atan((sqrt(c)*sqrt(a + im*a*tan(e + f*x)))/(sqrt(a)*sqrt(c - im*c*tan(e + f*x)))))/(sqrt(c)*f) - ((2*im)*a*sqrt(a + im*a*tan(e + f*x)))/(f*sqrt(c - im*c*tan(e + f*x)))]
    @test_int [(a + im*a*tan(e + f*x))^(1/2)/sqrt(c - im*c*tan(e + f*x)), x, 2, ((-im)*sqrt(a + im*a*tan(e + f*x)))/(f*sqrt(c - im*c*tan(e + f*x)))]
    @test_int [1/((a + im*a*tan(e + f*x))^(1/2)*sqrt(c - im*c*tan(e + f*x))), x, 2, tan(e + f*x)/(f*sqrt(a + im*a*tan(e + f*x))*sqrt(c - im*c*tan(e + f*x)))]
    @test_int [1/((a + im*a*tan(e + f*x))^(3/2)*sqrt(c - im*c*tan(e + f*x))), x, 3, im/(3*f*(a + im*a*tan(e + f*x))^(3/2)*sqrt(c - im*c*tan(e + f*x))) + (2*tan(e + f*x))/(3*a*f*sqrt(a + im*a*tan(e + f*x))*sqrt(c - im*c*tan(e + f*x)))]
    @test_int [1/((a + im*a*tan(e + f*x))^(5/2)*sqrt(c - im*c*tan(e + f*x))), x, 4, im/(5*f*(a + im*a*tan(e + f*x))^(5/2)*sqrt(c - im*c*tan(e + f*x))) + im/(5*a*f*(a + im*a*tan(e + f*x))^(3/2)*sqrt(c - im*c*tan(e + f*x))) + (2*tan(e + f*x))/(5*a^2*f*sqrt(a + im*a*tan(e + f*x))*sqrt(c - im*c*tan(e + f*x)))]
    @test_int [1/((a + im*a*tan(e + f*x))^(7/2)*sqrt(c - im*c*tan(e + f*x))), x, 5, im/(7*f*(a + im*a*tan(e + f*x))^(7/2)*sqrt(c - im*c*tan(e + f*x))) + (4*im)/(35*a*f*(a + im*a*tan(e + f*x))^(5/2)*sqrt(c - im*c*tan(e + f*x))) + (4*im)/(35*a^2*f*(a + im*a*tan(e + f*x))^(3/2)*sqrt(c - im*c*tan(e + f*x))) + (8*tan(e + f*x))/(35*a^3*f*sqrt(a + im*a*tan(e + f*x))*sqrt(c - im*c*tan(e + f*x)))]


    @test_int [(a + im*a*tan(e + f*x))^(9/2)/(c - im*c*tan(e + f*x))^(3/2), x, 8, -((35*im*a^(9/2)*atan((sqrt(c)*sqrt(a + im*a*tan(e + f*x)))/(sqrt(a)*sqrt(c - im*c*tan(e + f*x)))))/(c^(3/2)*f)) - (2*im*a*(a + im*a*tan(e + f*x))^(7/2))/(3*f*(c - im*c*tan(e + f*x))^(3/2)) + (14*im*a^2*(a + im*a*tan(e + f*x))^(5/2))/(3*c*f*sqrt(c - im*c*tan(e + f*x))) + (35*im*a^4*sqrt(a + im*a*tan(e + f*x))*sqrt(c - im*c*tan(e + f*x)))/(2*c^2*f) + (35*im*a^3*(a + im*a*tan(e + f*x))^(3/2)*sqrt(c - im*c*tan(e + f*x)))/(6*c^2*f)]
    @test_int [(a + im*a*tan(e + f*x))^(7/2)/(c - im*c*tan(e + f*x))^(3/2), x, 7, -((10*im*a^(7/2)*atan((sqrt(c)*sqrt(a + im*a*tan(e + f*x)))/(sqrt(a)*sqrt(c - im*c*tan(e + f*x)))))/(c^(3/2)*f)) - (2*im*a*(a + im*a*tan(e + f*x))^(5/2))/(3*f*(c - im*c*tan(e + f*x))^(3/2)) + (10*im*a^2*(a + im*a*tan(e + f*x))^(3/2))/(3*c*f*sqrt(c - im*c*tan(e + f*x))) + (5*im*a^3*sqrt(a + im*a*tan(e + f*x))*sqrt(c - im*c*tan(e + f*x)))/(c^2*f)]
    @test_int [(a + im*a*tan(e + f*x))^(5/2)/(c - im*c*tan(e + f*x))^(3/2), x, 6, ((-2*im)*a^(5/2)*atan((sqrt(c)*sqrt(a + im*a*tan(e + f*x)))/(sqrt(a)*sqrt(c - im*c*tan(e + f*x)))))/(c^(3/2)*f) - (((2*im)/3)*a*(a + im*a*tan(e + f*x))^(3/2))/(f*(c - im*c*tan(e + f*x))^(3/2)) + ((2*im)*a^2*sqrt(a + im*a*tan(e + f*x)))/(c*f*sqrt(c - im*c*tan(e + f*x)))]
    @test_int [(a + im*a*tan(e + f*x))^(3/2)/(c - im*c*tan(e + f*x))^(3/2), x, 2, ((-im/3)*(a + im*a*tan(e + f*x))^(3/2))/(f*(c - im*c*tan(e + f*x))^(3/2))]
    @test_int [(a + im*a*tan(e + f*x))^(1/2)/(c - im*c*tan(e + f*x))^(3/2), x, 3, ((-im/3)*sqrt(a + im*a*tan(e + f*x)))/(f*(c - im*c*tan(e + f*x))^(3/2)) - ((im/3)*sqrt(a + im*a*tan(e + f*x)))/(c*f*sqrt(c - im*c*tan(e + f*x)))]
    @test_int [1/((a + im*a*tan(e + f*x))^(1/2)*(c - im*c*tan(e + f*x))^(3/2)), x, 4, im/(f*sqrt(a + im*a*tan(e + f*x))*(c - im*c*tan(e + f*x))^(3/2)) - (2*im*sqrt(a + im*a*tan(e + f*x)))/(3*a*f*(c - im*c*tan(e + f*x))^(3/2)) - (2*im*sqrt(a + im*a*tan(e + f*x)))/(3*a*c*f*sqrt(c - im*c*tan(e + f*x)))]
    @test_int [1/((a + im*a*tan(e + f*x))^(3/2)*(c - im*c*tan(e + f*x))^(3/2)), x, 3, tan(e + f*x)/(3*f*(a + im*a*tan(e + f*x))^(3/2)*(c - im*c*tan(e + f*x))^(3/2)) + (2*tan(e + f*x))/(3*a*c*f*sqrt(a + im*a*tan(e + f*x))*sqrt(c - im*c*tan(e + f*x)))]
    @test_int [1/((a + im*a*tan(e + f*x))^(5/2)*(c - im*c*tan(e + f*x))^(3/2)), x, 4, im/(5*f*(a + im*a*tan(e + f*x))^(5/2)*(c - im*c*tan(e + f*x))^(3/2)) + (4*tan(e + f*x))/(15*a*f*(a + im*a*tan(e + f*x))^(3/2)*(c - im*c*tan(e + f*x))^(3/2)) + (8*tan(e + f*x))/(15*a^2*c*f*sqrt(a + im*a*tan(e + f*x))*sqrt(c - im*c*tan(e + f*x)))]
    @test_int [1/((a + im*a*tan(e + f*x))^(7/2)*(c - im*c*tan(e + f*x))^(3/2)), x, 5, im/(7*f*(a + im*a*tan(e + f*x))^(7/2)*(c - im*c*tan(e + f*x))^(3/2)) + im/(7*a*f*(a + im*a*tan(e + f*x))^(5/2)*(c - im*c*tan(e + f*x))^(3/2)) + (4*tan(e + f*x))/(21*a^2*f*(a + im*a*tan(e + f*x))^(3/2)*(c - im*c*tan(e + f*x))^(3/2)) + (8*tan(e + f*x))/(21*a^3*c*f*sqrt(a + im*a*tan(e + f*x))*sqrt(c - im*c*tan(e + f*x)))]


    @test_int [(a + im*a*tan(e + f*x))^(11/2)/(c - im*c*tan(e + f*x))^(5/2), x, 9, (63*im*a^(11/2)*atan((sqrt(c)*sqrt(a + im*a*tan(e + f*x)))/(sqrt(a)*sqrt(c - im*c*tan(e + f*x)))))/(c^(5/2)*f) - (2*im*a*(a + im*a*tan(e + f*x))^(9/2))/(5*f*(c - im*c*tan(e + f*x))^(5/2)) + (6*im*a^2*(a + im*a*tan(e + f*x))^(7/2))/(5*c*f*(c - im*c*tan(e + f*x))^(3/2)) - (42*im*a^3*(a + im*a*tan(e + f*x))^(5/2))/(5*c^2*f*sqrt(c - im*c*tan(e + f*x))) - (63*im*a^5*sqrt(a + im*a*tan(e + f*x))*sqrt(c - im*c*tan(e + f*x)))/(2*c^3*f) - (21*im*a^4*(a + im*a*tan(e + f*x))^(3/2)*sqrt(c - im*c*tan(e + f*x)))/(2*c^3*f)]
    @test_int [(a + im*a*tan(e + f*x))^(9/2)/(c - im*c*tan(e + f*x))^(5/2), x, 8, (14*im*a^(9/2)*atan((sqrt(c)*sqrt(a + im*a*tan(e + f*x)))/(sqrt(a)*sqrt(c - im*c*tan(e + f*x)))))/(c^(5/2)*f) - (2*im*a*(a + im*a*tan(e + f*x))^(7/2))/(5*f*(c - im*c*tan(e + f*x))^(5/2)) + (14*im*a^2*(a + im*a*tan(e + f*x))^(5/2))/(15*c*f*(c - im*c*tan(e + f*x))^(3/2)) - (14*im*a^3*(a + im*a*tan(e + f*x))^(3/2))/(3*c^2*f*sqrt(c - im*c*tan(e + f*x))) - (7*im*a^4*sqrt(a + im*a*tan(e + f*x))*sqrt(c - im*c*tan(e + f*x)))/(c^3*f)]
    @test_int [(a + im*a*tan(e + f*x))^(7/2)/(c - im*c*tan(e + f*x))^(5/2), x, 7, (2*im*a^(7/2)*atan((sqrt(c)*sqrt(a + im*a*tan(e + f*x)))/(sqrt(a)*sqrt(c - im*c*tan(e + f*x)))))/(c^(5/2)*f) - (2*im*a*(a + im*a*tan(e + f*x))^(5/2))/(5*f*(c - im*c*tan(e + f*x))^(5/2)) + (2*im*a^2*(a + im*a*tan(e + f*x))^(3/2))/(3*c*f*(c - im*c*tan(e + f*x))^(3/2)) - (2*im*a^3*sqrt(a + im*a*tan(e + f*x)))/(c^2*f*sqrt(c - im*c*tan(e + f*x)))]
    @test_int [(a + im*a*tan(e + f*x))^(5/2)/(c - im*c*tan(e + f*x))^(5/2), x, 2, ((-im/5)*(a + im*a*tan(e + f*x))^(5/2))/(f*(c - im*c*tan(e + f*x))^(5/2))]
    @test_int [(a + im*a*tan(e + f*x))^(3/2)/(c - im*c*tan(e + f*x))^(5/2), x, 3, ((-im/5)*(a + im*a*tan(e + f*x))^(3/2))/(f*(c - im*c*tan(e + f*x))^(5/2)) - ((im/15)*(a + im*a*tan(e + f*x))^(3/2))/(c*f*(c - im*c*tan(e + f*x))^(3/2))]
    @test_int [(a + im*a*tan(e + f*x))^(1/2)/(c - im*c*tan(e + f*x))^(5/2), x, 4, ((-im/5)*sqrt(a + im*a*tan(e + f*x)))/(f*(c - im*c*tan(e + f*x))^(5/2)) - (((2*im)/15)*sqrt(a + im*a*tan(e + f*x)))/(c*f*(c - im*c*tan(e + f*x))^(3/2)) - (((2*im)/15)*sqrt(a + im*a*tan(e + f*x)))/(c^2*f*sqrt(c - im*c*tan(e + f*x)))]
    @test_int [1/((a + im*a*tan(e + f*x))^(1/2)*(c - im*c*tan(e + f*x))^(5/2)), x, 5, im/(f*sqrt(a + im*a*tan(e + f*x))*(c - im*c*tan(e + f*x))^(5/2)) - (3*im*sqrt(a + im*a*tan(e + f*x)))/(5*a*f*(c - im*c*tan(e + f*x))^(5/2)) - (2*im*sqrt(a + im*a*tan(e + f*x)))/(5*a*c*f*(c - im*c*tan(e + f*x))^(3/2)) - (2*im*sqrt(a + im*a*tan(e + f*x)))/(5*a*c^2*f*sqrt(c - im*c*tan(e + f*x)))]
    @test_int [1/((a + im*a*tan(e + f*x))^(3/2)*(c - im*c*tan(e + f*x))^(5/2)), x, 6, im/(3*f*(a + im*a*tan(e + f*x))^(3/2)*(c - im*c*tan(e + f*x))^(5/2)) + (4*im)/(3*a*f*sqrt(a + im*a*tan(e + f*x))*(c - im*c*tan(e + f*x))^(5/2)) - (4*im*sqrt(a + im*a*tan(e + f*x)))/(5*a^2*f*(c - im*c*tan(e + f*x))^(5/2)) - (8*im*sqrt(a + im*a*tan(e + f*x)))/(15*a^2*c*f*(c - im*c*tan(e + f*x))^(3/2)) - (8*im*sqrt(a + im*a*tan(e + f*x)))/(15*a^2*c^2*f*sqrt(c - im*c*tan(e + f*x)))]
    @test_int [1/((a + im*a*tan(e + f*x))^(5/2)*(c - im*c*tan(e + f*x))^(5/2)), x, 4, tan(e + f*x)/(5*f*(a + im*a*tan(e + f*x))^(5/2)*(c - im*c*tan(e + f*x))^(5/2)) + (4*tan(e + f*x))/(15*a*c*f*(a + im*a*tan(e + f*x))^(3/2)*(c - im*c*tan(e + f*x))^(3/2)) + (8*tan(e + f*x))/(15*a^2*c^2*f*sqrt(a + im*a*tan(e + f*x))*sqrt(c - im*c*tan(e + f*x)))]
    @test_int [1/((a + im*a*tan(e + f*x))^(7/2)*(c - im*c*tan(e + f*x))^(5/2)), x, 5, im/(7*f*(a + im*a*tan(e + f*x))^(7/2)*(c - im*c*tan(e + f*x))^(5/2)) + (6*tan(e + f*x))/(35*a*f*(a + im*a*tan(e + f*x))^(5/2)*(c - im*c*tan(e + f*x))^(5/2)) + (8*tan(e + f*x))/(35*a^2*c*f*(a + im*a*tan(e + f*x))^(3/2)*(c - im*c*tan(e + f*x))^(3/2)) + (16*tan(e + f*x))/(35*a^3*c^2*f*sqrt(a + im*a*tan(e + f*x))*sqrt(c - im*c*tan(e + f*x)))]


    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*(a+im*a*tan(e+f*x))^m*(c-c*im*tan(e+f*x))^n*with*n*symbolic=#


    @test_int [(a + im*a*tan(e + f*x))^4*(c - im*c*tan(e + f*x))^n, x, 4, (8*im*a^4*(c - im*c*tan(e + f*x))^n)/(f*n) - (12*im*a^4*(c - im*c*tan(e + f*x))^(1 + n))/(c*f*(1 + n)) + (6*im*a^4*(c - im*c*tan(e + f*x))^(2 + n))/(c^2*f*(2 + n)) - (im*a^4*(c - im*c*tan(e + f*x))^(3 + n))/(c^3*f*(3 + n))]
    @test_int [(a + im*a*tan(e + f*x))^3*(c - im*c*tan(e + f*x))^n, x, 4, (4*im*a^3*(c - im*c*tan(e + f*x))^n)/(f*n) - (4*im*a^3*(c - im*c*tan(e + f*x))^(1 + n))/(c*f*(1 + n)) + (im*a^3*(c - im*c*tan(e + f*x))^(2 + n))/(c^2*f*(2 + n))]
    @test_int [(a + im*a*tan(e + f*x))^2*(c - im*c*tan(e + f*x))^n, x, 4, ((2*im)*a^2*(c - im*c*tan(e + f*x))^n)/(f*n) - (im*a^2*(c - im*c*tan(e + f*x))^(1 + n))/(c*f*(1 + n))]
    @test_int [(a + im*a*tan(e + f*x))^1*(c - im*c*tan(e + f*x))^n, x, 3, (im*a*(c - im*c*tan(e + f*x))^n)/(f*n)]
    @test_int [(c - im*c*tan(e + f*x))^n/(a + im*a*tan(e + f*x))^1, x, 3, (im*HypergeometricFunctions._₂F₁(2, n, 1 + n, (1/2)*(1 - im*tan(e + f*x)))*(c - im*c*tan(e + f*x))^n)/(4*a*f*n)]
    @test_int [(c - im*c*tan(e + f*x))^n/(a + im*a*tan(e + f*x))^2, x, 3, (im*HypergeometricFunctions._₂F₁(3, n, 1 + n, (1/2)*(1 - im*tan(e + f*x)))*(c - im*c*tan(e + f*x))^n)/(8*a^2*f*n)]
    @test_int [(c - im*c*tan(e + f*x))^n/(a + im*a*tan(e + f*x))^3, x, 3, (im*HypergeometricFunctions._₂F₁(4, n, 1 + n, (1/2)*(1 - im*tan(e + f*x)))*(c - im*c*tan(e + f*x))^n)/(16*a^3*f*n)]


    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*(a+im*a*tan(e+f*x))^m*(c-c*im*tan(e+f*x))^n*with*m*symbolic=#


    @test_int [(a + im*a*tan(e + f*x))^m*(c - im*c*tan(e + f*x))^n, x, 3, ((im/2)*HypergeometricFunctions._₂F₁(1, m + n, 1 + n, (1 - im*tan(e + f*x))/2)*(a + im*a*tan(e + f*x))^m*(c - im*c*tan(e + f*x))^n)/(f*n), -((1/(f*m))*((im*2^(-1 + n)*HypergeometricFunctions._₂F₁(m, 1 - n, 1 + m, (1/2)*(1 + im*tan(e + f*x)))*(a + im*a*tan(e + f*x))^m*(c - im*c*tan(e + f*x))^n)/(1 - im*tan(e + f*x))^n))]


    @test_int [(a + im*a*tan(e + f*x))^m*(c - im*c*tan(e + f*x))^4, x, 4, -((8*im*c^4*(a + im*a*tan(e + f*x))^m)/(f*m)) + (12*im*c^4*(a + im*a*tan(e + f*x))^(1 + m))/(a*f*(1 + m)) - (6*im*c^4*(a + im*a*tan(e + f*x))^(2 + m))/(a^2*f*(2 + m)) + (im*c^4*(a + im*a*tan(e + f*x))^(3 + m))/(a^3*f*(3 + m))]
    @test_int [(a + im*a*tan(e + f*x))^m*(c - im*c*tan(e + f*x))^3, x, 4, ((-4*im)*c^3*(a + im*a*tan(e + f*x))^m)/(f*m) + ((4*im)*c^3*(a + im*a*tan(e + f*x))^(1 + m))/(a*f*(1 + m)) - (im*c^3*(a + im*a*tan(e + f*x))^(2 + m))/(a^2*f*(2 + m))]
    @test_int [(a + im*a*tan(e + f*x))^m*(c - im*c*tan(e + f*x))^2, x, 4, ((-2*im)*c^2*(a + im*a*tan(e + f*x))^m)/(f*m) + (im*c^2*(a + im*a*tan(e + f*x))^(1 + m))/(a*f*(1 + m))]
    @test_int [(a + im*a*tan(e + f*x))^m*(c - im*c*tan(e + f*x))^1, x, 3, ((-im)*c*(a + im*a*tan(e + f*x))^m)/(f*m)]
    @test_int [(a + im*a*tan(e + f*x))^m/(c - im*c*tan(e + f*x))^1, x, 3, -((im*HypergeometricFunctions._₂F₁(2, m, 1 + m, (1/2)*(1 + im*tan(e + f*x)))*(a + im*a*tan(e + f*x))^m)/(4*c*f*m))]
    @test_int [(a + im*a*tan(e + f*x))^m/(c - im*c*tan(e + f*x))^2, x, 3, -((im*HypergeometricFunctions._₂F₁(3, m, 1 + m, (1/2)*(1 + im*tan(e + f*x)))*(a + im*a*tan(e + f*x))^m)/(8*c^2*f*m))]
    @test_int [(a + im*a*tan(e + f*x))^m/(c - im*c*tan(e + f*x))^3, x, 3, -((im*HypergeometricFunctions._₂F₁(4, m, 1 + m, (1/2)*(1 + im*tan(e + f*x)))*(a + im*a*tan(e + f*x))^m)/(16*c^3*f*m))]
    @test_int [(a + im*a*tan(e + f*x))^m/(c - im*c*tan(e + f*x))^4, x, 3, -((im*HypergeometricFunctions._₂F₁(5, m, 1 + m, (1/2)*(1 + im*tan(e + f*x)))*(a + im*a*tan(e + f*x))^m)/(32*c^4*f*m))]


    @test_int [(a + im*a*tan(e + f*x))^m*(c - im*c*tan(e + f*x))^(5/2), x, 3, ((im/5)*HypergeometricFunctions._₂F₁(1, 5/2 + m, 7/2, (1 - im*tan(e + f*x))/2)*(a + im*a*tan(e + f*x))^m*(c - im*c*tan(e + f*x))^(5/2))/f, (1/(5*f))*((im*2^m*HypergeometricFunctions._₂F₁(5/2, 1 - m, 7/2, (1/2)*(1 - im*tan(e + f*x)))*(a + im*a*tan(e + f*x))^m*(c - im*c*tan(e + f*x))^(5/2))/(1 + im*tan(e + f*x))^m)]
    @test_int [(a + im*a*tan(e + f*x))^m*(c - im*c*tan(e + f*x))^(3/2), x, 3, ((im/3)*HypergeometricFunctions._₂F₁(1, 3/2 + m, 5/2, (1 - im*tan(e + f*x))/2)*(a + im*a*tan(e + f*x))^m*(c - im*c*tan(e + f*x))^(3/2))/f, (1/(3*f))*((im*2^m*HypergeometricFunctions._₂F₁(3/2, 1 - m, 5/2, (1/2)*(1 - im*tan(e + f*x)))*(a + im*a*tan(e + f*x))^m*(c - im*c*tan(e + f*x))^(3/2))/(1 + im*tan(e + f*x))^m)]
    @test_int [(a + im*a*tan(e + f*x))^m*(c - im*c*tan(e + f*x))^(1/2), x, 3, (im*HypergeometricFunctions._₂F₁(1, 1/2 + m, 3/2, (1 - im*tan(e + f*x))/2)*(a + im*a*tan(e + f*x))^m*sqrt(c - im*c*tan(e + f*x)))/f, (1/f)*((im*2^m*HypergeometricFunctions._₂F₁(1/2, 1 - m, 3/2, (1/2)*(1 - im*tan(e + f*x)))*(a + im*a*tan(e + f*x))^m*sqrt(c - im*c*tan(e + f*x)))/(1 + im*tan(e + f*x))^m)]
    @test_int [(a + im*a*tan(e + f*x))^m/(c - im*c*tan(e + f*x))^(1/2), x, 3, ((-im)*HypergeometricFunctions._₂F₁(1, -1/2 + m, 1/2, (1 - im*tan(e + f*x))/2)*(a + im*a*tan(e + f*x))^m)/(f*sqrt(c - im*c*tan(e + f*x))), -((1/(f*sqrt(c - im*c*tan(e + f*x))))*((im*2^m*HypergeometricFunctions._₂F₁(-(1/2), 1 - m, 1/2, (1/2)*(1 - im*tan(e + f*x)))*(a + im*a*tan(e + f*x))^m)/(1 + im*tan(e + f*x))^m))]
    @test_int [(a + im*a*tan(e + f*x))^m/(c - im*c*tan(e + f*x))^(3/2), x, 3, ((-im/3)*HypergeometricFunctions._₂F₁(1, -3/2 + m, -1/2, (1 - im*tan(e + f*x))/2)*(a + im*a*tan(e + f*x))^m)/(f*(c - im*c*tan(e + f*x))^(3/2)), -((im*2^m*HypergeometricFunctions._₂F₁(-(3/2), 1 - m, -(1/2), (1/2)*(1 - im*tan(e + f*x)))*(a + im*a*tan(e + f*x))^m)/(1 + im*tan(e + f*x))^m/(3*f*(c - im*c*tan(e + f*x))^(3/2)))]
    @test_int [(a + im*a*tan(e + f*x))^m/(c - im*c*tan(e + f*x))^(5/2), x, 3, ((-im/5)*HypergeometricFunctions._₂F₁(1, -5/2 + m, -3/2, (1 - im*tan(e + f*x))/2)*(a + im*a*tan(e + f*x))^m)/(f*(c - im*c*tan(e + f*x))^(5/2)), -((im*2^m*HypergeometricFunctions._₂F₁(-(5/2), 1 - m, -(3/2), (1/2)*(1 - im*tan(e + f*x)))*(a + im*a*tan(e + f*x))^m)/(1 + im*tan(e + f*x))^m/(5*f*(c - im*c*tan(e + f*x))^(5/2)))]


    #= ::Section::Closed:: =#
    #=Integrands*of*the*form*(a+im*a*tan(e+f*x))^m*(c+d*tan(e+f*x))^n=#


    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*(a+im*a*tan(e+f*x))^m*(c+d*tan(e+f*x))^n=#


    #= ::Subsubsection::Closed:: =#
    #=n > 0=#


    @test_int [(a + im*a*tan(e + f*x))^3*(c + d*tan(e + f*x)), x, 4, 4*a^3*(c - im*d)*x - (4*a^3*(im*c + d)*log(cos(e + f*x)))/f - (2*a^3*(c - im*d)*tan(e + f*x))/f + (a*(im*c + d)*(a + im*a*tan(e + f*x))^2)/(2*f) + (d*(a + im*a*tan(e + f*x))^3)/(3*f)]
    @test_int [(a + im*a*tan(e + f*x))^2*(c + d*tan(e + f*x)), x, 3, 2*a^2*(c - im*d)*x - (2*a^2*(im*c + d)*log(cos(e + f*x)))/f - (a^2*(c - im*d)*tan(e + f*x))/f + (d*(a + im*a*tan(e + f*x))^2)/(2*f)]
    @test_int [(a + im*a*tan(e + f*x))^1*(c + d*tan(e + f*x)), x, 2, a*(c - im*d)*x - (a*(im*c + d)*log(cos(e + f*x)))/f + (im*a*d*tan(e + f*x))/f]
    @test_int [(c + d*tan(e + f*x))/(a + im*a*tan(e + f*x))^1, x, 2, ((c - im*d)*x)/(2*a) + (im*c - d)/(2*f*(a + im*a*tan(e + f*x)))]
    @test_int [(c + d*tan(e + f*x))/(a + im*a*tan(e + f*x))^2, x, 3, ((c - im*d)*x)/(4*a^2) + (im*c - d)/(4*f*(a + im*a*tan(e + f*x))^2) + (im*c + d)/(4*f*(a^2 + im*a^2*tan(e + f*x)))]
    @test_int [(c + d*tan(e + f*x))/(a + im*a*tan(e + f*x))^3, x, 4, ((c - im*d)*x)/(8*a^3) + (im*c - d)/(6*f*(a + im*a*tan(e + f*x))^3) + (im*c + d)/(8*a*f*(a + im*a*tan(e + f*x))^2) + (im*c + d)/(8*f*(a^3 + im*a^3*tan(e + f*x)))]


    @test_int [(a + im*a*tan(e + f*x))^3*(c + d*tan(e + f*x))^2, x, 5, 4*a^3*(c - im*d)^2*x - ((4*im)*a^3*(c - im*d)^2*log(cos(e + f*x)))/f - (2*a^3*(c - im*d)^2*tan(e + f*x))/f + ((im/2)*a*(c - im*d)^2*(a + im*a*tan(e + f*x))^2)/f + (2*c*d*(a + im*a*tan(e + f*x))^3)/(3*f) - ((im/4)*d^2*(a + im*a*tan(e + f*x))^4)/(a*f)]
    @test_int [(a + im*a*tan(e + f*x))^2*(c + d*tan(e + f*x))^2, x, 4, 2*a^2*(c - im*d)^2*x - ((2*im)*a^2*(c - im*d)^2*log(cos(e + f*x)))/f - (a^2*(c - im*d)^2*tan(e + f*x))/f + (c*d*(a + im*a*tan(e + f*x))^2)/f - ((im/3)*d^2*(a + im*a*tan(e + f*x))^3)/(a*f)]
    @test_int [(a + im*a*tan(e + f*x))^1*(c + d*tan(e + f*x))^2, x, 3, a*(c - im*d)^2*x - (im*a*(c - im*d)^2*log(cos(e + f*x)))/f + (a*d*(im*c + d)*tan(e + f*x))/f + ((im/2)*a*(c + d*tan(e + f*x))^2)/f]
    @test_int [(c + d*tan(e + f*x))^2/(a + im*a*tan(e + f*x))^1, x, 3, ((c^2 - 2*im*c*d + d^2)*x)/(2*a) + (im*d^2*log(cos(e + f*x)))/(a*f) + (im*(c + im*d)^2)/(2*f*(a + im*a*tan(e + f*x)))]
    @test_int [(c + d*tan(e + f*x))^2/(a + im*a*tan(e + f*x))^2, x, 3, ((c - im*d)^2*x)/(4*a^2) + ((c + im*d)*(im*c + 3*d))/(4*a^2*f*(1 + im*tan(e + f*x))) + (im*(c + im*d)^2)/(4*f*(a + im*a*tan(e + f*x))^2)]
    @test_int [(c + d*tan(e + f*x))^2/(a + im*a*tan(e + f*x))^3, x, 4, ((c - im*d)^2*x)/(8*a^3) + (im*(c + im*d)^2)/(6*f*(a + im*a*tan(e + f*x))^3) + ((c + im*d)*(im*c + 3*d))/(8*a*f*(a + im*a*tan(e + f*x))^2) + (im*(c - im*d)^2)/(8*f*(a^3 + im*a^3*tan(e + f*x)))]


    @test_int [(a + im*a*tan(e + f*x))^3*(c + d*tan(e + f*x))^3, x, 6, 4*a^3*(c - im*d)^3*x + (4*a^3*(im*c + d)^3*log(cos(e + f*x)))/f + ((4*im)*a^3*(c - im*d)^2*d*tan(e + f*x))/f + (2*a^3*(im*c + d)*(c + d*tan(e + f*x))^2)/f + (((4*im)/3)*a^3*(c + d*tan(e + f*x))^3)/f + (a^3*(im*c - 11*d)*(c + d*tan(e + f*x))^4)/(20*d^2*f) - ((a^3 + im*a^3*tan(e + f*x))*(c + d*tan(e + f*x))^4)/(5*d*f)]
    @test_int [(a + im*a*tan(e + f*x))^2*(c + d*tan(e + f*x))^3, x, 5, 2*a^2*(c - im*d)^3*x + (2*a^2*(im*c + d)^3*log(cos(e + f*x)))/f + ((2*im)*a^2*(c - im*d)^2*d*tan(e + f*x))/f + (a^2*(im*c + d)*(c + d*tan(e + f*x))^2)/f + (((2*im)/3)*a^2*(c + d*tan(e + f*x))^3)/f - (a^2*(c + d*tan(e + f*x))^4)/(4*d*f)]
    @test_int [(a + im*a*tan(e + f*x))^1*(c + d*tan(e + f*x))^3, x, 4, a*(c - im*d)^3*x + (a*(im*c + d)^3*log(cos(e + f*x)))/f + (im*a*(c - im*d)^2*d*tan(e + f*x))/f + (a*(im*c + d)*(c + d*tan(e + f*x))^2)/(2*f) + ((im/3)*a*(c + d*tan(e + f*x))^3)/f]
    @test_int [(c + d*tan(e + f*x))^3/(a + im*a*tan(e + f*x))^1, x, 3, ((c^3 - (3*im)*c^2*d + 3*c*d^2 + (3*im)*d^3)*x)/(2*a) + (((3*im)*c - d)*d^2*log(cos(e + f*x)))/(a*f) - ((c + (3*im)*d)*d^2*tan(e + f*x))/(2*a*f) + ((im*c - d)*(c + d*tan(e + f*x))^2)/(2*f*(a + im*a*tan(e + f*x)))]
    @test_int [(c + d*tan(e + f*x))^3/(a + im*a*tan(e + f*x))^2, x, 5, ((c^3 - (3*im)*c^2*d - 3*c*d^2 - (3*im)*d^3)*x)/(4*a^2) + (d^3*log(cos(e + f*x)))/(a^2*f) + ((c + im*d)^2*(im*c + 3*d))/(4*a^2*f*(1 + im*tan(e + f*x))) + ((im*c - d)*(c + d*tan(e + f*x))^2)/(4*f*(a + im*a*tan(e + f*x))^2)]
    @test_int [(c + d*tan(e + f*x))^3/(a + im*a*tan(e + f*x))^3, x, 4, ((c - im*d)^3*x)/(8*a^3) + ((c + im*d)*(c - 3*im*d)*(im*c + d))/(8*a^3*f*(1 + im*tan(e + f*x))) + ((c + im*d)^2*(im*c + d))/(8*a*f*(a + im*a*tan(e + f*x))^2) + (im*(c + d*tan(e + f*x))^3)/(6*f*(a + im*a*tan(e + f*x))^3)]


    #= ::Subsubsection::Closed:: =#
    #=n < 0=#


    @test_int [(a + im*a*tan(e + f*x))^3/(c + d*tan(e + f*x)), x, 5, (4*a^3*x)/(c - im*d) - (a^3*(im*c - 3*d)*log(cos(e + f*x)))/(d^2*f) - (a^3*(c + im*d)^2*log(c*cos(e + f*x) + d*sin(e + f*x)))/(d^2*(im*c + d)*f) - (a^3 + im*a^3*tan(e + f*x))/(d*f)]
    @test_int [(a + im*a*tan(e + f*x))^2/(c + d*tan(e + f*x)), x, 4, -((a^2*c*(c + im*d)*x)/((c - im*d)*d^2)) + (a^2*(c + (2*im)*d)*x)/d^2 + (a^2*log(cos(e + f*x)))/(d*f) - (a^2*(im*c - d)*log(c*cos(e + f*x) + d*sin(e + f*x)))/(d*(im*c + d)*f)]
    @test_int [(a + im*a*tan(e + f*x))^1/(c + d*tan(e + f*x)), x, 2, (a*x)/(c - im*d) + (a*log(c*cos(e + f*x) + d*sin(e + f*x)))/((im*c + d)*f)]
    @test_int [1/((a + im*a*tan(e + f*x))^1*(c + d*tan(e + f*x))), x, 5, x/(2*a*(c + im*d)) - (c*d*x)/(a*(im*c - d)*(c^2 + d^2)) - (d^2*log(c*cos(e + f*x) + d*sin(e + f*x)))/(a*(im*c - d)*(c^2 + d^2)*f) - 1/(2*(im*c - d)*f*(a + im*a*tan(e + f*x)))]
    @test_int [1/((a + im*a*tan(e + f*x))^2*(c + d*tan(e + f*x))), x, 4, ((c^3 + (3*im)*c^2*d - 3*c*d^2 + (3*im)*d^3)*x)/(4*a^2*(c - im*d)*(c + im*d)^3) - (d^3*log(c*cos(e + f*x) + d*sin(e + f*x)))/(a^2*(c - im*d)*(c + im*d)^3*f) + (im*c - 3*d)/(4*a^2*(c + im*d)^2*f*(1 + im*tan(e + f*x))) - 1/(4*(im*c - d)*f*(a + im*a*tan(e + f*x))^2)]
    @test_int [1/((a + im*a*tan(e + f*x))^3*(c + d*tan(e + f*x))), x, 5, ((c^4 + (4*im)*c^3*d - 6*c^2*d^2 - (4*im)*c*d^3 - 7*d^4)*x)/(8*a^3*(c - im*d)*(c + im*d)^4) + (d^4*log(c*cos(e + f*x) + d*sin(e + f*x)))/(a^3*(c + im*d)^4*(im*c + d)*f) - 1/(6*(im*c - d)*f*(a + im*a*tan(e + f*x))^3) + (im*c - 3*d)/(8*a*(c + im*d)^2*f*(a + im*a*tan(e + f*x))^2) + (c^2 + (4*im)*c*d - 7*d^2)/(8*(im*c - d)^3*f*(a^3 + im*a^3*tan(e + f*x)))]


    @test_int [(a + im*a*tan(e + f*x))^3/(c + d*tan(e + f*x))^2, x, 5, (4*a^3*x)/(c - im*d)^2 + (im*a^3*log(cos(e + f*x)))/(d^2*f) - (a^3*(im*c - d)*(c - (3*im)*d)*log(c*cos(e + f*x) + d*sin(e + f*x)))/((c - im*d)^2*d^2*f) + ((c + im*d)*(a^3 + im*a^3*tan(e + f*x)))/((c - im*d)*d*f*(c + d*tan(e + f*x)))]
    @test_int [(a + im*a*tan(e + f*x))^2/(c + d*tan(e + f*x))^2, x, 3, (2*a^2*x)/(c - im*d)^2 - ((2*im)*a^2*log(c*cos(e + f*x) + d*sin(e + f*x)))/((c - im*d)^2*f) + (a^2*(im*c - d))/(d*(im*c + d)*f*(c + d*tan(e + f*x)))]
    @test_int [(a + im*a*tan(e + f*x))^1/(c + d*tan(e + f*x))^2, x, 3, (a*x)/(c - im*d)^2 - (im*a*log(c*cos(e + f*x) + d*sin(e + f*x)))/((c - im*d)^2*f) - a/((im*c + d)*f*(c + d*tan(e + f*x)))]
    @test_int [1/((a + im*a*tan(e + f*x))^1*(c + d*tan(e + f*x))^2), x, 4, ((c^3 + (3*im)*c^2*d + 3*c*d^2 - (3*im)*d^3)*x)/(2*a*(c - im*d)^2*(c + im*d)^3) + ((3*c - im*d)*d^2*log(c*cos(e + f*x) + d*sin(e + f*x)))/(a*(im*c - d)^3*(c - im*d)^2*f) + ((c - (3*im)*d)*d)/(2*a*(c - im*d)*(c + im*d)^2*f*(c + d*tan(e + f*x))) - 1/(2*(im*c - d)*f*(a + im*a*tan(e + f*x))*(c + d*tan(e + f*x)))]
    @test_int [1/((a + im*a*tan(e + f*x))^2*(c + d*tan(e + f*x))^2), x, 5, ((c^4 + (4*im)*c^3*d - 6*c^2*d^2 + (12*im)*c*d^3 + 9*d^4)*x)/(4*a^2*(c - im*d)^2*(c + im*d)^4) - (2*(2*c - im*d)*d^3*log(c*cos(e + f*x) + d*sin(e + f*x)))/(a^2*(c - im*d)^2*(c + im*d)^4*f) + (d*(c^2 + (4*im)*c*d + 9*d^2))/(4*a^2*(c - im*d)*(c + im*d)^3*f*(c + d*tan(e + f*x))) + (im*c - 4*d)/(4*a^2*(c + im*d)^2*f*(1 + im*tan(e + f*x))*(c + d*tan(e + f*x))) - 1/(4*(im*c - d)*f*(a + im*a*tan(e + f*x))^2*(c + d*tan(e + f*x)))]
    @test_int [1/((a + im*a*tan(e + f*x))^3*(c + d*tan(e + f*x))^2), x, 6, ((c^5 + (5*im)*c^4*d - 10*c^3*d^2 - (10*im)*c^2*d^3 - 35*c*d^4 + (25*im)*d^5)*x)/(8*a^3*(c - im*d)^2*(c + im*d)^5) + ((5*c - (3*im)*d)*d^4*log(c*cos(e + f*x) + d*sin(e + f*x)))/(a^3*(im*c - d)^5*(c - im*d)^2*f) + (d*(c^3 + (5*im)*c^2*d - 11*c*d^2 + (25*im)*d^3))/(8*a^3*(c - im*d)*(c + im*d)^4*f*(c + d*tan(e + f*x))) - 1/(6*(im*c - d)*f*(a + im*a*tan(e + f*x))^3*(c + d*tan(e + f*x))) + ((3*im)*c - 11*d)/(24*a*(c + im*d)^2*f*(a + im*a*tan(e + f*x))^2*(c + d*tan(e + f*x))) + (c^2 + (5*im)*c*d - 12*d^2)/(8*(im*c - d)^3*f*(a^3 + im*a^3*tan(e + f*x))*(c + d*tan(e + f*x)))]


    @test_int [(a + im*a*tan(e + f*x))^3/(c + d*tan(e + f*x))^3, x, 4, (4*a^3*x)/(c - im*d)^3 - (4*a^3*log(c*cos(e + f*x) + d*sin(e + f*x)))/((im*c + d)^3*f) - (a*(a + im*a*tan(e + f*x))^2)/(2*(im*c + d)*f*(c + d*tan(e + f*x))^2) + (2*a^3*(c + im*d))/((c - im*d)^2*d*f*(c + d*tan(e + f*x)))]
    @test_int [(a + im*a*tan(e + f*x))^2/(c + d*tan(e + f*x))^3, x, 4, (2*a^2*x)/(c - im*d)^3 - (2*a^2*log(c*cos(e + f*x) + d*sin(e + f*x)))/((im*c + d)^3*f) + (a^2*(im*c - d))/(2*d*(im*c + d)*f*(c + d*tan(e + f*x))^2) + ((2*im)*a^2)/((c - im*d)^2*f*(c + d*tan(e + f*x)))]
    @test_int [(a + im*a*tan(e + f*x))^1/(c + d*tan(e + f*x))^3, x, 4, (a*x)/(c - im*d)^3 - (a*log(c*cos(e + f*x) + d*sin(e + f*x)))/((im*c + d)^3*f) - a/(2*(im*c + d)*f*(c + d*tan(e + f*x))^2) + (im*a)/((c - im*d)^2*f*(c + d*tan(e + f*x)))]
    @test_int [1/((a + im*a*tan(e + f*x))^1*(c + d*tan(e + f*x))^3), x, 5, ((c^4 + (4*im)*c^3*d + 6*c^2*d^2 - (12*im)*c*d^3 - 3*d^4)*x)/(2*a*(c - im*d)^3*(c + im*d)^4) + (2*d^2*(3*c^2 - (2*im)*c*d - d^2)*log(c*cos(e + f*x) + d*sin(e + f*x)))/(a*(c + im*d)^4*(im*c + d)^3*f) + ((c - (2*im)*d)*d)/(2*a*(c - im*d)*(c + im*d)^2*f*(c + d*tan(e + f*x))^2) - 1/(2*(im*c - d)*f*(a + im*a*tan(e + f*x))*(c + d*tan(e + f*x))^2) + (d*(c^2 - (8*im)*c*d - 3*d^2))/(2*a*(c - im*d)^2*(c + im*d)^3*f*(c + d*tan(e + f*x)))]
    @test_int [1/((a + im*a*tan(e + f*x))^2*(c + d*tan(e + f*x))^3), x, 6, ((c^5 + (5*im)*c^4*d - 10*c^3*d^2 + (30*im)*c^2*d^3 + 45*c*d^4 - (15*im)*d^5)*x)/(4*a^2*(c - im*d)^3*(c + im*d)^5) - (2*d^3*(5*c^2 - (5*im)*c*d - 2*d^2)*log(c*cos(e + f*x) + d*sin(e + f*x)))/(a^2*(im*c - d)^5*(im*c + d)^3*f) + (d*(c^2 + (5*im)*c*d + 8*d^2))/(4*a^2*(c - im*d)*(c + im*d)^3*f*(c + d*tan(e + f*x))^2) + (im*c - 5*d)/(4*a^2*(c + im*d)^2*f*(1 + im*tan(e + f*x))*(c + d*tan(e + f*x))^2) - 1/(4*(im*c - d)*f*(a + im*a*tan(e + f*x))^2*(c + d*tan(e + f*x))^2) + ((c - (3*im)*d)*d*(c^2 + (8*im)*c*d + 5*d^2))/(4*a^2*(c - im*d)^2*(c + im*d)^4*f*(c + d*tan(e + f*x)))]
    @test_int [1/((a + im*a*tan(e + f*x))^3*(c + d*tan(e + f*x))^3), x, 7, ((c^6 + (6*im)*c^5*d - 15*c^4*d^2 - (20*im)*c^3*d^3 - 105*c^2*d^4 + (150*im)*c*d^5 + 55*d^6)*x)/(8*a^3*(c - im*d)^3*(c + im*d)^6) - (d^4*(15*c^2 - (18*im)*c*d - 7*d^2)*log(c*cos(e + f*x) + d*sin(e + f*x)))/(a^3*(c + im*d)^6*(im*c + d)^3*f) + (d*(c^3 + (6*im)*c^2*d - 17*c*d^2 + (28*im)*d^3))/(8*a^3*(c - im*d)*(c + im*d)^4*f*(c + d*tan(e + f*x))^2) - 1/(6*(im*c - d)*f*(a + im*a*tan(e + f*x))^3*(c + d*tan(e + f*x))^2) + ((3*im)*c - 13*d)/(24*a*(c + im*d)^2*f*(a + im*a*tan(e + f*x))^2*(c + d*tan(e + f*x))^2) + (3*c^2 + (18*im)*c*d - 55*d^2)/(24*(im*c - d)^3*f*(a^3 + im*a^3*tan(e + f*x))*(c + d*tan(e + f*x))^2) + (d*(c^4 + (6*im)*c^3*d - 16*c^2*d^2 + (94*im)*c*d^3 + 55*d^4))/(8*a^3*(c - im*d)^2*(c + im*d)^5*f*(c + d*tan(e + f*x)))]


    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*(a+im*a*tan(e+f*x))^m*(c+d*tan(e+f*x))^(n/2)=#


    #= ::Subsubsection::Closed:: =#
    #=n > 0=#


    @test_int [(a + im*a*tan(e + f*x))^3*sqrt(c + d*tan(e + f*x)), x, 6, ((-8*im)*a^3*sqrt(c - im*d)*atanh(sqrt(c + d*tan(e + f*x))/sqrt(c - im*d)))/f + ((8*im)*a^3*sqrt(c + d*tan(e + f*x)))/f + (4*a^3*(im*c - 6*d)*(c + d*tan(e + f*x))^(3/2))/(15*d^2*f) - (2*(a^3 + im*a^3*tan(e + f*x))*(c + d*tan(e + f*x))^(3/2))/(5*d*f)]
    @test_int [(a + im*a*tan(e + f*x))^2*sqrt(c + d*tan(e + f*x)), x, 5, ((-4*im)*a^2*sqrt(c - im*d)*atanh(sqrt(c + d*tan(e + f*x))/sqrt(c - im*d)))/f + ((4*im)*a^2*sqrt(c + d*tan(e + f*x)))/f - (2*a^2*(c + d*tan(e + f*x))^(3/2))/(3*d*f)]
    @test_int [(a + im*a*tan(e + f*x))*sqrt(c + d*tan(e + f*x)), x, 4, ((-2*im)*a*sqrt(c - im*d)*atanh(sqrt(c + d*tan(e + f*x))/sqrt(c - im*d)))/f + ((2*im)*a*sqrt(c + d*tan(e + f*x)))/f]
    @test_int [sqrt(c + d*tan(e + f*x))/(a + im*a*tan(e + f*x)), x, 8, ((-im/2)*sqrt(c - im*d)*atanh(sqrt(c + d*tan(e + f*x))/sqrt(c - im*d)))/(a*f) + ((im/2)*c*atanh(sqrt(c + d*tan(e + f*x))/sqrt(c + im*d)))/(a*sqrt(c + im*d)*f) + ((im/2)*sqrt(c + d*tan(e + f*x)))/(f*(a + im*a*tan(e + f*x)))]
    @test_int [sqrt(c + d*tan(e + f*x))/(a + im*a*tan(e + f*x))^2, x, 9, ((-im/4)*sqrt(c - im*d)*atanh(sqrt(c + d*tan(e + f*x))/sqrt(c - im*d)))/(a^2*f) - ((2*c*d - im*(2*c^2 + d^2))*atanh(sqrt(c + d*tan(e + f*x))/sqrt(c + im*d)))/(8*a^2*(c + im*d)^(3/2)*f) + (((2*im)*c - d)*sqrt(c + d*tan(e + f*x)))/(8*a^2*(c + im*d)*f*(1 + im*tan(e + f*x))) + ((im/4)*sqrt(c + d*tan(e + f*x)))/(f*(a + im*a*tan(e + f*x))^2)]
    @test_int [sqrt(c + d*tan(e + f*x))/(a + im*a*tan(e + f*x))^3, x, 10, ((-im/8)*sqrt(c - im*d)*atanh(sqrt(c + d*tan(e + f*x))/sqrt(c - im*d)))/(a^3*f) + (((2*im)*c^3 - 4*c^2*d - im*c*d^2 - 2*d^3)*atanh(sqrt(c + d*tan(e + f*x))/sqrt(c + im*d)))/(16*a^3*(c + im*d)^(5/2)*f) + ((im/6)*sqrt(c + d*tan(e + f*x)))/(f*(a + im*a*tan(e + f*x))^3) + (((3*im)*c - 2*d)*sqrt(c + d*tan(e + f*x)))/(24*a*(c + im*d)*f*(a + im*a*tan(e + f*x))^2) + (c*((2*im)*c - 3*d)*sqrt(c + d*tan(e + f*x)))/(16*(c + im*d)^2*f*(a^3 + im*a^3*tan(e + f*x)))]


    @test_int [(a + im*a*tan(e + f*x))^3*(c + d*tan(e + f*x))^(3/2), x, 7, ((-8*im)*a^3*(c - im*d)^(3/2)*atanh(sqrt(c + d*tan(e + f*x))/sqrt(c - im*d)))/f + (8*a^3*(im*c + d)*sqrt(c + d*tan(e + f*x)))/f + (((8*im)/3)*a^3*(c + d*tan(e + f*x))^(3/2))/f + (4*a^3*(im*c - 8*d)*(c + d*tan(e + f*x))^(5/2))/(35*d^2*f) - (2*(a^3 + im*a^3*tan(e + f*x))*(c + d*tan(e + f*x))^(5/2))/(7*d*f)]
    @test_int [(a + im*a*tan(e + f*x))^2*(c + d*tan(e + f*x))^(3/2), x, 6, ((-4*im)*a^2*(c - im*d)^(3/2)*atanh(sqrt(c + d*tan(e + f*x))/sqrt(c - im*d)))/f + (4*a^2*(im*c + d)*sqrt(c + d*tan(e + f*x)))/f + (((4*im)/3)*a^2*(c + d*tan(e + f*x))^(3/2))/f - (2*a^2*(c + d*tan(e + f*x))^(5/2))/(5*d*f)]
    @test_int [(a + im*a*tan(e + f*x))*(c + d*tan(e + f*x))^(3/2), x, 5, ((-2*im)*a*(c - im*d)^(3/2)*atanh(sqrt(c + d*tan(e + f*x))/sqrt(c - im*d)))/f + (2*a*(im*c + d)*sqrt(c + d*tan(e + f*x)))/f + (((2*im)/3)*a*(c + d*tan(e + f*x))^(3/2))/f]
    @test_int [(c + d*tan(e + f*x))^(3/2)/(a + im*a*tan(e + f*x)), x, 8, ((-im/2)*(c - im*d)^(3/2)*atanh(sqrt(c + d*tan(e + f*x))/sqrt(c - im*d)))/(a*f) + (sqrt(c + im*d)*(im*c + 2*d)*atanh(sqrt(c + d*tan(e + f*x))/sqrt(c + im*d)))/(2*a*f) + ((im*c - d)*sqrt(c + d*tan(e + f*x)))/(2*f*(a + im*a*tan(e + f*x)))]
    @test_int [(c + d*tan(e + f*x))^(3/2)/(a + im*a*tan(e + f*x))^2, x, 9, ((-im/4)*(c - im*d)^(3/2)*atanh(sqrt(c + d*tan(e + f*x))/sqrt(c - im*d)))/(a^2*f) + ((2*c*d + im*(2*c^2 + d^2))*atanh(sqrt(c + d*tan(e + f*x))/sqrt(c + im*d)))/(8*a^2*sqrt(c + im*d)*f) + (((2*im)*c + 3*d)*sqrt(c + d*tan(e + f*x)))/(8*a^2*f*(1 + im*tan(e + f*x))) + ((im*c - d)*sqrt(c + d*tan(e + f*x)))/(4*f*(a + im*a*tan(e + f*x))^2)]
    @test_int [(c + d*tan(e + f*x))^(3/2)/(a + im*a*tan(e + f*x))^3, x, 10, ((-im/8)*(c - im*d)^(3/2)*atanh(sqrt(c + d*tan(e + f*x))/sqrt(c - im*d)))/(a^3*f) + ((im/16)*c*(2*c^2 + 3*d^2)*atanh(sqrt(c + d*tan(e + f*x))/sqrt(c + im*d)))/(a^3*(c + im*d)^(3/2)*f) + ((im*c - d)*sqrt(c + d*tan(e + f*x)))/(6*f*(a + im*a*tan(e + f*x))^3) + (((3*im)*c + 4*d)*sqrt(c + d*tan(e + f*x)))/(24*a*f*(a + im*a*tan(e + f*x))^2) - ((2*c^2 - im*c*d + 2*d^2)*sqrt(c + d*tan(e + f*x)))/(16*(im*c - d)*f*(a^3 + im*a^3*tan(e + f*x)))]


    @test_int [(a + im*a*tan(e + f*x))^3*(c + d*tan(e + f*x))^(5/2), x, 8, ((-8*im)*a^3*(c - im*d)^(5/2)*atanh(sqrt(c + d*tan(e + f*x))/sqrt(c - im*d)))/f + ((8*im)*a^3*(c - im*d)^2*sqrt(c + d*tan(e + f*x)))/f + (8*a^3*(im*c + d)*(c + d*tan(e + f*x))^(3/2))/(3*f) + (((8*im)/5)*a^3*(c + d*tan(e + f*x))^(5/2))/f + (4*a^3*(im*c - 10*d)*(c + d*tan(e + f*x))^(7/2))/(63*d^2*f) - (2*(a^3 + im*a^3*tan(e + f*x))*(c + d*tan(e + f*x))^(7/2))/(9*d*f)]
    @test_int [(a + im*a*tan(e + f*x))^2*(c + d*tan(e + f*x))^(5/2), x, 7, ((-4*im)*a^2*(c - im*d)^(5/2)*atanh(sqrt(c + d*tan(e + f*x))/sqrt(c - im*d)))/f + ((4*im)*a^2*(c - im*d)^2*sqrt(c + d*tan(e + f*x)))/f + (4*a^2*(im*c + d)*(c + d*tan(e + f*x))^(3/2))/(3*f) + (((4*im)/5)*a^2*(c + d*tan(e + f*x))^(5/2))/f - (2*a^2*(c + d*tan(e + f*x))^(7/2))/(7*d*f)]
    @test_int [(a + im*a*tan(e + f*x))*(c + d*tan(e + f*x))^(5/2), x, 6, ((-2*im)*a*(c - im*d)^(5/2)*atanh(sqrt(c + d*tan(e + f*x))/sqrt(c - im*d)))/f + ((2*im)*a*(c - im*d)^2*sqrt(c + d*tan(e + f*x)))/f + (2*a*(im*c + d)*(c + d*tan(e + f*x))^(3/2))/(3*f) + (((2*im)/5)*a*(c + d*tan(e + f*x))^(5/2))/f]
    @test_int [(c + d*tan(e + f*x))^(5/2)/(a + im*a*tan(e + f*x)), x, 9, ((-im/2)*(c - im*d)^(5/2)*atanh(sqrt(c + d*tan(e + f*x))/sqrt(c - im*d)))/(a*f) + ((c + im*d)^(3/2)*(im*c + 4*d)*atanh(sqrt(c + d*tan(e + f*x))/sqrt(c + im*d)))/(2*a*f) - ((c + (5*im)*d)*d*sqrt(c + d*tan(e + f*x)))/(2*a*f) + ((im*c - d)*(c + d*tan(e + f*x))^(3/2))/(2*f*(a + im*a*tan(e + f*x)))]
    @test_int [(c + d*tan(e + f*x))^(5/2)/(a + im*a*tan(e + f*x))^2, x, 9, ((-im/4)*(c - im*d)^(5/2)*atanh(sqrt(c + d*tan(e + f*x))/sqrt(c - im*d)))/(a^2*f) + (sqrt(c + im*d)*((2*im)*c^2 + 6*c*d - (7*im)*d^2)*atanh(sqrt(c + d*tan(e + f*x))/sqrt(c + im*d)))/(8*a^2*f) + ((c + im*d)*((2*im)*c + 5*d)*sqrt(c + d*tan(e + f*x)))/(8*a^2*f*(1 + im*tan(e + f*x))) + ((im*c - d)*(c + d*tan(e + f*x))^(3/2))/(4*f*(a + im*a*tan(e + f*x))^2)]
    @test_int [(c + d*tan(e + f*x))^(5/2)/(a + im*a*tan(e + f*x))^3, x, 10, ((-im/8)*(c - im*d)^(5/2)*atanh(sqrt(c + d*tan(e + f*x))/sqrt(c - im*d)))/(a^3*f) + (((2*im)*c^3 + 4*c^2*d - im*c*d^2 + 2*d^3)*atanh(sqrt(c + d*tan(e + f*x))/sqrt(c + im*d)))/(16*a^3*sqrt(c + im*d)*f) + ((c + im*d)*(im*c + 2*d)*sqrt(c + d*tan(e + f*x)))/(8*a*f*(a + im*a*tan(e + f*x))^2) + (((2*im)*c^2 + 5*c*d - (4*im)*d^2)*sqrt(c + d*tan(e + f*x)))/(16*f*(a^3 + im*a^3*tan(e + f*x))) + ((im*c - d)*(c + d*tan(e + f*x))^(3/2))/(6*f*(a + im*a*tan(e + f*x))^3)]


    #= ::Subsubsection::Closed:: =#
    #=n < 0=#


    @test_int [(a + im*a*tan(e + f*x))^3/sqrt(c + d*tan(e + f*x)), x, 5, ((-8*im)*a^3*atanh(sqrt(c + d*tan(e + f*x))/sqrt(c - im*d)))/(sqrt(c - im*d)*f) + (4*a^3*(im*c - 4*d)*sqrt(c + d*tan(e + f*x)))/(3*d^2*f) - (2*(a^3 + im*a^3*tan(e + f*x))*sqrt(c + d*tan(e + f*x)))/(3*d*f)]
    @test_int [(a + im*a*tan(e + f*x))^2/sqrt(c + d*tan(e + f*x)), x, 4, ((-4*im)*a^2*atanh(sqrt(c + d*tan(e + f*x))/sqrt(c - im*d)))/(sqrt(c - im*d)*f) - (2*a^2*sqrt(c + d*tan(e + f*x)))/(d*f)]
    @test_int [(a + im*a*tan(e + f*x))/sqrt(c + d*tan(e + f*x)), x, 3, ((-2*im)*a*atanh(sqrt(c + d*tan(e + f*x))/sqrt(c - im*d)))/(sqrt(c - im*d)*f)]
    @test_int [1/((a + im*a*tan(e + f*x))*sqrt(c + d*tan(e + f*x))), x, 8, ((-im/2)*atanh(sqrt(c + d*tan(e + f*x))/sqrt(c - im*d)))/(a*sqrt(c - im*d)*f) + ((im*c - 2*d)*atanh(sqrt(c + d*tan(e + f*x))/sqrt(c + im*d)))/(2*a*(c + im*d)^(3/2)*f) - sqrt(c + d*tan(e + f*x))/(2*(im*c - d)*f*(a + im*a*tan(e + f*x)))]
    @test_int [1/((a + im*a*tan(e + f*x))^2*sqrt(c + d*tan(e + f*x))), x, 9, ((-im/4)*atanh(sqrt(c + d*tan(e + f*x))/sqrt(c - im*d)))/(a^2*sqrt(c - im*d)*f) + (((2*im)*c^2 - 6*c*d - (7*im)*d^2)*atanh(sqrt(c + d*tan(e + f*x))/sqrt(c + im*d)))/(8*a^2*(c + im*d)^(5/2)*f) + (((2*im)*c - 5*d)*sqrt(c + d*tan(e + f*x)))/(8*a^2*(c + im*d)^2*f*(1 + im*tan(e + f*x))) - sqrt(c + d*tan(e + f*x))/(4*(im*c - d)*f*(a + im*a*tan(e + f*x))^2)]
    @test_int [1/((a + im*a*tan(e + f*x))^3*sqrt(c + d*tan(e + f*x))), x, 10, ((-im/8)*atanh(sqrt(c + d*tan(e + f*x))/sqrt(c - im*d)))/(a^3*sqrt(c - im*d)*f) + (((2*im)*c^3 - 8*c^2*d - (13*im)*c*d^2 + 12*d^3)*atanh(sqrt(c + d*tan(e + f*x))/sqrt(c + im*d)))/(16*a^3*(c + im*d)^(7/2)*f) - sqrt(c + d*tan(e + f*x))/(6*(im*c - d)*f*(a + im*a*tan(e + f*x))^3) + (((3*im)*c - 8*d)*sqrt(c + d*tan(e + f*x)))/(24*a*(c + im*d)^2*f*(a + im*a*tan(e + f*x))^2) + ((2*c^2 + (7*im)*c*d - 10*d^2)*sqrt(c + d*tan(e + f*x)))/(16*(im*c - d)^3*f*(a^3 + im*a^3*tan(e + f*x)))]


    @test_int [(a + im*a*tan(e + f*x))^3/(c + d*tan(e + f*x))^(3/2), x, 5, ((-8*im)*a^3*atanh(sqrt(c + d*tan(e + f*x))/sqrt(c - im*d)))/((c - im*d)^(3/2)*f) + (2*(c + im*d)*(a^3 + im*a^3*tan(e + f*x)))/((c - im*d)*d*f*sqrt(c + d*tan(e + f*x))) + (4*a^3*c*sqrt(c + d*tan(e + f*x)))/(d^2*(im*c + d)*f)]
    @test_int [(a + im*a*tan(e + f*x))^2/(c + d*tan(e + f*x))^(3/2), x, 4, ((-4*im)*a^2*atanh(sqrt(c + d*tan(e + f*x))/sqrt(c - im*d)))/((c - im*d)^(3/2)*f) + (2*a^2*(im*c - d))/(d*(im*c + d)*f*sqrt(c + d*tan(e + f*x)))]
    @test_int [(a + im*a*tan(e + f*x))/(c + d*tan(e + f*x))^(3/2), x, 4, ((-2*im)*a*atanh(sqrt(c + d*tan(e + f*x))/sqrt(c - im*d)))/((c - im*d)^(3/2)*f) - (2*a)/((im*c + d)*f*sqrt(c + d*tan(e + f*x)))]
    @test_int [1/((a + im*a*tan(e + f*x))*(c + d*tan(e + f*x))^(3/2)), x, 9, ((-im/2)*atanh(sqrt(c + d*tan(e + f*x))/sqrt(c - im*d)))/(a*(c - im*d)^(3/2)*f) + ((im*c - 4*d)*atanh(sqrt(c + d*tan(e + f*x))/sqrt(c + im*d)))/(2*a*(c + im*d)^(5/2)*f) + ((c - (5*im)*d)*d)/(2*a*(c - im*d)*(c + im*d)^2*f*sqrt(c + d*tan(e + f*x))) - 1/(2*(im*c - d)*f*(a + im*a*tan(e + f*x))*sqrt(c + d*tan(e + f*x)))]
    @test_int [1/((a + im*a*tan(e + f*x))^2*(c + d*tan(e + f*x))^(3/2)), x, 10, ((-im/4)*atanh(sqrt(c + d*tan(e + f*x))/sqrt(c - im*d)))/(a^2*(c - im*d)^(3/2)*f) + (((2*im)*c^2 - 10*c*d - (23*im)*d^2)*atanh(sqrt(c + d*tan(e + f*x))/sqrt(c + im*d)))/(8*a^2*(c + im*d)^(7/2)*f) + (d*(2*c^2 + (7*im)*c*d + 25*d^2))/(8*a^2*(c - im*d)*(c + im*d)^3*f*sqrt(c + d*tan(e + f*x))) + ((2*im)*c - 7*d)/(8*a^2*(c + im*d)^2*f*(1 + im*tan(e + f*x))*sqrt(c + d*tan(e + f*x))) - 1/(4*(im*c - d)*f*(a + im*a*tan(e + f*x))^2*sqrt(c + d*tan(e + f*x)))]
    @test_int [1/((a + im*a*tan(e + f*x))^3*(c + d*tan(e + f*x))^(3/2)), x, 11, ((-im/8)*atanh(sqrt(c + d*tan(e + f*x))/sqrt(c - im*d)))/(a^3*(c - im*d)^(3/2)*f) + (((2*im)*c^3 - 12*c^2*d - (33*im)*c*d^2 + 58*d^3)*atanh(sqrt(c + d*tan(e + f*x))/sqrt(c + im*d)))/(16*a^3*(c + im*d)^(9/2)*f) + (d*(2*c^3 + (9*im)*c^2*d - 17*c*d^2 + (60*im)*d^3))/(16*a^3*(c - im*d)*(c + im*d)^4*f*sqrt(c + d*tan(e + f*x))) - 1/(6*(im*c - d)*f*(a + im*a*tan(e + f*x))^3*sqrt(c + d*tan(e + f*x))) + ((3*im)*c - 10*d)/(24*a*(c + im*d)^2*f*(a + im*a*tan(e + f*x))^2*sqrt(c + d*tan(e + f*x))) + (6*c^2 + (27*im)*c*d - 56*d^2)/(48*(im*c - d)^3*f*(a^3 + im*a^3*tan(e + f*x))*sqrt(c + d*tan(e + f*x)))]


    @test_int [(a + im*a*tan(e + f*x))^3/(c + d*tan(e + f*x))^(5/2), x, 5, ((-8*im)*a^3*atanh(sqrt(c + d*tan(e + f*x))/sqrt(c - im*d)))/((c - im*d)^(5/2)*f) + (2*(c + im*d)*(a^3 + im*a^3*tan(e + f*x)))/(3*(c - im*d)*d*f*(c + d*tan(e + f*x))^(3/2)) + (4*a^3*(im*c - d)*(c - (4*im)*d))/(3*(c - im*d)^2*d^2*f*sqrt(c + d*tan(e + f*x)))]
    @test_int [(a + im*a*tan(e + f*x))^2/(c + d*tan(e + f*x))^(5/2), x, 5, ((-4*im)*a^2*atanh(sqrt(c + d*tan(e + f*x))/sqrt(c - im*d)))/((c - im*d)^(5/2)*f) + (2*a^2*(im*c - d))/(3*d*(im*c + d)*f*(c + d*tan(e + f*x))^(3/2)) + ((4*im)*a^2)/((c - im*d)^2*f*sqrt(c + d*tan(e + f*x)))]
    @test_int [(a + im*a*tan(e + f*x))/(c + d*tan(e + f*x))^(5/2), x, 5, ((-2*im)*a*atanh(sqrt(c + d*tan(e + f*x))/sqrt(c - im*d)))/((c - im*d)^(5/2)*f) - (2*a)/(3*(im*c + d)*f*(c + d*tan(e + f*x))^(3/2)) + ((2*im)*a)/((c - im*d)^2*f*sqrt(c + d*tan(e + f*x)))]
    @test_int [1/((a + im*a*tan(e + f*x))*(c + d*tan(e + f*x))^(5/2)), x, 10, ((-im/2)*atanh(sqrt(c + d*tan(e + f*x))/sqrt(c - im*d)))/(a*(c - im*d)^(5/2)*f) + ((im*c - 6*d)*atanh(sqrt(c + d*tan(e + f*x))/sqrt(c + im*d)))/(2*a*(c + im*d)^(7/2)*f) + (d*((3*im)*c + 7*d))/(6*a*(im*c - d)*(c^2 + d^2)*f*(c + d*tan(e + f*x))^(3/2)) - 1/(2*(im*c - d)*f*(a + im*a*tan(e + f*x))*(c + d*tan(e + f*x))^(3/2)) + (d*(c^2 - (14*im)*c*d - 5*d^2))/(2*a*(c - im*d)^2*(c + im*d)^3*f*sqrt(c + d*tan(e + f*x)))]
    @test_int [1/((a + im*a*tan(e + f*x))^2*(c + d*tan(e + f*x))^(5/2)), x, 11, ((-im/4)*atanh(sqrt(c + d*tan(e + f*x))/sqrt(c - im*d)))/(a^2*(c - im*d)^(5/2)*f) + (((2*im)*c^2 - 14*c*d - (47*im)*d^2)*atanh(sqrt(c + d*tan(e + f*x))/sqrt(c + im*d)))/(8*a^2*(c + im*d)^(9/2)*f) + (d*(6*c^2 + (27*im)*c*d + 49*d^2))/(24*a^2*(c - im*d)*(c + im*d)^3*f*(c + d*tan(e + f*x))^(3/2)) + ((2*im)*c - 9*d)/(8*a^2*(c + im*d)^2*f*(1 + im*tan(e + f*x))*(c + d*tan(e + f*x))^(3/2)) - 1/(4*(im*c - d)*f*(a + im*a*tan(e + f*x))^2*(c + d*tan(e + f*x))^(3/2)) + (d*(2*c^3 + (9*im)*c^2*d + 88*c*d^2 - (45*im)*d^3))/(8*a^2*(c - im*d)^2*(c + im*d)^4*f*sqrt(c + d*tan(e + f*x)))]
    @test_int [1/((a + im*a*tan(e + f*x))^3*(c + d*tan(e + f*x))^(5/2)), x, 12, ((-im/8)*atanh(sqrt(c + d*tan(e + f*x))/sqrt(c - im*d)))/(a^3*(c - im*d)^(5/2)*f) + (((2*im)*c^3 - 16*c^2*d - (61*im)*c*d^2 + 152*d^3)*atanh(sqrt(c + d*tan(e + f*x))/sqrt(c + im*d)))/(16*a^3*(c + im*d)^(11/2)*f) + (d*(6*c^3 + (33*im)*c^2*d - 83*c*d^2 + (154*im)*d^3))/(48*a^3*(c - im*d)*(c + im*d)^4*f*(c + d*tan(e + f*x))^(3/2)) - 1/(6*(im*c - d)*f*(a + im*a*tan(e + f*x))^3*(c + d*tan(e + f*x))^(3/2)) + (im*c - 4*d)/(8*a*(c + im*d)^2*f*(a + im*a*tan(e + f*x))^2*(c + d*tan(e + f*x))^(3/2)) + (2*c^2 + (11*im)*c*d - 30*d^2)/(16*(im*c - d)^3*f*(a^3 + im*a^3*tan(e + f*x))*(c + d*tan(e + f*x))^(3/2)) + (d*(2*c^4 + (11*im)*c^3*d - 26*c^2*d^2 + (253*im)*c*d^3 + 150*d^4))/(16*a^3*(c - im*d)^2*(c + im*d)^5*f*sqrt(c + d*tan(e + f*x)))]


    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*(a+im*a*tan(e+f*x))^(m/2)*(c+d*tan(e+f*x))^(n/2)=#


    #= ::Subsubsection::Closed:: =#
    #=n > 0=#


    @test_int [(a + im*a*tan(e + f*x))^(5/2)*sqrt(c + d*tan(e + f*x)), x, 9, -((-1)^(1/4)*a^(5/2)*(c^2 + (10*im)*c*d + 23*d^2)*atanh(((-1)^(3/4)*sqrt(d)*sqrt(a + im*a*tan(e + f*x)))/(sqrt(a)*sqrt(c + d*tan(e + f*x)))))/(4*d^(3/2)*f) - ((4*im)*sqrt(2)*a^(5/2)*sqrt(c - im*d)*atanh((sqrt(2)*sqrt(a)*sqrt(c + d*tan(e + f*x)))/(sqrt(c - im*d)*sqrt(a + im*a*tan(e + f*x)))))/f + (a^2*(c + (9*im)*d)*sqrt(a + im*a*tan(e + f*x))*sqrt(c + d*tan(e + f*x)))/(4*d*f) - (a^2*sqrt(a + im*a*tan(e + f*x))*(c + d*tan(e + f*x))^(3/2))/(2*d*f)]
    @test_int [(a + im*a*tan(e + f*x))^(3/2)*sqrt(c + d*tan(e + f*x)), x, 9, -(((-1)^(1/4)*a^(3/2)*(im*c + 3*d)*atanh(((-1)^(3/4)*sqrt(d)*sqrt(a + im*a*tan(e + f*x)))/(sqrt(a)*sqrt(c + d*tan(e + f*x)))))/(sqrt(d)*f)) - ((2*im)*sqrt(2)*a^(3/2)*sqrt(c - im*d)*atanh((sqrt(2)*sqrt(a)*sqrt(c + d*tan(e + f*x)))/(sqrt(c - im*d)*sqrt(a + im*a*tan(e + f*x)))))/f + (a^2*(c + im*d)*sqrt(c + d*tan(e + f*x)))/(d*f*sqrt(a + im*a*tan(e + f*x))) - (a^2*(c + d*tan(e + f*x))^(3/2))/(d*f*sqrt(a + im*a*tan(e + f*x)))]
    @test_int [sqrt(a + im*a*tan(e + f*x))*sqrt(c + d*tan(e + f*x)), x, 7, (-2*(-1)^(1/4)*sqrt(a)*sqrt(d)*atanh(((-1)^(3/4)*sqrt(d)*sqrt(a + im*a*tan(e + f*x)))/(sqrt(a)*sqrt(c + d*tan(e + f*x)))))/f - (im*sqrt(2)*sqrt(a)*sqrt(c - im*d)*atanh((sqrt(2)*sqrt(a)*sqrt(c + d*tan(e + f*x)))/(sqrt(c - im*d)*sqrt(a + im*a*tan(e + f*x)))))/f]
    @test_int [sqrt(c + d*tan(e + f*x))/sqrt(a + im*a*tan(e + f*x)), x, 3, ((-im)*sqrt(c - im*d)*atanh((sqrt(2)*sqrt(a)*sqrt(c + d*tan(e + f*x)))/(sqrt(c - im*d)*sqrt(a + im*a*tan(e + f*x)))))/(sqrt(2)*sqrt(a)*f) + (im*sqrt(c + d*tan(e + f*x)))/(f*sqrt(a + im*a*tan(e + f*x)))]
    @test_int [sqrt(c + d*tan(e + f*x))/(a + im*a*tan(e + f*x))^(3/2), x, 4, ((-im/2)*sqrt(c - im*d)*atanh((sqrt(2)*sqrt(a)*sqrt(c + d*tan(e + f*x)))/(sqrt(c - im*d)*sqrt(a + im*a*tan(e + f*x)))))/(sqrt(2)*a^(3/2)*f) + ((im/2)*sqrt(c + d*tan(e + f*x)))/(a*f*sqrt(a + im*a*tan(e + f*x))) - (c + d*tan(e + f*x))^(3/2)/(3*(im*c - d)*f*(a + im*a*tan(e + f*x))^(3/2))]
    @test_int [sqrt(c + d*tan(e + f*x))/(a + im*a*tan(e + f*x))^(5/2), x, 6, ((-im/4)*sqrt(c - im*d)*atanh((sqrt(2)*sqrt(a)*sqrt(c + d*tan(e + f*x)))/(sqrt(c - im*d)*sqrt(a + im*a*tan(e + f*x)))))/(sqrt(2)*a^(5/2)*f) + ((im/5)*sqrt(c + d*tan(e + f*x)))/(f*(a + im*a*tan(e + f*x))^(5/2)) + (((5*im)*c - 3*d)*sqrt(c + d*tan(e + f*x)))/(30*a*(c + im*d)*f*(a + im*a*tan(e + f*x))^(3/2)) - ((20*c*d - im*(15*c^2 + 3*d^2))*sqrt(c + d*tan(e + f*x)))/(60*a^2*(c + im*d)^2*f*sqrt(a + im*a*tan(e + f*x)))]


    @test_int [(a + im*a*tan(e + f*x))^(5/2)*(c + d*tan(e + f*x))^(3/2), x, 10, -((-1)^(1/4)*a^(5/2)*(c - (3*im)*d)*(c^2 + (18*im)*c*d + 15*d^2)*atanh(((-1)^(3/4)*sqrt(d)*sqrt(a + im*a*tan(e + f*x)))/(sqrt(a)*sqrt(c + d*tan(e + f*x)))))/(8*d^(3/2)*f) - ((4*im)*sqrt(2)*a^(5/2)*(c - im*d)^(3/2)*atanh((sqrt(2)*sqrt(a)*sqrt(c + d*tan(e + f*x)))/(sqrt(c - im*d)*sqrt(a + im*a*tan(e + f*x)))))/f + (a^2*(c^2 + (14*im)*c*d + 19*d^2)*sqrt(a + im*a*tan(e + f*x))*sqrt(c + d*tan(e + f*x)))/(8*d*f) + (a^2*(c + (13*im)*d)*sqrt(a + im*a*tan(e + f*x))*(c + d*tan(e + f*x))^(3/2))/(12*d*f) - (a^2*sqrt(a + im*a*tan(e + f*x))*(c + d*tan(e + f*x))^(5/2))/(3*d*f)]
    @test_int [(a + im*a*tan(e + f*x))^(3/2)*(c + d*tan(e + f*x))^(3/2), x, 10, -((-1)^(1/4)*a^(3/2)*((3*im)*c^2 + 18*c*d - (11*im)*d^2)*atanh(((-1)^(3/4)*sqrt(d)*sqrt(a + im*a*tan(e + f*x)))/(sqrt(a)*sqrt(c + d*tan(e + f*x)))))/(4*sqrt(d)*f) - ((2*im)*sqrt(2)*a^(3/2)*(c - im*d)^(3/2)*atanh((sqrt(2)*sqrt(a)*sqrt(c + d*tan(e + f*x)))/(sqrt(c - im*d)*sqrt(a + im*a*tan(e + f*x)))))/f + (a*((3*im)*c + 5*d)*sqrt(a + im*a*tan(e + f*x))*sqrt(c + d*tan(e + f*x)))/(4*f) + (a^2*(c + im*d)*(c + d*tan(e + f*x))^(3/2))/(2*d*f*sqrt(a + im*a*tan(e + f*x))) - (a^2*(c + d*tan(e + f*x))^(5/2))/(2*d*f*sqrt(a + im*a*tan(e + f*x)))]
    @test_int [sqrt(a + im*a*tan(e + f*x))*(c + d*tan(e + f*x))^(3/2), x, 8, -(((-1)^(1/4)*sqrt(a)*(3*c - im*d)*sqrt(d)*atanh(((-1)^(3/4)*sqrt(d)*sqrt(a + im*a*tan(e + f*x)))/(sqrt(a)*sqrt(c + d*tan(e + f*x)))))/f) - (im*sqrt(2)*sqrt(a)*(c - im*d)^(3/2)*atanh((sqrt(2)*sqrt(a)*sqrt(c + d*tan(e + f*x)))/(sqrt(c - im*d)*sqrt(a + im*a*tan(e + f*x)))))/f + (d*sqrt(a + im*a*tan(e + f*x))*sqrt(c + d*tan(e + f*x)))/f]
    @test_int [(c + d*tan(e + f*x))^(3/2)/sqrt(a + im*a*tan(e + f*x)), x, 8, (2*(-1)^(3/4)*d^(3/2)*atanh(((-1)^(3/4)*sqrt(d)*sqrt(a + im*a*tan(e + f*x)))/(sqrt(a)*sqrt(c + d*tan(e + f*x)))))/(sqrt(a)*f) - (im*(c - im*d)^(3/2)*atanh((sqrt(2)*sqrt(a)*sqrt(c + d*tan(e + f*x)))/(sqrt(c - im*d)*sqrt(a + im*a*tan(e + f*x)))))/(sqrt(2)*sqrt(a)*f) + ((im*c - d)*sqrt(c + d*tan(e + f*x)))/(f*sqrt(a + im*a*tan(e + f*x)))]
    @test_int [(c + d*tan(e + f*x))^(3/2)/(a + im*a*tan(e + f*x))^(3/2), x, 4, ((-im/2)*(c - im*d)^(3/2)*atanh((sqrt(2)*sqrt(a)*sqrt(c + d*tan(e + f*x)))/(sqrt(c - im*d)*sqrt(a + im*a*tan(e + f*x)))))/(sqrt(2)*a^(3/2)*f) + ((im*c + d)*sqrt(c + d*tan(e + f*x)))/(2*a*f*sqrt(a + im*a*tan(e + f*x))) + ((im/3)*(c + d*tan(e + f*x))^(3/2))/(f*(a + im*a*tan(e + f*x))^(3/2))]
    @test_int [(c + d*tan(e + f*x))^(3/2)/(a + im*a*tan(e + f*x))^(5/2), x, 5, ((-im/4)*(c - im*d)^(3/2)*atanh((sqrt(2)*sqrt(a)*sqrt(c + d*tan(e + f*x)))/(sqrt(c - im*d)*sqrt(a + im*a*tan(e + f*x)))))/(sqrt(2)*a^(5/2)*f) + ((im*c + d)*sqrt(c + d*tan(e + f*x)))/(4*a^2*f*sqrt(a + im*a*tan(e + f*x))) + ((im/6)*(c + d*tan(e + f*x))^(3/2))/(a*f*(a + im*a*tan(e + f*x))^(3/2)) - (c + d*tan(e + f*x))^(5/2)/(5*(im*c - d)*f*(a + im*a*tan(e + f*x))^(5/2))]


    @test_int [(a + im*a*tan(e + f*x))^(5/2)*(c + d*tan(e + f*x))^(5/2), x, 11, -((-1)^(1/4)*a^(5/2)*(5*c^4 + (100*im)*c^3*d + 690*c^2*d^2 - (900*im)*c*d^3 - 363*d^4)*atanh(((-1)^(3/4)*sqrt(d)*sqrt(a + im*a*tan(e + f*x)))/(sqrt(a)*sqrt(c + d*tan(e + f*x)))))/(64*d^(3/2)*f) - ((4*im)*sqrt(2)*a^(5/2)*(c - im*d)^(5/2)*atanh((sqrt(2)*sqrt(a)*sqrt(c + d*tan(e + f*x)))/(sqrt(c - im*d)*sqrt(a + im*a*tan(e + f*x)))))/f + (a^2*(5*c^3 + (95*im)*c^2*d + 273*c*d^2 - (149*im)*d^3)*sqrt(a + im*a*tan(e + f*x))*sqrt(c + d*tan(e + f*x)))/(64*d*f) + (a^2*(5*c^2 + (90*im)*c*d + 107*d^2)*sqrt(a + im*a*tan(e + f*x))*(c + d*tan(e + f*x))^(3/2))/(96*d*f) + (a^2*(c + (17*im)*d)*sqrt(a + im*a*tan(e + f*x))*(c + d*tan(e + f*x))^(5/2))/(24*d*f) - (a^2*sqrt(a + im*a*tan(e + f*x))*(c + d*tan(e + f*x))^(7/2))/(4*d*f)]
    @test_int [(a + im*a*tan(e + f*x))^(3/2)*(c + d*tan(e + f*x))^(5/2), x, 11, -((-1)^(1/4)*a^(3/2)*((5*im)*c^3 + 45*c^2*d - (55*im)*c*d^2 - 23*d^3)*atanh(((-1)^(3/4)*sqrt(d)*sqrt(a + im*a*tan(e + f*x)))/(sqrt(a)*sqrt(c + d*tan(e + f*x)))))/(8*sqrt(d)*f) - ((2*im)*sqrt(2)*a^(3/2)*(c - im*d)^(5/2)*atanh((sqrt(2)*sqrt(a)*sqrt(c + d*tan(e + f*x)))/(sqrt(c - im*d)*sqrt(a + im*a*tan(e + f*x)))))/f + (a*(c - (3*im)*d)*((5*im)*c + 3*d)*sqrt(a + im*a*tan(e + f*x))*sqrt(c + d*tan(e + f*x)))/(8*f) + (a*((5*im)*c + 7*d)*sqrt(a + im*a*tan(e + f*x))*(c + d*tan(e + f*x))^(3/2))/(12*f) + (a^2*(c + im*d)*(c + d*tan(e + f*x))^(5/2))/(3*d*f*sqrt(a + im*a*tan(e + f*x))) - (a^2*(c + d*tan(e + f*x))^(7/2))/(3*d*f*sqrt(a + im*a*tan(e + f*x)))]
    @test_int [sqrt(a + im*a*tan(e + f*x))*(c + d*tan(e + f*x))^(5/2), x, 9, -((-1)^(1/4)*sqrt(a)*sqrt(d)*(15*c^2 - (10*im)*c*d - 7*d^2)*atanh(((-1)^(3/4)*sqrt(d)*sqrt(a + im*a*tan(e + f*x)))/(sqrt(a)*sqrt(c + d*tan(e + f*x)))))/(4*f) - (im*sqrt(2)*sqrt(a)*(c - im*d)^(5/2)*atanh((sqrt(2)*sqrt(a)*sqrt(c + d*tan(e + f*x)))/(sqrt(c - im*d)*sqrt(a + im*a*tan(e + f*x)))))/f + ((7*c - im*d)*d*sqrt(a + im*a*tan(e + f*x))*sqrt(c + d*tan(e + f*x)))/(4*f) + (d*sqrt(a + im*a*tan(e + f*x))*(c + d*tan(e + f*x))^(3/2))/(2*f)]
    @test_int [(c + d*tan(e + f*x))^(5/2)/sqrt(a + im*a*tan(e + f*x)), x, 9, ((-1)^(1/4)*(5*im*c - d)*d^(3/2)*atanh(((-1)^(3/4)*sqrt(d)*sqrt(a + im*a*tan(e + f*x)))/(sqrt(a)*sqrt(c + d*tan(e + f*x)))))/(sqrt(a)*f) - (im*(c - im*d)^(5/2)*atanh((sqrt(2)*sqrt(a)*sqrt(c + d*tan(e + f*x)))/(sqrt(c - im*d)*sqrt(a + im*a*tan(e + f*x)))))/(sqrt(2)*sqrt(a)*f) - ((c + 2*im*d)*d*sqrt(a + im*a*tan(e + f*x))*sqrt(c + d*tan(e + f*x)))/(a*f) + ((im*c - d)*(c + d*tan(e + f*x))^(3/2))/(f*sqrt(a + im*a*tan(e + f*x)))]
    @test_int [(c + d*tan(e + f*x))^(5/2)/(a + im*a*tan(e + f*x))^(3/2), x, 9, (2*(-1)^(1/4)*d^(5/2)*atanh(((-1)^(3/4)*sqrt(d)*sqrt(a + im*a*tan(e + f*x)))/(sqrt(a)*sqrt(c + d*tan(e + f*x)))))/(a^(3/2)*f) - ((im/2)*(c - im*d)^(5/2)*atanh((sqrt(2)*sqrt(a)*sqrt(c + d*tan(e + f*x)))/(sqrt(c - im*d)*sqrt(a + im*a*tan(e + f*x)))))/(sqrt(2)*a^(3/2)*f) + ((c + im*d)*(im*c + 3*d)*sqrt(c + d*tan(e + f*x)))/(2*a*f*sqrt(a + im*a*tan(e + f*x))) + ((im*c - d)*(c + d*tan(e + f*x))^(3/2))/(3*f*(a + im*a*tan(e + f*x))^(3/2))]
    @test_int [(c + d*tan(e + f*x))^(5/2)/(a + im*a*tan(e + f*x))^(5/2), x, 5, ((-im/4)*(c - im*d)^(5/2)*atanh((sqrt(2)*sqrt(a)*sqrt(c + d*tan(e + f*x)))/(sqrt(c - im*d)*sqrt(a + im*a*tan(e + f*x)))))/(sqrt(2)*a^(5/2)*f) + ((im/4)*(c - im*d)^2*sqrt(c + d*tan(e + f*x)))/(a^2*f*sqrt(a + im*a*tan(e + f*x))) + ((im*c + d)*(c + d*tan(e + f*x))^(3/2))/(6*a*f*(a + im*a*tan(e + f*x))^(3/2)) + ((im/5)*(c + d*tan(e + f*x))^(5/2))/(f*(a + im*a*tan(e + f*x))^(5/2))]


    #= ::Subsubsection::Closed:: =#
    #=n < 0=#


    @test_int [(a + im*a*tan(e + f*x))^(5/2)/sqrt(c + d*tan(e + f*x)), x, 8, -(((-1)^(1/4)*a^(5/2)*(c + (5*im)*d)*atanh(((-1)^(3/4)*sqrt(d)*sqrt(a + im*a*tan(e + f*x)))/(sqrt(a)*sqrt(c + d*tan(e + f*x)))))/(d^(3/2)*f)) - ((4*im)*sqrt(2)*a^(5/2)*atanh((sqrt(2)*sqrt(a)*sqrt(c + d*tan(e + f*x)))/(sqrt(c - im*d)*sqrt(a + im*a*tan(e + f*x)))))/(sqrt(c - im*d)*f) - (a^2*sqrt(a + im*a*tan(e + f*x))*sqrt(c + d*tan(e + f*x)))/(d*f)]
    @test_int [(a + im*a*tan(e + f*x))^(3/2)/sqrt(c + d*tan(e + f*x)), x, 7, (-2*(-1)^(3/4)*a^(3/2)*atanh(((-1)^(3/4)*sqrt(d)*sqrt(a + im*a*tan(e + f*x)))/(sqrt(a)*sqrt(c + d*tan(e + f*x)))))/(sqrt(d)*f) - ((2*im)*sqrt(2)*a^(3/2)*atanh((sqrt(2)*sqrt(a)*sqrt(c + d*tan(e + f*x)))/(sqrt(c - im*d)*sqrt(a + im*a*tan(e + f*x)))))/(sqrt(c - im*d)*f)]
    @test_int [sqrt(a + im*a*tan(e + f*x))/sqrt(c + d*tan(e + f*x)), x, 2, ((-im)*sqrt(2)*sqrt(a)*atanh((sqrt(2)*sqrt(a)*sqrt(c + d*tan(e + f*x)))/(sqrt(c - im*d)*sqrt(a + im*a*tan(e + f*x)))))/(sqrt(c - im*d)*f)]
    @test_int [1/(sqrt(a + im*a*tan(e + f*x))*sqrt(c + d*tan(e + f*x))), x, 4, ((-im)*atanh((sqrt(2)*sqrt(a)*sqrt(c + d*tan(e + f*x)))/(sqrt(c - im*d)*sqrt(a + im*a*tan(e + f*x)))))/(sqrt(2)*sqrt(a)*sqrt(c - im*d)*f) - sqrt(c + d*tan(e + f*x))/((im*c + d)*f*sqrt(a + im*a*tan(e + f*x))) + (2*d*sqrt(c + d*tan(e + f*x)))/((c^2 + d^2)*f*sqrt(a + im*a*tan(e + f*x)))]
    @test_int [1/((a + im*a*tan(e + f*x))^(3/2)*sqrt(c + d*tan(e + f*x))), x, 5, ((-im/2)*atanh((sqrt(2)*sqrt(a)*sqrt(c + d*tan(e + f*x)))/(sqrt(c - im*d)*sqrt(a + im*a*tan(e + f*x)))))/(sqrt(2)*a^(3/2)*sqrt(c - im*d)*f) - sqrt(c + d*tan(e + f*x))/(3*(im*c - d)*f*(a + im*a*tan(e + f*x))^(3/2)) + (((3*im)*c - 7*d)*sqrt(c + d*tan(e + f*x)))/(6*a*(c + im*d)^2*f*sqrt(a + im*a*tan(e + f*x)))]
    @test_int [1/((a + im*a*tan(e + f*x))^(5/2)*sqrt(c + d*tan(e + f*x))), x, 6, ((-im/4)*atanh((sqrt(2)*sqrt(a)*sqrt(c + d*tan(e + f*x)))/(sqrt(c - im*d)*sqrt(a + im*a*tan(e + f*x)))))/(sqrt(2)*a^(5/2)*sqrt(c - im*d)*f) - sqrt(c + d*tan(e + f*x))/(5*(im*c - d)*f*(a + im*a*tan(e + f*x))^(5/2)) + (((5*im)*c - 13*d)*sqrt(c + d*tan(e + f*x)))/(30*a*(c + im*d)^2*f*(a + im*a*tan(e + f*x))^(3/2)) + ((15*c^2 + (50*im)*c*d - 67*d^2)*sqrt(c + d*tan(e + f*x)))/(60*a^2*(im*c - d)^3*f*sqrt(a + im*a*tan(e + f*x)))]


    @test_int [(a + im*a*tan(e + f*x))^(5/2)/(c + d*tan(e + f*x))^(3/2), x, 8, (2*(-1)^(1/4)*a^(5/2)*atanh(((-1)^(3/4)*sqrt(d)*sqrt(a + im*a*tan(e + f*x)))/(sqrt(a)*sqrt(c + d*tan(e + f*x)))))/(d^(3/2)*f) - ((4*im)*sqrt(2)*a^(5/2)*atanh((sqrt(2)*sqrt(a)*sqrt(c + d*tan(e + f*x)))/(sqrt(c - im*d)*sqrt(a + im*a*tan(e + f*x)))))/((c - im*d)^(3/2)*f) + (2*a^2*(c + im*d)*sqrt(a + im*a*tan(e + f*x)))/((c - im*d)*d*f*sqrt(c + d*tan(e + f*x)))]
    @test_int [(a + im*a*tan(e + f*x))^(3/2)/(c + d*tan(e + f*x))^(3/2), x, 3, ((-2*im)*sqrt(2)*a^(3/2)*atanh((sqrt(2)*sqrt(a)*sqrt(c + d*tan(e + f*x)))/(sqrt(c - im*d)*sqrt(a + im*a*tan(e + f*x)))))/((c - im*d)^(3/2)*f) - (2*a*sqrt(a + im*a*tan(e + f*x)))/((im*c + d)*f*sqrt(c + d*tan(e + f*x)))]
    @test_int [sqrt(a + im*a*tan(e + f*x))/(c + d*tan(e + f*x))^(3/2), x, 3, ((-im)*sqrt(2)*sqrt(a)*atanh((sqrt(2)*sqrt(a)*sqrt(c + d*tan(e + f*x)))/(sqrt(c - im*d)*sqrt(a + im*a*tan(e + f*x)))))/((c - im*d)^(3/2)*f) - (2*d*sqrt(a + im*a*tan(e + f*x)))/((c^2 + d^2)*f*sqrt(c + d*tan(e + f*x)))]
    @test_int [1/(sqrt(a + im*a*tan(e + f*x))*(c + d*tan(e + f*x))^(3/2)), x, 5, -((im*atanh((sqrt(2)*sqrt(a)*sqrt(c + d*tan(e + f*x)))/(sqrt(c - im*d)*sqrt(a + im*a*tan(e + f*x)))))/(sqrt(2)*sqrt(a)*(c - im*d)^(3/2)*f)) - 1/((im*c - d)*f*sqrt(a + im*a*tan(e + f*x))*sqrt(c + d*tan(e + f*x))) + ((c - 3*im*d)*d*sqrt(a + im*a*tan(e + f*x)))/(a*(c - im*d)*(c + im*d)^2*f*sqrt(c + d*tan(e + f*x)))]
    @test_int [1/((a + im*a*tan(e + f*x))^(3/2)*(c + d*tan(e + f*x))^(3/2)), x, 6, ((-im/2)*atanh((sqrt(2)*sqrt(a)*sqrt(c + d*tan(e + f*x)))/(sqrt(c - im*d)*sqrt(a + im*a*tan(e + f*x)))))/(sqrt(2)*a^(3/2)*(c - im*d)^(3/2)*f) - 1/(3*(im*c - d)*f*(a + im*a*tan(e + f*x))^(3/2)*sqrt(c + d*tan(e + f*x))) + ((3*im)*c - 11*d)/(6*a*(c + im*d)^2*f*sqrt(a + im*a*tan(e + f*x))*sqrt(c + d*tan(e + f*x))) + ((3*c - (5*im)*d)*(c + (5*im)*d)*d*sqrt(a + im*a*tan(e + f*x)))/(6*a^2*(c - im*d)*(c + im*d)^3*f*sqrt(c + d*tan(e + f*x)))]
    @test_int [1/((a + im*a*tan(e + f*x))^(5/2)*(c + d*tan(e + f*x))^(3/2)), x, 7, ((-im/4)*atanh((sqrt(2)*sqrt(a)*sqrt(c + d*tan(e + f*x)))/(sqrt(c - im*d)*sqrt(a + im*a*tan(e + f*x)))))/(sqrt(2)*a^(5/2)*(c - im*d)^(3/2)*f) - 1/(5*(im*c - d)*f*(a + im*a*tan(e + f*x))^(5/2)*sqrt(c + d*tan(e + f*x))) + ((5*im)*c - 17*d)/(30*a*(c + im*d)^2*f*(a + im*a*tan(e + f*x))^(3/2)*sqrt(c + d*tan(e + f*x))) + (15*c^2 + (70*im)*c*d - 151*d^2)/(60*a^2*(im*c - d)^3*f*sqrt(a + im*a*tan(e + f*x))*sqrt(c + d*tan(e + f*x))) + (d*(15*c^3 + (65*im)*c^2*d - 117*c*d^2 + (317*im)*d^3)*sqrt(a + im*a*tan(e + f*x)))/(60*a^3*(c - im*d)*(c + im*d)^4*f*sqrt(c + d*tan(e + f*x)))]


    @test_int [(a + im*a*tan(e + f*x))^(5/2)/(c + d*tan(e + f*x))^(5/2), x, 4, ((-4*im)*sqrt(2)*a^(5/2)*atanh((sqrt(2)*sqrt(a)*sqrt(c + d*tan(e + f*x)))/(sqrt(c - im*d)*sqrt(a + im*a*tan(e + f*x)))))/((c - im*d)^(5/2)*f) - (2*a*(a + im*a*tan(e + f*x))^(3/2))/(3*(im*c + d)*f*(c + d*tan(e + f*x))^(3/2)) + ((4*im)*a^2*sqrt(a + im*a*tan(e + f*x)))/((c - im*d)^2*f*sqrt(c + d*tan(e + f*x)))]
    @test_int [(a + im*a*tan(e + f*x))^(3/2)/(c + d*tan(e + f*x))^(5/2), x, 4, ((-2*im)*sqrt(2)*a^(3/2)*atanh((sqrt(2)*sqrt(a)*sqrt(c + d*tan(e + f*x)))/(sqrt(c - im*d)*sqrt(a + im*a*tan(e + f*x)))))/((c - im*d)^(5/2)*f) - (2*d*(a + im*a*tan(e + f*x))^(3/2))/(3*(c^2 + d^2)*f*(c + d*tan(e + f*x))^(3/2)) + ((2*im)*a*sqrt(a + im*a*tan(e + f*x)))/((c - im*d)^2*f*sqrt(c + d*tan(e + f*x)))]
    @test_int [sqrt(a + im*a*tan(e + f*x))/(c + d*tan(e + f*x))^(5/2), x, 5, ((-im)*sqrt(2)*sqrt(a)*atanh((sqrt(2)*sqrt(a)*sqrt(c + d*tan(e + f*x)))/(sqrt(c - im*d)*sqrt(a + im*a*tan(e + f*x)))))/((c - im*d)^(5/2)*f) - (2*d*sqrt(a + im*a*tan(e + f*x)))/(3*(c^2 + d^2)*f*(c + d*tan(e + f*x))^(3/2)) - (2*(5*c + im*d)*d*sqrt(a + im*a*tan(e + f*x)))/(3*(c^2 + d^2)^2*f*sqrt(c + d*tan(e + f*x)))]
    @test_int [1/(sqrt(a + im*a*tan(e + f*x))*(c + d*tan(e + f*x))^(5/2)), x, 6, -((im*atanh((sqrt(2)*sqrt(a)*sqrt(c + d*tan(e + f*x)))/(sqrt(c - im*d)*sqrt(a + im*a*tan(e + f*x)))))/(sqrt(2)*sqrt(a)*(c - im*d)^(5/2)*f)) - 1/((im*c - d)*f*sqrt(a + im*a*tan(e + f*x))*(c + d*tan(e + f*x))^(3/2)) + (d*(3*im*c + 5*d)*sqrt(a + im*a*tan(e + f*x)))/(3*a*(im*c - d)*(c^2 + d^2)*f*(c + d*tan(e + f*x))^(3/2)) + ((3*c - im*d)*(c - 7*im*d)*d*sqrt(a + im*a*tan(e + f*x)))/(3*a*(c - im*d)^2*(c + im*d)^3*f*sqrt(c + d*tan(e + f*x)))]
    @test_int [1/((a + im*a*tan(e + f*x))^(3/2)*(c + d*tan(e + f*x))^(5/2)), x, 7, ((-im/2)*atanh((sqrt(2)*sqrt(a)*sqrt(c + d*tan(e + f*x)))/(sqrt(c - im*d)*sqrt(a + im*a*tan(e + f*x)))))/(sqrt(2)*a^(3/2)*(c - im*d)^(5/2)*f) - 1/(3*(im*c - d)*f*(a + im*a*tan(e + f*x))^(3/2)*(c + d*tan(e + f*x))^(3/2)) + (im*c - 5*d)/(2*a*(c + im*d)^2*f*sqrt(a + im*a*tan(e + f*x))*(c + d*tan(e + f*x))^(3/2)) + (d*(3*c^2 + (14*im)*c*d + 21*d^2)*sqrt(a + im*a*tan(e + f*x)))/(6*a^2*(c - im*d)*(c + im*d)^3*f*(c + d*tan(e + f*x))^(3/2)) + ((c - (3*im)*d)*d*(3*c^2 + (22*im)*c*d + 13*d^2)*sqrt(a + im*a*tan(e + f*x)))/(6*a^2*(c - im*d)^2*(c + im*d)^4*f*sqrt(c + d*tan(e + f*x)))]
    @test_int [1/((a + im*a*tan(e + f*x))^(5/2)*(c + d*tan(e + f*x))^(5/2)), x, 8, ((-im/4)*atanh((sqrt(2)*sqrt(a)*sqrt(c + d*tan(e + f*x)))/(sqrt(c - im*d)*sqrt(a + im*a*tan(e + f*x)))))/(sqrt(2)*a^(5/2)*(c - im*d)^(5/2)*f) - 1/(5*(im*c - d)*f*(a + im*a*tan(e + f*x))^(5/2)*(c + d*tan(e + f*x))^(3/2)) + ((5*im)*c - 21*d)/(30*a*(c + im*d)^2*f*(a + im*a*tan(e + f*x))^(3/2)*(c + d*tan(e + f*x))^(3/2)) + (5*c^2 + (30*im)*c*d - 89*d^2)/(20*a^2*(im*c - d)^3*f*sqrt(a + im*a*tan(e + f*x))*(c + d*tan(e + f*x))^(3/2)) + (d*(15*c^3 + (85*im)*c^2*d - 221*c*d^2 + (361*im)*d^3)*sqrt(a + im*a*tan(e + f*x)))/(60*a^3*(c - im*d)*(c + im*d)^4*f*(c + d*tan(e + f*x))^(3/2)) + (d*(15*c^4 + (80*im)*c^3*d - 182*c^2*d^2 + (1224*im)*c*d^3 + 707*d^4)*sqrt(a + im*a*tan(e + f*x)))/(60*a^3*(c - im*d)^2*(c + im*d)^5*f*sqrt(c + d*tan(e + f*x)))]


    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*(a+im*a*tan(e+f*x))^m*(c+d*tan(e+f*x))^n*with*n*symbolic=#


    @test_int [(a + im*a*tan(e + f*x))^m*(c + d*tan(e + f*x))^n, x, 3, -((1/(2*f*m))*((im*AppellF1(m, -n, 1, 1 + m, -((d*(1 + im*tan(e + f*x)))/(im*c - d)), (1/2)*(1 + im*tan(e + f*x)))*(a + im*a*tan(e + f*x))^m*(c + d*tan(e + f*x))^n)/((c + d*tan(e + f*x))/(c + im*d))^n))]


    @test_int [(a + im*a*tan(e + f*x))^3*(c + d*tan(e + f*x))^n, x, 4, (a^3*(im*c - d*(5 + 2*n))*(c + d*tan(e + f*x))^(1 + n))/(d^2*f*(1 + n)*(2 + n)) + (4*a^3*HypergeometricFunctions._₂F₁(1, 1 + n, 2 + n, (c + d*tan(e + f*x))/(c - im*d))*(c + d*tan(e + f*x))^(1 + n))/((im*c + d)*f*(1 + n)) - ((a^3 + im*a^3*tan(e + f*x))*(c + d*tan(e + f*x))^(1 + n))/(d*f*(2 + n))]
    @test_int [(a + im*a*tan(e + f*x))^2*(c + d*tan(e + f*x))^n, x, 3, -((a^2*(c + d*tan(e + f*x))^(1 + n))/(d*f*(1 + n))) + (2*a^2*HypergeometricFunctions._₂F₁(1, 1 + n, 2 + n, (c + d*tan(e + f*x))/(c - im*d))*(c + d*tan(e + f*x))^(1 + n))/((im*c + d)*f*(1 + n))]
    @test_int [(a + im*a*tan(e + f*x))^1*(c + d*tan(e + f*x))^n, x, 2, (a*HypergeometricFunctions._₂F₁(1, 1 + n, 2 + n, (c + d*tan(e + f*x))/(c - im*d))*(c + d*tan(e + f*x))^(1 + n))/((im*c + d)*f*(1 + n))]
    @test_int [(c + d*tan(e + f*x))^n/(a + im*a*tan(e + f*x))^1, x, 6, (HypergeometricFunctions._₂F₁(1, 1 + n, 2 + n, (c + d*tan(e + f*x))/(c - im*d))*(c + d*tan(e + f*x))^(1 + n))/(4*a*(im*c + d)*f*(1 + n)) + ((im*c - d + 2*d*n)*HypergeometricFunctions._₂F₁(1, 1 + n, 2 + n, (c + d*tan(e + f*x))/(c + im*d))*(c + d*tan(e + f*x))^(1 + n))/(4*a*(c + im*d)^2*f*(1 + n)) - (c + d*tan(e + f*x))^(1 + n)/(2*(im*c - d)*f*(a + im*a*tan(e + f*x)))]
    @test_int [(c + d*tan(e + f*x))^n/(a + im*a*tan(e + f*x))^2, x, 7, (HypergeometricFunctions._₂F₁(1, 1 + n, 2 + n, (c + d*tan(e + f*x))/(c - im*d))*(c + d*tan(e + f*x))^(1 + n))/(8*a^2*(im*c + d)*f*(1 + n)) + ((c^2 + 2*im*c*d*(1 - n) - d^2*(1 - 4*n + 2*n^2))*HypergeometricFunctions._₂F₁(1, 1 + n, 2 + n, (c + d*tan(e + f*x))/(c + im*d))*(c + d*tan(e + f*x))^(1 + n))/(8*a^2*(im*c - d)^3*f*(1 + n)) + ((im*c - d*(2 - n))*(c + d*tan(e + f*x))^(1 + n))/(4*a^2*(c + im*d)^2*f*(1 + im*tan(e + f*x))) - (c + d*tan(e + f*x))^(1 + n)/(4*(im*c - d)*f*(a + im*a*tan(e + f*x))^2)]
    @test_int [(c + d*tan(e + f*x))^n/(a + im*a*tan(e + f*x))^3, x, 8, (HypergeometricFunctions._₂F₁(1, 1 + n, 2 + n, (c + d*tan(e + f*x))/(c - im*d))*(c + d*tan(e + f*x))^(1 + n))/(16*a^3*(im*c + d)*f*(1 + n)) + ((3*im*c^3 - c^2*d*(9 - 6*n) - 3*im*c*d^2*(3 - 6*n + 2*n^2) + d^3*(3 - 20*n + 18*n^2 - 4*n^3))*HypergeometricFunctions._₂F₁(1, 1 + n, 2 + n, (c + d*tan(e + f*x))/(c + im*d))*(c + d*tan(e + f*x))^(1 + n))/(48*a^3*(c + im*d)^4*f*(1 + n)) - (c + d*tan(e + f*x))^(1 + n)/(6*(im*c - d)*f*(a + im*a*tan(e + f*x))^3) + ((3*im*c - d*(7 - 2*n))*(c + d*tan(e + f*x))^(1 + n))/(24*a*(c + im*d)^2*f*(a + im*a*tan(e + f*x))^2) + ((3*im*c^2 - 3*c*d*(3 - n) - im*d^2*(10 - 9*n + 2*n^2))*(c + d*tan(e + f*x))^(1 + n))/(24*(c + im*d)^3*f*(a^3 + im*a^3*tan(e + f*x)))]


    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*(a+im*a*tan(e+f*x))^m*(c+d*tan(e+f*x))^n*with*m*symbolic=#


    @test_int [(a + im*a*tan(e + f*x))^m*(c + d*tan(e + f*x))^3, x, 5, (-2*d*(d^2 + im*c*d*m - c^2*(3 + m))*(a + im*a*tan(e + f*x))^m)/(f*m*(2 + m)) + ((im*c + d)^3*HypergeometricFunctions._₂F₁(1, m, 1 + m, (1 + im*tan(e + f*x))/2)*(a + im*a*tan(e + f*x))^m)/(2*f*m) - (d^2*(d*m + im*c*(4 + m))*(a + im*a*tan(e + f*x))^(1 + m))/(a*f*(1 + m)*(2 + m)) + (d*(a + im*a*tan(e + f*x))^m*(c + d*tan(e + f*x))^2)/(f*(2 + m))]
    @test_int [(a + im*a*tan(e + f*x))^m*(c + d*tan(e + f*x))^2, x, 4, (2*c*d*(a + im*a*tan(e + f*x))^m)/(f*m) - ((im/2)*(c - im*d)^2*HypergeometricFunctions._₂F₁(1, m, 1 + m, (1 + im*tan(e + f*x))/2)*(a + im*a*tan(e + f*x))^m)/(f*m) - (im*d^2*(a + im*a*tan(e + f*x))^(1 + m))/(a*f*(1 + m))]
    @test_int [(a + im*a*tan(e + f*x))^m*(c + d*tan(e + f*x))^1, x, 3, (d*(a + im*a*tan(e + f*x))^m)/(f*m) - ((im*c + d)*HypergeometricFunctions._₂F₁(1, m, 1 + m, (1 + im*tan(e + f*x))/2)*(a + im*a*tan(e + f*x))^m)/(2*f*m)]
    @test_int [(a + im*a*tan(e + f*x))^m/(c + d*tan(e + f*x))^1, x, 5, (HypergeometricFunctions._₂F₁(1, m, 1 + m, (1/2)*(1 + im*tan(e + f*x)))*(a + im*a*tan(e + f*x))^m)/(2*(im*c + d)*f*m) - (d*HypergeometricFunctions._₂F₁(1, m, 1 + m, -((d*(1 + im*tan(e + f*x)))/(im*c - d)))*(a + im*a*tan(e + f*x))^m)/((c^2 + d^2)*f*m)]
    @test_int [(a + im*a*tan(e + f*x))^m/(c + d*tan(e + f*x))^2, x, 6, -((im*HypergeometricFunctions._₂F₁(1, m, 1 + m, (1/2)*(1 + im*tan(e + f*x)))*(a + im*a*tan(e + f*x))^m)/(2*(c - im*d)^2*f*m)) - (d*(c*(2 - m) + im*d*m)*HypergeometricFunctions._₂F₁(1, m, 1 + m, -((d*(1 + im*tan(e + f*x)))/(im*c - d)))*(a + im*a*tan(e + f*x))^m)/((c^2 + d^2)^2*f*m) - (d*(a + im*a*tan(e + f*x))^m)/((c^2 + d^2)*f*(c + d*tan(e + f*x)))]
    @test_int [(a + im*a*tan(e + f*x))^m/(c + d*tan(e + f*x))^3, x, 7, -((HypergeometricFunctions._₂F₁(1, m, 1 + m, (1/2)*(1 + im*tan(e + f*x)))*(a + im*a*tan(e + f*x))^m)/(2*(im*c + d)^3*f*m)) - (d*(2*im*c*d*(3 - m)*m + c^2*(6 - 5*m + m^2) - d^2*(2 - m + m^2))*HypergeometricFunctions._₂F₁(1, m, 1 + m, -((d*(1 + im*tan(e + f*x)))/(im*c - d)))*(a + im*a*tan(e + f*x))^m)/(2*(c^2 + d^2)^3*f*m) - (d*(a + im*a*tan(e + f*x))^m)/(2*(c^2 + d^2)*f*(c + d*tan(e + f*x))^2) - (d*(c*(4 - m) + im*d*m)*(a + im*a*tan(e + f*x))^m)/(2*(c^2 + d^2)^2*f*(c + d*tan(e + f*x)))]


    @test_int [(a + im*a*tan(e + f*x))^m*(c + d*tan(e + f*x))^(3/2), x, 3, -(((im*c - d)*AppellF1(m, -(3/2), 1, 1 + m, -((d*(1 + im*tan(e + f*x)))/(im*c - d)), (1/2)*(1 + im*tan(e + f*x)))*(a + im*a*tan(e + f*x))^m*sqrt(c + d*tan(e + f*x)))/(2*f*m*sqrt((c + d*tan(e + f*x))/(c + im*d))))]
    @test_int [(a + im*a*tan(e + f*x))^m*(c + d*tan(e + f*x))^(1/2), x, 3, -((im*AppellF1(m, -(1/2), 1, 1 + m, -((d*(1 + im*tan(e + f*x)))/(im*c - d)), (1/2)*(1 + im*tan(e + f*x)))*(a + im*a*tan(e + f*x))^m*sqrt(c + d*tan(e + f*x)))/(2*f*m*sqrt((c + d*tan(e + f*x))/(c + im*d))))]
    @test_int [(a + im*a*tan(e + f*x))^m/(c + d*tan(e + f*x))^(1/2), x, 3, -((im*AppellF1(m, 1/2, 1, 1 + m, -((d*(1 + im*tan(e + f*x)))/(im*c - d)), (1/2)*(1 + im*tan(e + f*x)))*(a + im*a*tan(e + f*x))^m*sqrt((c + d*tan(e + f*x))/(c + im*d)))/(2*f*m*sqrt(c + d*tan(e + f*x))))]
    @test_int [(a + im*a*tan(e + f*x))^m/(c + d*tan(e + f*x))^(3/2), x, 3, (AppellF1(m, 3/2, 1, 1 + m, -((d*(1 + im*tan(e + f*x)))/(im*c - d)), (1/2)*(1 + im*tan(e + f*x)))*(a + im*a*tan(e + f*x))^m*sqrt((c + d*tan(e + f*x))/(c + im*d)))/(2*(im*c - d)*f*m*sqrt(c + d*tan(e + f*x)))]
    @test_int [(a + im*a*tan(e + f*x))^m/(c + d*tan(e + f*x))^(5/2), x, 3, -((im*AppellF1(m, 5/2, 1, 1 + m, -((d*(1 + im*tan(e + f*x)))/(im*c - d)), (1/2)*(1 + im*tan(e + f*x)))*(a + im*a*tan(e + f*x))^m*sqrt((c + d*tan(e + f*x))/(c + im*d)))/(2*(c + im*d)^2*f*m*sqrt(c + d*tan(e + f*x))))]


    #= ::Section::Closed:: =#
    #=Integrands*of*the*form*(a+b*tan(e+f*x))^m*(c+d*tan(e+f*x))^n=#


    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*(a+b*tan(e+f*x))^m*(c+d*tan(e+f*x))^n=#


    #= ::Subsubsection::Closed:: =#
    #=n > 0=#


    @test_int [(a + b*tan(e + f*x))^3*(c + d*tan(e + f*x)), x, 4, (a^3*c - 3*a*b^2*c - 3*a^2*b*d + b^3*d)*x - ((3*a^2*b*c - b^3*c + a^3*d - 3*a*b^2*d)*log(cos(e + f*x)))/f + (b*(2*a*b*c + a^2*d - b^2*d)*tan(e + f*x))/f + ((b*c + a*d)*(a + b*tan(e + f*x))^2)/(2*f) + (d*(a + b*tan(e + f*x))^3)/(3*f)]
    @test_int [(a + b*tan(e + f*x))^2*(c + d*tan(e + f*x)), x, 3, (a^2*c - b^2*c - 2*a*b*d)*x - ((2*a*b*c + a^2*d - b^2*d)*log(cos(e + f*x)))/f + (b*(b*c + a*d)*tan(e + f*x))/f + (d*(a + b*tan(e + f*x))^2)/(2*f)]
    @test_int [(a + b*tan(e + f*x))*(c + d*tan(e + f*x)), x, 2, (a*c - b*d)*x - ((b*c + a*d)*log(cos(e + f*x)))/f + (b*d*tan(e + f*x))/f]
    @test_int [(c + d*tan(e + f*x))/(a + b*tan(e + f*x)), x, 2, ((a*c + b*d)*x)/(a^2 + b^2) + ((b*c - a*d)*log(a*cos(e + f*x) + b*sin(e + f*x)))/((a^2 + b^2)*f)]
    @test_int [(c + d*tan(e + f*x))/(a + b*tan(e + f*x))^2, x, 3, ((a^2*c - b^2*c + 2*a*b*d)*x)/(a^2 + b^2)^2 + ((2*a*b*c - a^2*d + b^2*d)*log(a*cos(e + f*x) + b*sin(e + f*x)))/((a^2 + b^2)^2*f) - (b*c - a*d)/((a^2 + b^2)*f*(a + b*tan(e + f*x)))]
    @test_int [(c + d*tan(e + f*x))/(a + b*tan(e + f*x))^3, x, 4, ((a^3*c - 3*a*b^2*c + 3*a^2*b*d - b^3*d)*x)/(a^2 + b^2)^3 + ((3*a^2*b*c - b^3*c - a^3*d + 3*a*b^2*d)*log(a*cos(e + f*x) + b*sin(e + f*x)))/((a^2 + b^2)^3*f) - (b*c - a*d)/(2*(a^2 + b^2)*f*(a + b*tan(e + f*x))^2) - (2*a*b*c - a^2*d + b^2*d)/((a^2 + b^2)^2*f*(a + b*tan(e + f*x)))]


    @test_int [(a + b*tan(e + f*x))^3*(c + d*tan(e + f*x))^2, x, 5, -((6*a^2*b*c*d - 2*b^3*c*d - a^3*(c^2 - d^2) + 3*a*b^2*(c^2 - d^2))*x) - ((2*a^3*c*d - 6*a*b^2*c*d + 3*a^2*b*(c^2 - d^2) - b^3*(c^2 - d^2))*log(cos(e + f*x)))/f + (2*b*(b*c + a*d)*(a*c - b*d)*tan(e + f*x))/f + ((2*a*c*d + b*(c^2 - d^2))*(a + b*tan(e + f*x))^2)/(2*f) + (2*c*d*(a + b*tan(e + f*x))^3)/(3*f) + (d^2*(a + b*tan(e + f*x))^4)/(4*b*f)]
    @test_int [(a + b*tan(e + f*x))^2*(c + d*tan(e + f*x))^2, x, 4, (a*c - b*c - a*d - b*d)*(a*c + b*c + a*d - b*d)*x - (2*(b*c + a*d)*(a*c - b*d)*log(cos(e + f*x)))/f + (b*(2*a*c*d + b*(c^2 - d^2))*tan(e + f*x))/f + (c*d*(a + b*tan(e + f*x))^2)/f + (d^2*(a + b*tan(e + f*x))^3)/(3*b*f)]
    @test_int [(a + b*tan(e + f*x))*(c + d*tan(e + f*x))^2, x, 3, -((2*b*c*d - a*(c^2 - d^2))*x) - ((2*a*c*d + b*(c^2 - d^2))*log(cos(e + f*x)))/f + (d*(b*c + a*d)*tan(e + f*x))/f + (b*(c + d*tan(e + f*x))^2)/(2*f)]
    @test_int [(c + d*tan(e + f*x))^2/(a + b*tan(e + f*x)), x, 4, (a*(b*c - a*d)^2*x)/(b^2*(a^2 + b^2)) + (d*(2*b*c - a*d)*x)/b^2 - (d^2*log(cos(e + f*x)))/(b*f) + ((b*c - a*d)^2*log(a*cos(e + f*x) + b*sin(e + f*x)))/(b*(a^2 + b^2)*f)]
    @test_int [(c + d*tan(e + f*x))^2/(a + b*tan(e + f*x))^2, x, 3, -(((b*(c - d) - a*(c + d))*(a*(c - d) + b*(c + d))*x)/(a^2 + b^2)^2) + (2*(b*c - a*d)*(a*c + b*d)*log(a*cos(e + f*x) + b*sin(e + f*x)))/((a^2 + b^2)^2*f) - (b*c - a*d)^2/(b*(a^2 + b^2)*f*(a + b*tan(e + f*x)))]
    @test_int [(c + d*tan(e + f*x))^2/(a + b*tan(e + f*x))^3, x, 4, ((6*a^2*b*c*d - 2*b^3*c*d + a^3*(c^2 - d^2) - 3*a*b^2*(c^2 - d^2))*x)/(a^2 + b^2)^3 - ((2*a^3*c*d - 6*a*b^2*c*d - 3*a^2*b*(c^2 - d^2) + b^3*(c^2 - d^2))*log(a*cos(e + f*x) + b*sin(e + f*x)))/((a^2 + b^2)^3*f) - (b*c - a*d)^2/(2*b*(a^2 + b^2)*f*(a + b*tan(e + f*x))^2) - (2*(b*c - a*d)*(a*c + b*d))/((a^2 + b^2)^2*f*(a + b*tan(e + f*x)))]


    @test_int [(a + b*tan(e + f*x))^3*(c + d*tan(e + f*x))^3, x, 6, -((a*c - b*d)*(8*a*b*c*d - a^2*(c^2 - 3*d^2) + b^2*(3*c^2 - d^2))*x) + ((b*c + a*d)*(8*a*b*c*d + b^2*(c^2 - 3*d^2) - a^2*(3*c^2 - d^2))*log(cos(e + f*x)))/f + (d*(2*a^3*c*d - 6*a*b^2*c*d + 3*a^2*b*(c^2 - d^2) - b^3*(c^2 - d^2))*tan(e + f*x))/f + ((3*a^2*b*c - b^3*c + a^3*d - 3*a*b^2*d)*(c + d*tan(e + f*x))^2)/(2*f) + (b*(3*a^2 - b^2)*(c + d*tan(e + f*x))^3)/(3*f) - (b^2*(b*c - 11*a*d)*(c + d*tan(e + f*x))^4)/(20*d^2*f) + (b^2*(a + b*tan(e + f*x))*(c + d*tan(e + f*x))^4)/(5*d*f)]
    @test_int [(a + b*tan(e + f*x))^2*(c + d*tan(e + f*x))^3, x, 5, -((b^2*c*(c^2 - 3*d^2) + 2*a*b*d*(3*c^2 - d^2) - a^2*(c^3 - 3*c*d^2))*x) - ((2*a*b*c*(c^2 - 3*d^2) - b^2*d*(3*c^2 - d^2) + a^2*(3*c^2*d - d^3))*log(cos(e + f*x)))/f + (2*d*(b*c + a*d)*(a*c - b*d)*tan(e + f*x))/f + ((2*a*b*c + a^2*d - b^2*d)*(c + d*tan(e + f*x))^2)/(2*f) + (2*a*b*(c + d*tan(e + f*x))^3)/(3*f) + (b^2*(c + d*tan(e + f*x))^4)/(4*d*f)]
    @test_int [(a + b*tan(e + f*x))*(c + d*tan(e + f*x))^3, x, 4, -((b*d*(3*c^2 - d^2) - a*(c^3 - 3*c*d^2))*x) - ((b*c^3 + 3*a*c^2*d - 3*b*c*d^2 - a*d^3)*log(cos(e + f*x)))/f + (d*(2*a*c*d + b*(c^2 - d^2))*tan(e + f*x))/f + ((b*c + a*d)*(c + d*tan(e + f*x))^2)/(2*f) + (b*(c + d*tan(e + f*x))^3)/(3*f)]
    @test_int [(c + d*tan(e + f*x))^3/(a + b*tan(e + f*x)), x, 5, ((a*c^3 + 3*b*c^2*d - 3*a*c*d^2 - b*d^3)*x)/(a^2 + b^2) + ((b*c^3 - 3*a*c^2*d - 3*b*c*d^2 + a*d^3)*log(cos(e + f*x)))/((a^2 + b^2)*f) + ((b*c - a*d)^3*log(a + b*tan(e + f*x)))/(b^2*(a^2 + b^2)*f) + (d^2*(c + d*tan(e + f*x)))/(b*f)]
    @test_int [(c + d*tan(e + f*x))^3/(a + b*tan(e + f*x))^2, x, 5, -(((b^2*c*(c^2 - 3*d^2) - 2*a*b*d*(3*c^2 - d^2) - a^2*(c^3 - 3*c*d^2))*x)/(a^2 + b^2)^2) + ((2*a*b*c*(c^2 - 3*d^2) + b^2*d*(3*c^2 - d^2) - a^2*(3*c^2*d - d^3))*log(cos(e + f*x)))/((a^2 + b^2)^2*f) + ((b*c - a*d)^2*(2*a*b*c + a^2*d + 3*b^2*d)*log(a + b*tan(e + f*x)))/(b^2*(a^2 + b^2)^2*f) - ((b*c - a*d)^2*(c + d*tan(e + f*x)))/(b*(a^2 + b^2)*f*(a + b*tan(e + f*x)))]
    @test_int [(c + d*tan(e + f*x))^3/(a + b*tan(e + f*x))^3, x, 4, ((a*c + b*d)*(8*a*b*c*d + a^2*(c^2 - 3*d^2) - b^2*(3*c^2 - d^2))*x)/(a^2 + b^2)^3 + ((b*c - a*d)*(8*a*b*c*d - b^2*(c^2 - 3*d^2) + a^2*(3*c^2 - d^2))*log(a*cos(e + f*x) + b*sin(e + f*x)))/((a^2 + b^2)^3*f) - ((b*c - a*d)^2*(4*a*b*c + a^2*d + 5*b^2*d))/(2*b^2*(a^2 + b^2)^2*f*(a + b*tan(e + f*x))) - ((b*c - a*d)^2*(c + d*tan(e + f*x)))/(2*b*(a^2 + b^2)*f*(a + b*tan(e + f*x))^2)]


    #= ::Subsubsection::Closed:: =#
    #=n < 0=#


    @test_int [(a + b*tan(e + f*x))^4/(c + d*tan(e + f*x)), x, 6, ((a^4*c - 6*a^2*b^2*c + b^4*c + 4*a^3*b*d - 4*a*b^3*d)*x)/(c^2 + d^2) - ((4*a^3*b*c - 4*a*b^3*c - a^4*d + 6*a^2*b^2*d - b^4*d)*log(cos(e + f*x)))/((c^2 + d^2)*f) + ((b*c - a*d)^4*log(c + d*tan(e + f*x)))/(d^3*(c^2 + d^2)*f) - (b^3*(b*c - 3*a*d)*tan(e + f*x))/(d^2*f) + (b^2*(a + b*tan(e + f*x))^2)/(2*d*f)]
    @test_int [(a + b*tan(e + f*x))^3/(c + d*tan(e + f*x)), x, 5, ((a^3*c - 3*a*b^2*c + 3*a^2*b*d - b^3*d)*x)/(c^2 + d^2) - ((3*a^2*b*c - b^3*c - a^3*d + 3*a*b^2*d)*log(cos(e + f*x)))/((c^2 + d^2)*f) - ((b*c - a*d)^3*log(c + d*tan(e + f*x)))/(d^2*(c^2 + d^2)*f) + (b^2*(a + b*tan(e + f*x)))/(d*f)]
    @test_int [(a + b*tan(e + f*x))^2/(c + d*tan(e + f*x)), x, 4, -((b*(b*c - 2*a*d)*x)/d^2) + (c*(b*c - a*d)^2*x)/(d^2*(c^2 + d^2)) - (b^2*log(cos(e + f*x)))/(d*f) + ((b*c - a*d)^2*log(c*cos(e + f*x) + d*sin(e + f*x)))/(d*(c^2 + d^2)*f)]
    @test_int [(a + b*tan(e + f*x))/(c + d*tan(e + f*x)), x, 2, ((a*c + b*d)*x)/(c^2 + d^2) - ((b*c - a*d)*log(c*cos(e + f*x) + d*sin(e + f*x)))/((c^2 + d^2)*f)]
    @test_int [1/((a + b*tan(e + f*x))*(c + d*tan(e + f*x))), x, 3, ((a*c - b*d)*x)/((a^2 + b^2)*(c^2 + d^2)) + (b^2*log(a*cos(e + f*x) + b*sin(e + f*x)))/((a^2 + b^2)*(b*c - a*d)*f) - (d^2*log(c*cos(e + f*x) + d*sin(e + f*x)))/((b*c - a*d)*(c^2 + d^2)*f)]
    @test_int [1/((a + b*tan(e + f*x))^2*(c + d*tan(e + f*x))), x, 4, ((a^2*c - b^2*c - 2*a*b*d)*x)/((a^2 + b^2)^2*(c^2 + d^2)) + (b^2*(2*a*b*c - 3*a^2*d - b^2*d)*log(a*cos(e + f*x) + b*sin(e + f*x)))/((a^2 + b^2)^2*(b*c - a*d)^2*f) + (d^3*log(c*cos(e + f*x) + d*sin(e + f*x)))/((b*c - a*d)^2*(c^2 + d^2)*f) - b^2/((a^2 + b^2)*(b*c - a*d)*f*(a + b*tan(e + f*x)))]
    @test_int [1/((a + b*tan(e + f*x))^3*(c + d*tan(e + f*x))), x, 5, ((a^3*c - 3*a*b^2*c - 3*a^2*b*d + b^3*d)*x)/((a^2 + b^2)^3*(c^2 + d^2)) - (b^2*(8*a^3*b*c*d - 6*a^4*d^2 + b^4*(c^2 - d^2) - 3*a^2*b^2*(c^2 + d^2))*log(a*cos(e + f*x) + b*sin(e + f*x)))/((a^2 + b^2)^3*(b*c - a*d)^3*f) - (d^4*log(c*cos(e + f*x) + d*sin(e + f*x)))/((b*c - a*d)^3*(c^2 + d^2)*f) - b^2/(2*(a^2 + b^2)*(b*c - a*d)*f*(a + b*tan(e + f*x))^2) - (b^2*(2*a*b*c - 3*a^2*d - b^2*d))/((a^2 + b^2)^2*(b*c - a*d)^2*f*(a + b*tan(e + f*x)))]


    @test_int [(a + b*tan(e + f*x))^4/(c + d*tan(e + f*x))^2, x, 6, ((8*a^3*b*c*d - 8*a*b^3*c*d + a^4*(c^2 - d^2) - 6*a^2*b^2*(c^2 - d^2) + b^4*(c^2 - d^2))*x)/(c^2 + d^2)^2 - (2*(a^2*c - b^2*c + 2*a*b*d)*(2*a*b*c - a^2*d + b^2*d)*log(cos(e + f*x)))/((c^2 + d^2)^2*f) - (2*(b*c - a*d)^3*(a*c*d + b*(c^2 + 2*d^2))*log(c + d*tan(e + f*x)))/(d^3*(c^2 + d^2)^2*f) - (b^2*(a*d*(2*b*c - a*d) - b^2*(2*c^2 + d^2))*tan(e + f*x))/(d^2*(c^2 + d^2)*f) - ((b*c - a*d)^2*(a + b*tan(e + f*x))^2)/(d*(c^2 + d^2)*f*(c + d*tan(e + f*x)))]
    @test_int [(a + b*tan(e + f*x))^3/(c + d*tan(e + f*x))^2, x, 5, ((6*a^2*b*c*d - 2*b^3*c*d + a^3*(c^2 - d^2) - 3*a*b^2*(c^2 - d^2))*x)/(c^2 + d^2)^2 + ((2*a^3*c*d - 6*a*b^2*c*d - 3*a^2*b*(c^2 - d^2) + b^3*(c^2 - d^2))*log(cos(e + f*x)))/((c^2 + d^2)^2*f) + ((b*c - a*d)^2*(2*a*c*d + b*(c^2 + 3*d^2))*log(c + d*tan(e + f*x)))/(d^2*(c^2 + d^2)^2*f) - ((b*c - a*d)^2*(a + b*tan(e + f*x)))/(d*(c^2 + d^2)*f*(c + d*tan(e + f*x)))]
    @test_int [(a + b*tan(e + f*x))^2/(c + d*tan(e + f*x))^2, x, 3, -(((b*(c - d) - a*(c + d))*(a*(c - d) + b*(c + d))*x)/(c^2 + d^2)^2) - (2*(b*c - a*d)*(a*c + b*d)*log(c*cos(e + f*x) + d*sin(e + f*x)))/((c^2 + d^2)^2*f) - (b*c - a*d)^2/(d*(c^2 + d^2)*f*(c + d*tan(e + f*x)))]
    @test_int [(a + b*tan(e + f*x))/(c + d*tan(e + f*x))^2, x, 3, ((2*b*c*d + a*(c^2 - d^2))*x)/(c^2 + d^2)^2 + ((2*a*c*d - b*(c^2 - d^2))*log(c*cos(e + f*x) + d*sin(e + f*x)))/((c^2 + d^2)^2*f) + (b*c - a*d)/((c^2 + d^2)*f*(c + d*tan(e + f*x)))]
    @test_int [1/((a + b*tan(e + f*x))*(c + d*tan(e + f*x))^2), x, 4, -(((2*b*c*d - a*(c^2 - d^2))*x)/((a^2 + b^2)*(c^2 + d^2)^2)) + (b^3*log(a*cos(e + f*x) + b*sin(e + f*x)))/((a^2 + b^2)*(b*c - a*d)^2*f) + (d^2*(2*a*c*d - b*(3*c^2 + d^2))*log(c*cos(e + f*x) + d*sin(e + f*x)))/((b*c - a*d)^2*(c^2 + d^2)^2*f) + d^2/((b*c - a*d)*(c^2 + d^2)*f*(c + d*tan(e + f*x)))]
    @test_int [1/((a + b*tan(e + f*x))^2*(c + d*tan(e + f*x))^2), x, 5, ((b*(c - d) + a*(c + d))*(a*(c - d) - b*(c + d))*x)/((a^2 + b^2)^2*(c^2 + d^2)^2) + (2*b^3*(a*b*c - 2*a^2*d - b^2*d)*log(a*cos(e + f*x) + b*sin(e + f*x)))/((a^2 + b^2)^2*(b*c - a*d)^3*f) - (2*d^3*(a*c*d - b*(2*c^2 + d^2))*log(c*cos(e + f*x) + d*sin(e + f*x)))/((b*c - a*d)^3*(c^2 + d^2)^2*f) - (d*(a^2*d^2 + b^2*(c^2 + 2*d^2)))/((a^2 + b^2)*(b*c - a*d)^2*(c^2 + d^2)*f*(c + d*tan(e + f*x))) - b^2/((a^2 + b^2)*(b*c - a*d)*f*(a + b*tan(e + f*x))*(c + d*tan(e + f*x)))]
    @test_int [1/((a + b*tan(e + f*x))^3*(c + d*tan(e + f*x))^2), x, 6, -(((6*a^2*b*c*d - 2*b^3*c*d - a^3*(c^2 - d^2) + 3*a*b^2*(c^2 - d^2))*x)/((a^2 + b^2)^3*(c^2 + d^2)^2)) - (b^3*(10*a^3*b*c*d + 2*a*b^3*c*d - 10*a^4*d^2 + b^4*(c^2 - 3*d^2) - 3*a^2*b^2*(c^2 + 3*d^2))*log(a*cos(e + f*x) + b*sin(e + f*x)))/((a^2 + b^2)^3*(b*c - a*d)^4*f) - (d^4*(5*b*c^2 - 2*a*c*d + 3*b*d^2)*log(c*cos(e + f*x) + d*sin(e + f*x)))/((b*c - a*d)^4*(c^2 + d^2)^2*f) + (d*(a^4*d^3 - 2*a*b^3*c*(c^2 + d^2) + 2*a^2*b^2*d*(2*c^2 + 3*d^2) + b^4*d*(2*c^2 + 3*d^2)))/((a^2 + b^2)^2*(b*c - a*d)^3*(c^2 + d^2)*f*(c + d*tan(e + f*x))) - b^2/(2*(a^2 + b^2)*(b*c - a*d)*f*(a + b*tan(e + f*x))^2*(c + d*tan(e + f*x))) - (b^2*(4*a*b*c - 7*a^2*d - 3*b^2*d))/(2*(a^2 + b^2)^2*(b*c - a*d)^2*f*(a + b*tan(e + f*x))*(c + d*tan(e + f*x)))]


    @test_int [(a + b*tan(e + f*x))^4/(c + d*tan(e + f*x))^3, x, 6, -(((6*a^2*b^2*c*(c^2 - 3*d^2) - b^4*c*(c^2 - 3*d^2) - 4*a^3*b*d*(3*c^2 - d^2) + 4*a*b^3*d*(3*c^2 - d^2) - a^4*(c^3 - 3*c*d^2))*x)/(c^2 + d^2)^3) - ((4*a^3*b*c*(c^2 - 3*d^2) - 4*a*b^3*c*(c^2 - 3*d^2) + 6*a^2*b^2*d*(3*c^2 - d^2) - b^4*d*(3*c^2 - d^2) - a^4*(3*c^2*d - d^3))*log(cos(e + f*x)))/((c^2 + d^2)^3*f) + ((b*c - a*d)^2*(a^2*d^2*(3*c^2 - d^2) + 2*a*b*c*d*(c^2 + 5*d^2) + b^2*(c^4 + 3*c^2*d^2 + 6*d^4))*log(c + d*tan(e + f*x)))/(d^3*(c^2 + d^2)^3*f) - ((b*c - a*d)^2*(a + b*tan(e + f*x))^2)/(2*d*(c^2 + d^2)*f*(c + d*tan(e + f*x))^2) + ((b*c - a*d)^3*(2*a*c*d + b*(c^2 + 3*d^2)))/(d^3*(c^2 + d^2)^2*f*(c + d*tan(e + f*x)))]
    @test_int [(a + b*tan(e + f*x))^3/(c + d*tan(e + f*x))^3, x, 4, ((a*c + b*d)*(8*a*b*c*d + a^2*(c^2 - 3*d^2) - b^2*(3*c^2 - d^2))*x)/(c^2 + d^2)^3 - ((b*c - a*d)*(8*a*b*c*d - b^2*(c^2 - 3*d^2) + a^2*(3*c^2 - d^2))*log(c*cos(e + f*x) + d*sin(e + f*x)))/((c^2 + d^2)^3*f) - ((b*c - a*d)^2*(a + b*tan(e + f*x)))/(2*d*(c^2 + d^2)*f*(c + d*tan(e + f*x))^2) - ((b*c - a*d)^2*(4*a*c*d + b*(c^2 + 5*d^2)))/(2*d^2*(c^2 + d^2)^2*f*(c + d*tan(e + f*x)))]
    @test_int [(a + b*tan(e + f*x))^2/(c + d*tan(e + f*x))^3, x, 4, -(((b^2*c*(c^2 - 3*d^2) - 2*a*b*d*(3*c^2 - d^2) - a^2*(c^3 - 3*c*d^2))*x)/(c^2 + d^2)^3) - ((2*a*b*c*(c^2 - 3*d^2) + b^2*d*(3*c^2 - d^2) - a^2*(3*c^2*d - d^3))*log(c*cos(e + f*x) + d*sin(e + f*x)))/((c^2 + d^2)^3*f) - (b*c - a*d)^2/(2*d*(c^2 + d^2)*f*(c + d*tan(e + f*x))^2) + (2*(b*c - a*d)*(a*c + b*d))/((c^2 + d^2)^2*f*(c + d*tan(e + f*x)))]
    @test_int [(a + b*tan(e + f*x))/(c + d*tan(e + f*x))^3, x, 4, ((a*c^3 + 3*b*c^2*d - 3*a*c*d^2 - b*d^3)*x)/(c^2 + d^2)^3 + ((a*d*(3*c^2 - d^2) - b*(c^3 - 3*c*d^2))*log(c*cos(e + f*x) + d*sin(e + f*x)))/((c^2 + d^2)^3*f) + (b*c - a*d)/(2*(c^2 + d^2)*f*(c + d*tan(e + f*x))^2) - (2*a*c*d - b*(c^2 - d^2))/((c^2 + d^2)^2*f*(c + d*tan(e + f*x)))]
    @test_int [1/((a + b*tan(e + f*x))*(c + d*tan(e + f*x))^3), x, 5, -(((b*d*(3*c^2 - d^2) - a*(c^3 - 3*c*d^2))*x)/((a^2 + b^2)*(c^2 + d^2)^3)) + (b^4*log(a*cos(e + f*x) + b*sin(e + f*x)))/((a^2 + b^2)*(b*c - a*d)^3*f) + (d^2*(8*a*b*c^3*d - a^2*d^2*(3*c^2 - d^2) - b^2*(6*c^4 + 3*c^2*d^2 + d^4))*log(c*cos(e + f*x) + d*sin(e + f*x)))/((b*c - a*d)^3*(c^2 + d^2)^3*f) + d^2/(2*(b*c - a*d)*(c^2 + d^2)*f*(c + d*tan(e + f*x))^2) - (d^2*(2*a*c*d - b*(3*c^2 + d^2)))/((b*c - a*d)^2*(c^2 + d^2)^2*f*(c + d*tan(e + f*x)))]
    @test_int [1/((a + b*tan(e + f*x))^2*(c + d*tan(e + f*x))^3), x, 6, -(((b^2*c*(c^2 - 3*d^2) - a^2*(c^3 - 3*c*d^2) + a*b*(6*c^2*d - 2*d^3))*x)/((a^2 + b^2)^2*(c^2 + d^2)^3)) + (b^4*(2*a*b*c - 5*a^2*d - 3*b^2*d)*log(a*cos(e + f*x) + b*sin(e + f*x)))/((a^2 + b^2)^2*(b*c - a*d)^4*f) + (d^3*(a^2*d^2*(3*c^2 - d^2) - 2*a*b*c*d*(5*c^2 + d^2) + b^2*(10*c^4 + 9*c^2*d^2 + 3*d^4))*log(c*cos(e + f*x) + d*sin(e + f*x)))/((b*c - a*d)^4*(c^2 + d^2)^3*f) - (d*(a^2*d^2 + b^2*(2*c^2 + 3*d^2)))/(2*(a^2 + b^2)*(b*c - a*d)^2*(c^2 + d^2)*f*(c + d*tan(e + f*x))^2) - b^2/((a^2 + b^2)*(b*c - a*d)*f*(a + b*tan(e + f*x))*(c + d*tan(e + f*x))^2) + (d*(2*a^3*c*d^3 + 2*a*b^2*c*d^3 - 2*a^2*b*d^2*(2*c^2 + d^2) - b^3*(c^4 + 6*c^2*d^2 + 3*d^4)))/((a^2 + b^2)*(b*c - a*d)^3*(c^2 + d^2)^2*f*(c + d*tan(e + f*x)))]


    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*(a+b*tan(e+f*x))^m*(c+d*tan(e+f*x))^(n/2)=#


    #= ::Subsubsection::Closed:: =#
    #=n > 0=#


    @test_int [(a + b*tan(e + f*x))^3*sqrt(c + d*tan(e + f*x)), x, 10, ((im*a + b)^3*sqrt(c - im*d)*atanh(sqrt(c + d*tan(e + f*x))/sqrt(c - im*d)))/f - ((im*a - b)^3*sqrt(c + im*d)*atanh(sqrt(c + d*tan(e + f*x))/sqrt(c + im*d)))/f + (2*b*(3*a^2 - b^2)*sqrt(c + d*tan(e + f*x)))/f - (4*b^2*(b*c - 6*a*d)*(c + d*tan(e + f*x))^(3/2))/(15*d^2*f) + (2*b^2*(a + b*tan(e + f*x))*(c + d*tan(e + f*x))^(3/2))/(5*d*f)]
    @test_int [(a + b*tan(e + f*x))^2*sqrt(c + d*tan(e + f*x)), x, 9, ((-im)*(a - im*b)^2*sqrt(c - im*d)*atanh(sqrt(c + d*tan(e + f*x))/sqrt(c - im*d)))/f + (im*(a + im*b)^2*sqrt(c + im*d)*atanh(sqrt(c + d*tan(e + f*x))/sqrt(c + im*d)))/f + (4*a*b*sqrt(c + d*tan(e + f*x)))/f + (2*b^2*(c + d*tan(e + f*x))^(3/2))/(3*d*f)]
    @test_int [(a + b*tan(e + f*x))*sqrt(c + d*tan(e + f*x)), x, 8, -(((im*a + b)*sqrt(c - im*d)*atanh(sqrt(c + d*tan(e + f*x))/sqrt(c - im*d)))/f) + ((im*a - b)*sqrt(c + im*d)*atanh(sqrt(c + d*tan(e + f*x))/sqrt(c + im*d)))/f + (2*b*sqrt(c + d*tan(e + f*x)))/f]
    @test_int [sqrt(c + d*tan(e + f*x))/(a + b*tan(e + f*x)), x, 11, (sqrt(c - im*d)*atanh(sqrt(c + d*tan(e + f*x))/sqrt(c - im*d)))/((im*a + b)*f) - (sqrt(c + im*d)*atanh(sqrt(c + d*tan(e + f*x))/sqrt(c + im*d)))/((im*a - b)*f) - (2*sqrt(b)*sqrt(b*c - a*d)*atanh((sqrt(b)*sqrt(c + d*tan(e + f*x)))/sqrt(b*c - a*d)))/((a^2 + b^2)*f)]
    @test_int [sqrt(c + d*tan(e + f*x))/(a + b*tan(e + f*x))^2, x, 12, ((-im)*sqrt(c - im*d)*atanh(sqrt(c + d*tan(e + f*x))/sqrt(c - im*d)))/((a - im*b)^2*f) + (im*sqrt(c + im*d)*atanh(sqrt(c + d*tan(e + f*x))/sqrt(c + im*d)))/((a + im*b)^2*f) - (sqrt(b)*(4*a*b*c - 3*a^2*d + b^2*d)*atanh((sqrt(b)*sqrt(c + d*tan(e + f*x)))/sqrt(b*c - a*d)))/((a^2 + b^2)^2*sqrt(b*c - a*d)*f) - (b*sqrt(c + d*tan(e + f*x)))/((a^2 + b^2)*f*(a + b*tan(e + f*x)))]
    @test_int [sqrt(c + d*tan(e + f*x))/(a + b*tan(e + f*x))^3, x, 13, -((sqrt(c - im*d)*atanh(sqrt(c + d*tan(e + f*x))/sqrt(c - im*d)))/((im*a + b)^3*f)) + (sqrt(c + im*d)*atanh(sqrt(c + d*tan(e + f*x))/sqrt(c + im*d)))/((im*a - b)^3*f) + (sqrt(b)*(40*a^3*b*c*d - 24*a*b^3*c*d - 15*a^4*d^2 - 6*a^2*b^2*(4*c^2 - 3*d^2) + b^4*(8*c^2 + d^2))*atanh((sqrt(b)*sqrt(c + d*tan(e + f*x)))/sqrt(b*c - a*d)))/(4*(a^2 + b^2)^3*(b*c - a*d)^(3/2)*f) - (b*sqrt(c + d*tan(e + f*x)))/(2*(a^2 + b^2)*f*(a + b*tan(e + f*x))^2) - (b*(8*a*b*c - 7*a^2*d + b^2*d)*sqrt(c + d*tan(e + f*x)))/(4*(a^2 + b^2)^2*(b*c - a*d)*f*(a + b*tan(e + f*x)))]


    @test_int [(a + b*tan(e + f*x))^3*(c + d*tan(e + f*x))^(3/2), x, 11, ((im*a + b)^3*(c - im*d)^(3/2)*atanh(sqrt(c + d*tan(e + f*x))/sqrt(c - im*d)))/f - ((im*a - b)^3*(c + im*d)^(3/2)*atanh(sqrt(c + d*tan(e + f*x))/sqrt(c + im*d)))/f + (2*(3*a^2*b*c - b^3*c + a^3*d - 3*a*b^2*d)*sqrt(c + d*tan(e + f*x)))/f + (2*b*(3*a^2 - b^2)*(c + d*tan(e + f*x))^(3/2))/(3*f) - (4*b^2*(b*c - 8*a*d)*(c + d*tan(e + f*x))^(5/2))/(35*d^2*f) + (2*b^2*(a + b*tan(e + f*x))*(c + d*tan(e + f*x))^(5/2))/(7*d*f)]
    @test_int [(a + b*tan(e + f*x))^2*(c + d*tan(e + f*x))^(3/2), x, 10, ((-im)*(a - im*b)^2*(c - im*d)^(3/2)*atanh(sqrt(c + d*tan(e + f*x))/sqrt(c - im*d)))/f + (im*(a + im*b)^2*(c + im*d)^(3/2)*atanh(sqrt(c + d*tan(e + f*x))/sqrt(c + im*d)))/f + (2*(2*a*b*c + a^2*d - b^2*d)*sqrt(c + d*tan(e + f*x)))/f + (4*a*b*(c + d*tan(e + f*x))^(3/2))/(3*f) + (2*b^2*(c + d*tan(e + f*x))^(5/2))/(5*d*f)]
    @test_int [(a + b*tan(e + f*x))*(c + d*tan(e + f*x))^(3/2), x, 9, -(((im*a + b)*(c - im*d)^(3/2)*atanh(sqrt(c + d*tan(e + f*x))/sqrt(c - im*d)))/f) + ((im*a - b)*(c + im*d)^(3/2)*atanh(sqrt(c + d*tan(e + f*x))/sqrt(c + im*d)))/f + (2*(b*c + a*d)*sqrt(c + d*tan(e + f*x)))/f + (2*b*(c + d*tan(e + f*x))^(3/2))/(3*f)]
    @test_int [(c + d*tan(e + f*x))^(3/2)/(a + b*tan(e + f*x)), x, 11, ((c - im*d)^(3/2)*atanh(sqrt(c + d*tan(e + f*x))/sqrt(c - im*d)))/((im*a + b)*f) - ((c + im*d)^(3/2)*atanh(sqrt(c + d*tan(e + f*x))/sqrt(c + im*d)))/((im*a - b)*f) - (2*(b*c - a*d)^(3/2)*atanh((sqrt(b)*sqrt(c + d*tan(e + f*x)))/sqrt(b*c - a*d)))/(sqrt(b)*(a^2 + b^2)*f)]
    @test_int [(c + d*tan(e + f*x))^(3/2)/(a + b*tan(e + f*x))^2, x, 12, ((-im)*(c - im*d)^(3/2)*atanh(sqrt(c + d*tan(e + f*x))/sqrt(c - im*d)))/((a - im*b)^2*f) + (im*(c + im*d)^(3/2)*atanh(sqrt(c + d*tan(e + f*x))/sqrt(c + im*d)))/((a + im*b)^2*f) - (sqrt(b*c - a*d)*(4*a*b*c - a^2*d + 3*b^2*d)*atanh((sqrt(b)*sqrt(c + d*tan(e + f*x)))/sqrt(b*c - a*d)))/(sqrt(b)*(a^2 + b^2)^2*f) - ((b*c - a*d)*sqrt(c + d*tan(e + f*x)))/((a^2 + b^2)*f*(a + b*tan(e + f*x)))]
    @test_int [(c + d*tan(e + f*x))^(3/2)/(a + b*tan(e + f*x))^3, x, 13, -(((c - im*d)^(3/2)*atanh(sqrt(c + d*tan(e + f*x))/sqrt(c - im*d)))/((im*a + b)^3*f)) + ((c + im*d)^(3/2)*atanh(sqrt(c + d*tan(e + f*x))/sqrt(c + im*d)))/((im*a - b)^3*f) + ((24*a^3*b*c*d - 40*a*b^3*c*d - 3*a^4*d^2 - 2*a^2*b^2*(12*c^2 - 13*d^2) + b^4*(8*c^2 - 3*d^2))*atanh((sqrt(b)*sqrt(c + d*tan(e + f*x)))/sqrt(b*c - a*d)))/(4*sqrt(b)*(a^2 + b^2)^3*sqrt(b*c - a*d)*f) - ((b*c - a*d)*sqrt(c + d*tan(e + f*x)))/(2*(a^2 + b^2)*f*(a + b*tan(e + f*x))^2) - ((8*a*b*c - 3*a^2*d + 5*b^2*d)*sqrt(c + d*tan(e + f*x)))/(4*(a^2 + b^2)^2*f*(a + b*tan(e + f*x)))]


    @test_int [(a + b*tan(e + f*x))^3*(c + d*tan(e + f*x))^(5/2), x, 12, ((im*a + b)^3*(c - im*d)^(5/2)*atanh(sqrt(c + d*tan(e + f*x))/sqrt(c - im*d)))/f - ((im*a - b)^3*(c + im*d)^(5/2)*atanh(sqrt(c + d*tan(e + f*x))/sqrt(c + im*d)))/f + (2*(2*a^3*c*d - 6*a*b^2*c*d + 3*a^2*b*(c^2 - d^2) - b^3*(c^2 - d^2))*sqrt(c + d*tan(e + f*x)))/f + (2*(3*a^2*b*c - b^3*c + a^3*d - 3*a*b^2*d)*(c + d*tan(e + f*x))^(3/2))/(3*f) + (2*b*(3*a^2 - b^2)*(c + d*tan(e + f*x))^(5/2))/(5*f) - (4*b^2*(b*c - 10*a*d)*(c + d*tan(e + f*x))^(7/2))/(63*d^2*f) + (2*b^2*(a + b*tan(e + f*x))*(c + d*tan(e + f*x))^(7/2))/(9*d*f)]
    @test_int [(a + b*tan(e + f*x))^2*(c + d*tan(e + f*x))^(5/2), x, 11, ((-im)*(a - im*b)^2*(c - im*d)^(5/2)*atanh(sqrt(c + d*tan(e + f*x))/sqrt(c - im*d)))/f + (im*(a + im*b)^2*(c + im*d)^(5/2)*atanh(sqrt(c + d*tan(e + f*x))/sqrt(c + im*d)))/f + (4*(b*c + a*d)*(a*c - b*d)*sqrt(c + d*tan(e + f*x)))/f + (2*(2*a*b*c + a^2*d - b^2*d)*(c + d*tan(e + f*x))^(3/2))/(3*f) + (4*a*b*(c + d*tan(e + f*x))^(5/2))/(5*f) + (2*b^2*(c + d*tan(e + f*x))^(7/2))/(7*d*f)]
    @test_int [(a + b*tan(e + f*x))*(c + d*tan(e + f*x))^(5/2), x, 10, -(((im*a + b)*(c - im*d)^(5/2)*atanh(sqrt(c + d*tan(e + f*x))/sqrt(c - im*d)))/f) + ((im*a - b)*(c + im*d)^(5/2)*atanh(sqrt(c + d*tan(e + f*x))/sqrt(c + im*d)))/f + (2*(2*a*c*d + b*(c^2 - d^2))*sqrt(c + d*tan(e + f*x)))/f + (2*(b*c + a*d)*(c + d*tan(e + f*x))^(3/2))/(3*f) + (2*b*(c + d*tan(e + f*x))^(5/2))/(5*f)]
    @test_int [(c + d*tan(e + f*x))^(5/2)/(a + b*tan(e + f*x)), x, 12, ((c - im*d)^(5/2)*atanh(sqrt(c + d*tan(e + f*x))/sqrt(c - im*d)))/((im*a + b)*f) - ((c + im*d)^(5/2)*atanh(sqrt(c + d*tan(e + f*x))/sqrt(c + im*d)))/((im*a - b)*f) - (2*(b*c - a*d)^(5/2)*atanh((sqrt(b)*sqrt(c + d*tan(e + f*x)))/sqrt(b*c - a*d)))/(b^(3/2)*(a^2 + b^2)*f) + (2*d^2*sqrt(c + d*tan(e + f*x)))/(b*f)]
    @test_int [(c + d*tan(e + f*x))^(5/2)/(a + b*tan(e + f*x))^2, x, 12, ((-im)*(c - im*d)^(5/2)*atanh(sqrt(c + d*tan(e + f*x))/sqrt(c - im*d)))/((a - im*b)^2*f) + (im*(c + im*d)^(5/2)*atanh(sqrt(c + d*tan(e + f*x))/sqrt(c + im*d)))/((a + im*b)^2*f) - ((b*c - a*d)^(3/2)*(4*a*b*c + a^2*d + 5*b^2*d)*atanh((sqrt(b)*sqrt(c + d*tan(e + f*x)))/sqrt(b*c - a*d)))/(b^(3/2)*(a^2 + b^2)^2*f) - ((b*c - a*d)^2*sqrt(c + d*tan(e + f*x)))/(b*(a^2 + b^2)*f*(a + b*tan(e + f*x)))]
    @test_int [(c + d*tan(e + f*x))^(5/2)/(a + b*tan(e + f*x))^3, x, 13, -(((c - im*d)^(5/2)*atanh(sqrt(c + d*tan(e + f*x))/sqrt(c - im*d)))/((im*a + b)^3*f)) + ((c + im*d)^(5/2)*atanh(sqrt(c + d*tan(e + f*x))/sqrt(c + im*d)))/((im*a - b)^3*f) + (sqrt(b*c - a*d)*(8*a^3*b*c*d - 56*a*b^3*c*d + a^4*d^2 + b^4*(8*c^2 - 15*d^2) - 6*a^2*b^2*(4*c^2 - 3*d^2))*atanh((sqrt(b)*sqrt(c + d*tan(e + f*x)))/sqrt(b*c - a*d)))/(4*b^(3/2)*(a^2 + b^2)^3*f) - ((b*c - a*d)^2*sqrt(c + d*tan(e + f*x)))/(2*b*(a^2 + b^2)*f*(a + b*tan(e + f*x))^2) - ((b*c - a*d)*(8*a*b*c + a^2*d + 9*b^2*d)*sqrt(c + d*tan(e + f*x)))/(4*b*(a^2 + b^2)^2*f*(a + b*tan(e + f*x)))]


    #= ::Subsubsection::Closed:: =#
    #=n < 0=#


    @test_int [(a + b*tan(e + f*x))^4/sqrt(c + d*tan(e + f*x)), x, 10, -((im*(a - im*b)^4*atanh(sqrt(c + d*tan(e + f*x))/sqrt(c - im*d)))/(sqrt(c - im*d)*f)) + (im*(a + im*b)^4*atanh(sqrt(c + d*tan(e + f*x))/sqrt(c + im*d)))/(sqrt(c + im*d)*f) - (2*b^2*(40*a*b*c*d - 87*a^2*d^2 - b^2*(8*c^2 - 15*d^2))*sqrt(c + d*tan(e + f*x)))/(15*d^3*f) - (4*b^3*(2*b*c - 7*a*d)*tan(e + f*x)*sqrt(c + d*tan(e + f*x)))/(15*d^2*f) + (2*b^2*(a + b*tan(e + f*x))^2*sqrt(c + d*tan(e + f*x)))/(5*d*f)]
    @test_int [(a + b*tan(e + f*x))^3/sqrt(c + d*tan(e + f*x)), x, 9, ((im*a + b)^3*atanh(sqrt(c + d*tan(e + f*x))/sqrt(c - im*d)))/(sqrt(c - im*d)*f) - ((im*a - b)^3*atanh(sqrt(c + d*tan(e + f*x))/sqrt(c + im*d)))/(sqrt(c + im*d)*f) - (4*b^2*(b*c - 4*a*d)*sqrt(c + d*tan(e + f*x)))/(3*d^2*f) + (2*b^2*(a + b*tan(e + f*x))*sqrt(c + d*tan(e + f*x)))/(3*d*f)]
    @test_int [(a + b*tan(e + f*x))^2/sqrt(c + d*tan(e + f*x)), x, 8, ((-im)*(a - im*b)^2*atanh(sqrt(c + d*tan(e + f*x))/sqrt(c - im*d)))/(sqrt(c - im*d)*f) + (im*(a + im*b)^2*atanh(sqrt(c + d*tan(e + f*x))/sqrt(c + im*d)))/(sqrt(c + im*d)*f) + (2*b^2*sqrt(c + d*tan(e + f*x)))/(d*f)]
    @test_int [(a + b*tan(e + f*x))/sqrt(c + d*tan(e + f*x)), x, 7, -(((im*a + b)*atanh(sqrt(c + d*tan(e + f*x))/sqrt(c - im*d)))/(sqrt(c - im*d)*f)) + ((im*a - b)*atanh(sqrt(c + d*tan(e + f*x))/sqrt(c + im*d)))/(sqrt(c + im*d)*f)]
    @test_int [1/((a + b*tan(e + f*x))*sqrt(c + d*tan(e + f*x))), x, 11, atanh(sqrt(c + d*tan(e + f*x))/sqrt(c - im*d))/((im*a + b)*sqrt(c - im*d)*f) - atanh(sqrt(c + d*tan(e + f*x))/sqrt(c + im*d))/((im*a - b)*sqrt(c + im*d)*f) - (2*b^(3/2)*atanh((sqrt(b)*sqrt(c + d*tan(e + f*x)))/sqrt(b*c - a*d)))/((a^2 + b^2)*sqrt(b*c - a*d)*f)]
    @test_int [1/((a + b*tan(e + f*x))^2*sqrt(c + d*tan(e + f*x))), x, 12, ((-im)*atanh(sqrt(c + d*tan(e + f*x))/sqrt(c - im*d)))/((a - im*b)^2*sqrt(c - im*d)*f) + (im*atanh(sqrt(c + d*tan(e + f*x))/sqrt(c + im*d)))/((a + im*b)^2*sqrt(c + im*d)*f) - (b^(3/2)*(4*a*b*c - 5*a^2*d - b^2*d)*atanh((sqrt(b)*sqrt(c + d*tan(e + f*x)))/sqrt(b*c - a*d)))/((a^2 + b^2)^2*(b*c - a*d)^(3/2)*f) - (b^2*sqrt(c + d*tan(e + f*x)))/((a^2 + b^2)*(b*c - a*d)*f*(a + b*tan(e + f*x)))]


    @test_int [(a + b*tan(e + f*x))^4/(c + d*tan(e + f*x))^(3/2), x, 10, -((im*(a - im*b)^4*atanh(sqrt(c + d*tan(e + f*x))/sqrt(c - im*d)))/((c - im*d)^(3/2)*f)) + (im*(a + im*b)^4*atanh(sqrt(c + d*tan(e + f*x))/sqrt(c + im*d)))/((c + im*d)^(3/2)*f) - (2*(b*c - a*d)^2*(a + b*tan(e + f*x))^2)/(d*(c^2 + d^2)*f*sqrt(c + d*tan(e + f*x))) - (2*b*(15*a^2*b*c*d^2 - 6*a^3*d^3 - 12*a*b^2*d*(2*c^2 + d^2) + b^3*(8*c^3 + 5*c*d^2))*sqrt(c + d*tan(e + f*x)))/(3*d^3*(c^2 + d^2)*f) - (2*b^2*(3*a*d*(2*b*c - a*d) - b^2*(4*c^2 + d^2))*tan(e + f*x)*sqrt(c + d*tan(e + f*x)))/(3*d^2*(c^2 + d^2)*f)]
    @test_int [(a + b*tan(e + f*x))^3/(c + d*tan(e + f*x))^(3/2), x, 9, ((im*a + b)^3*atanh(sqrt(c + d*tan(e + f*x))/sqrt(c - im*d)))/((c - im*d)^(3/2)*f) - ((im*a - b)^3*atanh(sqrt(c + d*tan(e + f*x))/sqrt(c + im*d)))/((c + im*d)^(3/2)*f) - (2*(b*c - a*d)^2*(a + b*tan(e + f*x)))/(d*(c^2 + d^2)*f*sqrt(c + d*tan(e + f*x))) - (2*b*(a*d*(2*b*c - a*d) - b^2*(2*c^2 + d^2))*sqrt(c + d*tan(e + f*x)))/(d^2*(c^2 + d^2)*f)]
    @test_int [(a + b*tan(e + f*x))^2/(c + d*tan(e + f*x))^(3/2), x, 8, ((-im)*(a - im*b)^2*atanh(sqrt(c + d*tan(e + f*x))/sqrt(c - im*d)))/((c - im*d)^(3/2)*f) + (im*(a + im*b)^2*atanh(sqrt(c + d*tan(e + f*x))/sqrt(c + im*d)))/((c + im*d)^(3/2)*f) - (2*(b*c - a*d)^2)/(d*(c^2 + d^2)*f*sqrt(c + d*tan(e + f*x)))]
    @test_int [(a + b*tan(e + f*x))/(c + d*tan(e + f*x))^(3/2), x, 8, -(((im*a + b)*atanh(sqrt(c + d*tan(e + f*x))/sqrt(c - im*d)))/((c - im*d)^(3/2)*f)) + ((im*a - b)*atanh(sqrt(c + d*tan(e + f*x))/sqrt(c + im*d)))/((c + im*d)^(3/2)*f) + (2*(b*c - a*d))/((c^2 + d^2)*f*sqrt(c + d*tan(e + f*x)))]
    @test_int [1/((a + b*tan(e + f*x))*(c + d*tan(e + f*x))^(3/2)), x, 12, atanh(sqrt(c + d*tan(e + f*x))/sqrt(c - im*d))/((im*a + b)*(c - im*d)^(3/2)*f) - atanh(sqrt(c + d*tan(e + f*x))/sqrt(c + im*d))/((im*a - b)*(c + im*d)^(3/2)*f) - (2*b^(5/2)*atanh((sqrt(b)*sqrt(c + d*tan(e + f*x)))/sqrt(b*c - a*d)))/((a^2 + b^2)*(b*c - a*d)^(3/2)*f) + (2*d^2)/((b*c - a*d)*(c^2 + d^2)*f*sqrt(c + d*tan(e + f*x)))]
    @test_int [1/((a + b*tan(e + f*x))^2*(c + d*tan(e + f*x))^(3/2)), x, 13, ((-im)*atanh(sqrt(c + d*tan(e + f*x))/sqrt(c - im*d)))/((a - im*b)^2*(c - im*d)^(3/2)*f) + (im*atanh(sqrt(c + d*tan(e + f*x))/sqrt(c + im*d)))/((a + im*b)^2*(c + im*d)^(3/2)*f) - (b^(5/2)*(4*a*b*c - 7*a^2*d - 3*b^2*d)*atanh((sqrt(b)*sqrt(c + d*tan(e + f*x)))/sqrt(b*c - a*d)))/((a^2 + b^2)^2*(b*c - a*d)^(5/2)*f) - (d*(2*a^2*d^2 + b^2*(c^2 + 3*d^2)))/((a^2 + b^2)*(b*c - a*d)^2*(c^2 + d^2)*f*sqrt(c + d*tan(e + f*x))) - b^2/((a^2 + b^2)*(b*c - a*d)*f*(a + b*tan(e + f*x))*sqrt(c + d*tan(e + f*x)))]


    @test_int [(a + b*tan(e + f*x))^4/(c + d*tan(e + f*x))^(5/2), x, 10, -((im*(a - im*b)^4*atanh(sqrt(c + d*tan(e + f*x))/sqrt(c - im*d)))/((c - im*d)^(5/2)*f)) + (im*(a + im*b)^4*atanh(sqrt(c + d*tan(e + f*x))/sqrt(c + im*d)))/((c + im*d)^(5/2)*f) - (2*(b*c - a*d)^2*(a + b*tan(e + f*x))^2)/(3*d*(c^2 + d^2)*f*(c + d*tan(e + f*x))^(3/2)) + (4*(b*c - a*d)^3*(2*b*c^2 + 3*a*c*d + 5*b*d^2))/(3*d^3*(c^2 + d^2)^2*f*sqrt(c + d*tan(e + f*x))) - (2*b^2*(a*d*(2*b*c - a*d) - b^2*(4*c^2 + 3*d^2))*sqrt(c + d*tan(e + f*x)))/(3*d^3*(c^2 + d^2)*f)]
    @test_int [(a + b*tan(e + f*x))^3/(c + d*tan(e + f*x))^(5/2), x, 9, ((im*a + b)^3*atanh(sqrt(c + d*tan(e + f*x))/sqrt(c - im*d)))/((c - im*d)^(5/2)*f) - ((im*a - b)^3*atanh(sqrt(c + d*tan(e + f*x))/sqrt(c + im*d)))/((c + im*d)^(5/2)*f) - (2*(b*c - a*d)^2*(a + b*tan(e + f*x)))/(3*d*(c^2 + d^2)*f*(c + d*tan(e + f*x))^(3/2)) - (4*(b*c - a*d)^2*(3*a*c*d + b*(c^2 + 4*d^2)))/(3*d^2*(c^2 + d^2)^2*f*sqrt(c + d*tan(e + f*x)))]
    @test_int [(a + b*tan(e + f*x))^2/(c + d*tan(e + f*x))^(5/2), x, 9, ((-im)*(a - im*b)^2*atanh(sqrt(c + d*tan(e + f*x))/sqrt(c - im*d)))/((c - im*d)^(5/2)*f) + (im*(a + im*b)^2*atanh(sqrt(c + d*tan(e + f*x))/sqrt(c + im*d)))/((c + im*d)^(5/2)*f) - (2*(b*c - a*d)^2)/(3*d*(c^2 + d^2)*f*(c + d*tan(e + f*x))^(3/2)) + (4*(b*c - a*d)*(a*c + b*d))/((c^2 + d^2)^2*f*sqrt(c + d*tan(e + f*x)))]
    @test_int [(a + b*tan(e + f*x))/(c + d*tan(e + f*x))^(5/2), x, 9, -(((im*a + b)*atanh(sqrt(c + d*tan(e + f*x))/sqrt(c - im*d)))/((c - im*d)^(5/2)*f)) + ((im*a - b)*atanh(sqrt(c + d*tan(e + f*x))/sqrt(c + im*d)))/((c + im*d)^(5/2)*f) + (2*(b*c - a*d))/(3*(c^2 + d^2)*f*(c + d*tan(e + f*x))^(3/2)) - (2*(2*a*c*d - b*(c^2 - d^2)))/((c^2 + d^2)^2*f*sqrt(c + d*tan(e + f*x)))]
    @test_int [1/((a + b*tan(e + f*x))*(c + d*tan(e + f*x))^(5/2)), x, 13, atanh(sqrt(c + d*tan(e + f*x))/sqrt(c - im*d))/((im*a + b)*(c - im*d)^(5/2)*f) - atanh(sqrt(c + d*tan(e + f*x))/sqrt(c + im*d))/((im*a - b)*(c + im*d)^(5/2)*f) - (2*b^(7/2)*atanh((sqrt(b)*sqrt(c + d*tan(e + f*x)))/sqrt(b*c - a*d)))/((a^2 + b^2)*(b*c - a*d)^(5/2)*f) + (2*d^2)/(3*(b*c - a*d)*(c^2 + d^2)*f*(c + d*tan(e + f*x))^(3/2)) - (2*d^2*(2*a*c*d - b*(3*c^2 + d^2)))/((b*c - a*d)^2*(c^2 + d^2)^2*f*sqrt(c + d*tan(e + f*x)))]
    @test_int [1/((a + b*tan(e + f*x))^2*(c + d*tan(e + f*x))^(5/2)), x, 14, ((-im)*atanh(sqrt(c + d*tan(e + f*x))/sqrt(c - im*d)))/((a - im*b)^2*(c - im*d)^(5/2)*f) + (im*atanh(sqrt(c + d*tan(e + f*x))/sqrt(c + im*d)))/((a + im*b)^2*(c + im*d)^(5/2)*f) - (b^(7/2)*(4*a*b*c - 9*a^2*d - 5*b^2*d)*atanh((sqrt(b)*sqrt(c + d*tan(e + f*x)))/sqrt(b*c - a*d)))/((a^2 + b^2)^2*(b*c - a*d)^(7/2)*f) - (d*(2*a^2*d^2 + b^2*(3*c^2 + 5*d^2)))/(3*(a^2 + b^2)*(b*c - a*d)^2*(c^2 + d^2)*f*(c + d*tan(e + f*x))^(3/2)) - b^2/((a^2 + b^2)*(b*c - a*d)*f*(a + b*tan(e + f*x))*(c + d*tan(e + f*x))^(3/2)) + (d*(4*a^3*c*d^3 + 4*a*b^2*c*d^3 - 4*a^2*b*d^2*(2*c^2 + d^2) - b^3*(c^4 + 10*c^2*d^2 + 5*d^4)))/((a^2 + b^2)*(b*c - a*d)^3*(c^2 + d^2)^2*f*sqrt(c + d*tan(e + f*x)))]


    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*(a+b*tan(e+f*x))^(m/2)*(c+d*tan(e+f*x))^(n/2)=#


    #= ::Subsubsection::Closed:: =#
    #=n > 0=#


    @test_int [(a + b*tan(e + f*x))^(5/2)*sqrt(c + d*tan(e + f*x)), x, 14, -((im*(a - im*b)^(5/2)*sqrt(c - im*d)*atanh((sqrt(c - im*d)*sqrt(a + b*tan(e + f*x)))/(sqrt(a - im*b)*sqrt(c + d*tan(e + f*x)))))/f) + (im*(a + im*b)^(5/2)*sqrt(c + im*d)*atanh((sqrt(c + im*d)*sqrt(a + b*tan(e + f*x)))/(sqrt(a + im*b)*sqrt(c + d*tan(e + f*x)))))/f + (sqrt(b)*(10*a*b*c*d + 15*a^2*d^2 - b^2*(c^2 + 8*d^2))*atanh((sqrt(d)*sqrt(a + b*tan(e + f*x)))/(sqrt(b)*sqrt(c + d*tan(e + f*x)))))/(4*d^(3/2)*f) - (b*(b*c - 9*a*d)*sqrt(a + b*tan(e + f*x))*sqrt(c + d*tan(e + f*x)))/(4*d*f) + (b^2*sqrt(a + b*tan(e + f*x))*(c + d*tan(e + f*x))^(3/2))/(2*d*f)]
    @test_int [(a + b*tan(e + f*x))^(3/2)*sqrt(c + d*tan(e + f*x)), x, 13, -((im*(a - im*b)^(3/2)*sqrt(c - im*d)*atanh((sqrt(c - im*d)*sqrt(a + b*tan(e + f*x)))/(sqrt(a - im*b)*sqrt(c + d*tan(e + f*x)))))/f) + (im*(a + im*b)^(3/2)*sqrt(c + im*d)*atanh((sqrt(c + im*d)*sqrt(a + b*tan(e + f*x)))/(sqrt(a + im*b)*sqrt(c + d*tan(e + f*x)))))/f + (sqrt(b)*(b*c + 3*a*d)*atanh((sqrt(d)*sqrt(a + b*tan(e + f*x)))/(sqrt(b)*sqrt(c + d*tan(e + f*x)))))/(sqrt(d)*f) + (b*sqrt(a + b*tan(e + f*x))*sqrt(c + d*tan(e + f*x)))/f]
    @test_int [sqrt(a + b*tan(e + f*x))*sqrt(c + d*tan(e + f*x)), x, 11, -((im*sqrt(a - im*b)*sqrt(c - im*d)*atanh((sqrt(c - im*d)*sqrt(a + b*tan(e + f*x)))/(sqrt(a - im*b)*sqrt(c + d*tan(e + f*x)))))/f) + (im*sqrt(a + im*b)*sqrt(c + im*d)*atanh((sqrt(c + im*d)*sqrt(a + b*tan(e + f*x)))/(sqrt(a + im*b)*sqrt(c + d*tan(e + f*x)))))/f + (2*sqrt(b)*sqrt(d)*atanh((sqrt(d)*sqrt(a + b*tan(e + f*x)))/(sqrt(b)*sqrt(c + d*tan(e + f*x)))))/f]
    @test_int [sqrt(c + d*tan(e + f*x))/sqrt(a + b*tan(e + f*x)), x, 7, -((im*sqrt(c - im*d)*atanh((sqrt(c - im*d)*sqrt(a + b*tan(e + f*x)))/(sqrt(a - im*b)*sqrt(c + d*tan(e + f*x)))))/(sqrt(a - im*b)*f)) + (im*sqrt(c + im*d)*atanh((sqrt(c + im*d)*sqrt(a + b*tan(e + f*x)))/(sqrt(a + im*b)*sqrt(c + d*tan(e + f*x)))))/(sqrt(a + im*b)*f)]
    @test_int [sqrt(c + d*tan(e + f*x))/(a + b*tan(e + f*x))^(3/2), x, 8, ((-im)*sqrt(c - im*d)*atanh((sqrt(c - im*d)*sqrt(a + b*tan(e + f*x)))/(sqrt(a - im*b)*sqrt(c + d*tan(e + f*x)))))/((a - im*b)^(3/2)*f) + (im*sqrt(c + im*d)*atanh((sqrt(c + im*d)*sqrt(a + b*tan(e + f*x)))/(sqrt(a + im*b)*sqrt(c + d*tan(e + f*x)))))/((a + im*b)^(3/2)*f) - (2*b*sqrt(c + d*tan(e + f*x)))/((a^2 + b^2)*f*sqrt(a + b*tan(e + f*x)))]
    @test_int [sqrt(c + d*tan(e + f*x))/(a + b*tan(e + f*x))^(5/2), x, 9, ((-im)*sqrt(c - im*d)*atanh((sqrt(c - im*d)*sqrt(a + b*tan(e + f*x)))/(sqrt(a - im*b)*sqrt(c + d*tan(e + f*x)))))/((a - im*b)^(5/2)*f) + (im*sqrt(c + im*d)*atanh((sqrt(c + im*d)*sqrt(a + b*tan(e + f*x)))/(sqrt(a + im*b)*sqrt(c + d*tan(e + f*x)))))/((a + im*b)^(5/2)*f) - (2*b*sqrt(c + d*tan(e + f*x)))/(3*(a^2 + b^2)*f*(a + b*tan(e + f*x))^(3/2)) - (2*b*(6*a*b*c - 5*a^2*d + b^2*d)*sqrt(c + d*tan(e + f*x)))/(3*(a^2 + b^2)^2*(b*c - a*d)*f*sqrt(a + b*tan(e + f*x)))]


    @test_int [(a + b*tan(e + f*x))^(3/2)*(c + d*tan(e + f*x))^(3/2), x, 14, -((im*(a - im*b)^(3/2)*(c - im*d)^(3/2)*atanh((sqrt(c - im*d)*sqrt(a + b*tan(e + f*x)))/(sqrt(a - im*b)*sqrt(c + d*tan(e + f*x)))))/f) + (im*(a + im*b)^(3/2)*(c + im*d)^(3/2)*atanh((sqrt(c + im*d)*sqrt(a + b*tan(e + f*x)))/(sqrt(a + im*b)*sqrt(c + d*tan(e + f*x)))))/f + ((18*a*b*c*d + 3*a^2*d^2 + b^2*(3*c^2 - 8*d^2))*atanh((sqrt(d)*sqrt(a + b*tan(e + f*x)))/(sqrt(b)*sqrt(c + d*tan(e + f*x)))))/(4*sqrt(b)*sqrt(d)*f) + ((3*b*c + 5*a*d)*sqrt(a + b*tan(e + f*x))*sqrt(c + d*tan(e + f*x)))/(4*f) + (b*sqrt(a + b*tan(e + f*x))*(c + d*tan(e + f*x))^(3/2))/(2*f)]
    @test_int [sqrt(a + b*tan(e + f*x))*(c + d*tan(e + f*x))^(3/2), x, 13, -((im*sqrt(a - im*b)*(c - im*d)^(3/2)*atanh((sqrt(c - im*d)*sqrt(a + b*tan(e + f*x)))/(sqrt(a - im*b)*sqrt(c + d*tan(e + f*x)))))/f) + (im*sqrt(a + im*b)*(c + im*d)^(3/2)*atanh((sqrt(c + im*d)*sqrt(a + b*tan(e + f*x)))/(sqrt(a + im*b)*sqrt(c + d*tan(e + f*x)))))/f + (sqrt(d)*(3*b*c + a*d)*atanh((sqrt(d)*sqrt(a + b*tan(e + f*x)))/(sqrt(b)*sqrt(c + d*tan(e + f*x)))))/(sqrt(b)*f) + (d*sqrt(a + b*tan(e + f*x))*sqrt(c + d*tan(e + f*x)))/f]
    @test_int [(c + d*tan(e + f*x))^(3/2)/sqrt(a + b*tan(e + f*x)), x, 12, -((im*(c - im*d)^(3/2)*atanh((sqrt(c - im*d)*sqrt(a + b*tan(e + f*x)))/(sqrt(a - im*b)*sqrt(c + d*tan(e + f*x)))))/(sqrt(a - im*b)*f)) + (im*(c + im*d)^(3/2)*atanh((sqrt(c + im*d)*sqrt(a + b*tan(e + f*x)))/(sqrt(a + im*b)*sqrt(c + d*tan(e + f*x)))))/(sqrt(a + im*b)*f) + (2*d^(3/2)*atanh((sqrt(d)*sqrt(a + b*tan(e + f*x)))/(sqrt(b)*sqrt(c + d*tan(e + f*x)))))/(sqrt(b)*f)]
    @test_int [(c + d*tan(e + f*x))^(3/2)/(a + b*tan(e + f*x))^(3/2), x, 8, ((-im)*(c - im*d)^(3/2)*atanh((sqrt(c - im*d)*sqrt(a + b*tan(e + f*x)))/(sqrt(a - im*b)*sqrt(c + d*tan(e + f*x)))))/((a - im*b)^(3/2)*f) + (im*(c + im*d)^(3/2)*atanh((sqrt(c + im*d)*sqrt(a + b*tan(e + f*x)))/(sqrt(a + im*b)*sqrt(c + d*tan(e + f*x)))))/((a + im*b)^(3/2)*f) - (2*(b*c - a*d)*sqrt(c + d*tan(e + f*x)))/((a^2 + b^2)*f*sqrt(a + b*tan(e + f*x)))]
    @test_int [(c + d*tan(e + f*x))^(3/2)/(a + b*tan(e + f*x))^(5/2), x, 9, ((-im)*(c - im*d)^(3/2)*atanh((sqrt(c - im*d)*sqrt(a + b*tan(e + f*x)))/(sqrt(a - im*b)*sqrt(c + d*tan(e + f*x)))))/((a - im*b)^(5/2)*f) + (im*(c + im*d)^(3/2)*atanh((sqrt(c + im*d)*sqrt(a + b*tan(e + f*x)))/(sqrt(a + im*b)*sqrt(c + d*tan(e + f*x)))))/((a + im*b)^(5/2)*f) - (2*(b*c - a*d)*sqrt(c + d*tan(e + f*x)))/(3*(a^2 + b^2)*f*(a + b*tan(e + f*x))^(3/2)) - (4*(3*a*b*c - a^2*d + 2*b^2*d)*sqrt(c + d*tan(e + f*x)))/(3*(a^2 + b^2)^2*f*sqrt(a + b*tan(e + f*x)))]
    @test_int [(c + d*tan(e + f*x))^(3/2)/(a + b*tan(e + f*x))^(7/2), x, 10, -((im*(c - im*d)^(3/2)*atanh((sqrt(c - im*d)*sqrt(a + b*tan(e + f*x)))/(sqrt(a - im*b)*sqrt(c + d*tan(e + f*x)))))/((a - im*b)^(7/2)*f)) + (im*(c + im*d)^(3/2)*atanh((sqrt(c + im*d)*sqrt(a + b*tan(e + f*x)))/(sqrt(a + im*b)*sqrt(c + d*tan(e + f*x)))))/((a + im*b)^(7/2)*f) - (2*(b*c - a*d)*sqrt(c + d*tan(e + f*x)))/(5*(a^2 + b^2)*f*(a + b*tan(e + f*x))^(5/2)) - (4*(5*a*b*c - 2*a^2*d + 3*b^2*d)*sqrt(c + d*tan(e + f*x)))/(15*(a^2 + b^2)^2*f*(a + b*tan(e + f*x))^(3/2)) + (2*(50*a^3*b*c*d - 70*a*b^3*c*d - 8*a^4*d^2 - a^2*b^2*(45*c^2 - 49*d^2) + 3*b^4*(5*c^2 - d^2))*sqrt(c + d*tan(e + f*x)))/(15*(a^2 + b^2)^3*(b*c - a*d)*f*sqrt(a + b*tan(e + f*x)))]


    @test_int [(a + b*tan(e + f*x))^(3/2)*(c + d*tan(e + f*x))^(5/2), x, 15, -((im*(a - im*b)^(3/2)*(c - im*d)^(5/2)*atanh((sqrt(c - im*d)*sqrt(a + b*tan(e + f*x)))/(sqrt(a - im*b)*sqrt(c + d*tan(e + f*x)))))/f) + (im*(a + im*b)^(3/2)*(c + im*d)^(5/2)*atanh((sqrt(c + im*d)*sqrt(a + b*tan(e + f*x)))/(sqrt(a + im*b)*sqrt(c + d*tan(e + f*x)))))/f + ((15*a^2*b*c*d^2 - a^3*d^3 + 3*a*b^2*d*(15*c^2 - 8*d^2) + 5*b^3*(c^3 - 8*c*d^2))*atanh((sqrt(d)*sqrt(a + b*tan(e + f*x)))/(sqrt(b)*sqrt(c + d*tan(e + f*x)))))/(8*b^(3/2)*sqrt(d)*f) + ((14*a*b*c*d - a^2*d^2 + b^2*(11*c^2 - 8*d^2))*sqrt(a + b*tan(e + f*x))*sqrt(c + d*tan(e + f*x)))/(8*b*f) + (d*(13*b*c - a*d)*(a + b*tan(e + f*x))^(3/2)*sqrt(c + d*tan(e + f*x)))/(12*b*f) + (d^2*(a + b*tan(e + f*x))^(5/2)*sqrt(c + d*tan(e + f*x)))/(3*b*f)]
    @test_int [sqrt(a + b*tan(e + f*x))*(c + d*tan(e + f*x))^(5/2), x, 14, -((im*sqrt(a - im*b)*(c - im*d)^(5/2)*atanh((sqrt(c - im*d)*sqrt(a + b*tan(e + f*x)))/(sqrt(a - im*b)*sqrt(c + d*tan(e + f*x)))))/f) + (im*sqrt(a + im*b)*(c + im*d)^(5/2)*atanh((sqrt(c + im*d)*sqrt(a + b*tan(e + f*x)))/(sqrt(a + im*b)*sqrt(c + d*tan(e + f*x)))))/f + (sqrt(d)*(10*a*b*c*d - a^2*d^2 + b^2*(15*c^2 - 8*d^2))*atanh((sqrt(d)*sqrt(a + b*tan(e + f*x)))/(sqrt(b)*sqrt(c + d*tan(e + f*x)))))/(4*b^(3/2)*f) + (d*(9*b*c - a*d)*sqrt(a + b*tan(e + f*x))*sqrt(c + d*tan(e + f*x)))/(4*b*f) + (d^2*(a + b*tan(e + f*x))^(3/2)*sqrt(c + d*tan(e + f*x)))/(2*b*f)]
    @test_int [(c + d*tan(e + f*x))^(5/2)/sqrt(a + b*tan(e + f*x)), x, 13, -((im*(c - im*d)^(5/2)*atanh((sqrt(c - im*d)*sqrt(a + b*tan(e + f*x)))/(sqrt(a - im*b)*sqrt(c + d*tan(e + f*x)))))/(sqrt(a - im*b)*f)) + (im*(c + im*d)^(5/2)*atanh((sqrt(c + im*d)*sqrt(a + b*tan(e + f*x)))/(sqrt(a + im*b)*sqrt(c + d*tan(e + f*x)))))/(sqrt(a + im*b)*f) + (d^(3/2)*(5*b*c - a*d)*atanh((sqrt(d)*sqrt(a + b*tan(e + f*x)))/(sqrt(b)*sqrt(c + d*tan(e + f*x)))))/(b^(3/2)*f) + (d^2*sqrt(a + b*tan(e + f*x))*sqrt(c + d*tan(e + f*x)))/(b*f)]
    @test_int [(c + d*tan(e + f*x))^(5/2)/(a + b*tan(e + f*x))^(3/2), x, 13, -((im*(c - im*d)^(5/2)*atanh((sqrt(c - im*d)*sqrt(a + b*tan(e + f*x)))/(sqrt(a - im*b)*sqrt(c + d*tan(e + f*x)))))/((a - im*b)^(3/2)*f)) + (im*(c + im*d)^(5/2)*atanh((sqrt(c + im*d)*sqrt(a + b*tan(e + f*x)))/(sqrt(a + im*b)*sqrt(c + d*tan(e + f*x)))))/((a + im*b)^(3/2)*f) + (2*d^(5/2)*atanh((sqrt(d)*sqrt(a + b*tan(e + f*x)))/(sqrt(b)*sqrt(c + d*tan(e + f*x)))))/(b^(3/2)*f) - (2*(b*c - a*d)^2*sqrt(c + d*tan(e + f*x)))/(b*(a^2 + b^2)*f*sqrt(a + b*tan(e + f*x)))]
    @test_int [(c + d*tan(e + f*x))^(5/2)/(a + b*tan(e + f*x))^(5/2), x, 9, ((-im)*(c - im*d)^(5/2)*atanh((sqrt(c - im*d)*sqrt(a + b*tan(e + f*x)))/(sqrt(a - im*b)*sqrt(c + d*tan(e + f*x)))))/((a - im*b)^(5/2)*f) + (im*(c + im*d)^(5/2)*atanh((sqrt(c + im*d)*sqrt(a + b*tan(e + f*x)))/(sqrt(a + im*b)*sqrt(c + d*tan(e + f*x)))))/((a + im*b)^(5/2)*f) - (2*(b*c - a*d)^2*sqrt(c + d*tan(e + f*x)))/(3*b*(a^2 + b^2)*f*(a + b*tan(e + f*x))^(3/2)) - (2*(b*c - a*d)*(6*a*b*c + a^2*d + 7*b^2*d)*sqrt(c + d*tan(e + f*x)))/(3*b*(a^2 + b^2)^2*f*sqrt(a + b*tan(e + f*x)))]
    @test_int [(c + d*tan(e + f*x))^(5/2)/(a + b*tan(e + f*x))^(7/2), x, 10, -((im*(c - im*d)^(5/2)*atanh((sqrt(c - im*d)*sqrt(a + b*tan(e + f*x)))/(sqrt(a - im*b)*sqrt(c + d*tan(e + f*x)))))/((a - im*b)^(7/2)*f)) + (im*(c + im*d)^(5/2)*atanh((sqrt(c + im*d)*sqrt(a + b*tan(e + f*x)))/(sqrt(a + im*b)*sqrt(c + d*tan(e + f*x)))))/((a + im*b)^(7/2)*f) - (2*(b*c - a*d)^2*sqrt(c + d*tan(e + f*x)))/(5*b*(a^2 + b^2)*f*(a + b*tan(e + f*x))^(5/2)) - (2*(b*c - a*d)*(10*a*b*c + a^2*d + 11*b^2*d)*sqrt(c + d*tan(e + f*x)))/(15*b*(a^2 + b^2)^2*f*(a + b*tan(e + f*x))^(3/2)) + (2*(20*a^3*b*c*d - 100*a*b^3*c*d + 2*a^4*d^2 + b^4*(15*c^2 - 23*d^2) - 3*a^2*b^2*(15*c^2 - 13*d^2))*sqrt(c + d*tan(e + f*x)))/(15*b*(a^2 + b^2)^3*f*sqrt(a + b*tan(e + f*x)))]


    #= ::Subsubsection::Closed:: =#
    #=n < 0=#


    @test_int [(a + b*tan(e + f*x))^(5/2)/sqrt(c + d*tan(e + f*x)), x, 13, -((im*(a - im*b)^(5/2)*atanh((sqrt(c - im*d)*sqrt(a + b*tan(e + f*x)))/(sqrt(a - im*b)*sqrt(c + d*tan(e + f*x)))))/(sqrt(c - im*d)*f)) + (im*(a + im*b)^(5/2)*atanh((sqrt(c + im*d)*sqrt(a + b*tan(e + f*x)))/(sqrt(a + im*b)*sqrt(c + d*tan(e + f*x)))))/(sqrt(c + im*d)*f) - (b^(3/2)*(b*c - 5*a*d)*atanh((sqrt(d)*sqrt(a + b*tan(e + f*x)))/(sqrt(b)*sqrt(c + d*tan(e + f*x)))))/(d^(3/2)*f) + (b^2*sqrt(a + b*tan(e + f*x))*sqrt(c + d*tan(e + f*x)))/(d*f)]
    @test_int [(a + b*tan(e + f*x))^(3/2)/sqrt(c + d*tan(e + f*x)), x, 12, -((im*(a - im*b)^(3/2)*atanh((sqrt(c - im*d)*sqrt(a + b*tan(e + f*x)))/(sqrt(a - im*b)*sqrt(c + d*tan(e + f*x)))))/(sqrt(c - im*d)*f)) + (im*(a + im*b)^(3/2)*atanh((sqrt(c + im*d)*sqrt(a + b*tan(e + f*x)))/(sqrt(a + im*b)*sqrt(c + d*tan(e + f*x)))))/(sqrt(c + im*d)*f) + (2*b^(3/2)*atanh((sqrt(d)*sqrt(a + b*tan(e + f*x)))/(sqrt(b)*sqrt(c + d*tan(e + f*x)))))/(sqrt(d)*f)]
    @test_int [sqrt(a + b*tan(e + f*x))/sqrt(c + d*tan(e + f*x)), x, 7, -((im*sqrt(a - im*b)*atanh((sqrt(c - im*d)*sqrt(a + b*tan(e + f*x)))/(sqrt(a - im*b)*sqrt(c + d*tan(e + f*x)))))/(sqrt(c - im*d)*f)) + (im*sqrt(a + im*b)*atanh((sqrt(c + im*d)*sqrt(a + b*tan(e + f*x)))/(sqrt(a + im*b)*sqrt(c + d*tan(e + f*x)))))/(sqrt(c + im*d)*f)]
    @test_int [1/(sqrt(a + b*tan(e + f*x))*sqrt(c + d*tan(e + f*x))), x, 7, -((im*atanh((sqrt(c - im*d)*sqrt(a + b*tan(e + f*x)))/(sqrt(a - im*b)*sqrt(c + d*tan(e + f*x)))))/(sqrt(a - im*b)*sqrt(c - im*d)*f)) + (im*atanh((sqrt(c + im*d)*sqrt(a + b*tan(e + f*x)))/(sqrt(a + im*b)*sqrt(c + d*tan(e + f*x)))))/(sqrt(a + im*b)*sqrt(c + im*d)*f)]
    @test_int [1/((a + b*tan(e + f*x))^(3/2)*sqrt(c + d*tan(e + f*x))), x, 8, ((-im)*atanh((sqrt(c - im*d)*sqrt(a + b*tan(e + f*x)))/(sqrt(a - im*b)*sqrt(c + d*tan(e + f*x)))))/((a - im*b)^(3/2)*sqrt(c - im*d)*f) + (im*atanh((sqrt(c + im*d)*sqrt(a + b*tan(e + f*x)))/(sqrt(a + im*b)*sqrt(c + d*tan(e + f*x)))))/((a + im*b)^(3/2)*sqrt(c + im*d)*f) - (2*b^2*sqrt(c + d*tan(e + f*x)))/((a^2 + b^2)*(b*c - a*d)*f*sqrt(a + b*tan(e + f*x)))]
    @test_int [1/((a + b*tan(e + f*x))^(5/2)*sqrt(c + d*tan(e + f*x))), x, 9, ((-im)*atanh((sqrt(c - im*d)*sqrt(a + b*tan(e + f*x)))/(sqrt(a - im*b)*sqrt(c + d*tan(e + f*x)))))/((a - im*b)^(5/2)*sqrt(c - im*d)*f) + (im*atanh((sqrt(c + im*d)*sqrt(a + b*tan(e + f*x)))/(sqrt(a + im*b)*sqrt(c + d*tan(e + f*x)))))/((a + im*b)^(5/2)*sqrt(c + im*d)*f) - (2*b^2*sqrt(c + d*tan(e + f*x)))/(3*(a^2 + b^2)*(b*c - a*d)*f*(a + b*tan(e + f*x))^(3/2)) - (4*b^2*(3*a*b*c - 4*a^2*d - b^2*d)*sqrt(c + d*tan(e + f*x)))/(3*(a^2 + b^2)^2*(b*c - a*d)^2*f*sqrt(a + b*tan(e + f*x)))]


    @test_int [(a + b*tan(e + f*x))^(7/2)/(c + d*tan(e + f*x))^(3/2), x, 14, -((im*(a - im*b)^(7/2)*atanh((sqrt(c - im*d)*sqrt(a + b*tan(e + f*x)))/(sqrt(a - im*b)*sqrt(c + d*tan(e + f*x)))))/((c - im*d)^(3/2)*f)) + (im*(a + im*b)^(7/2)*atanh((sqrt(c + im*d)*sqrt(a + b*tan(e + f*x)))/(sqrt(a + im*b)*sqrt(c + d*tan(e + f*x)))))/((c + im*d)^(3/2)*f) - (b^(5/2)*(3*b*c - 7*a*d)*atanh((sqrt(d)*sqrt(a + b*tan(e + f*x)))/(sqrt(b)*sqrt(c + d*tan(e + f*x)))))/(d^(5/2)*f) - (2*(b*c - a*d)^2*(a + b*tan(e + f*x))^(3/2))/(d*(c^2 + d^2)*f*sqrt(c + d*tan(e + f*x))) - (b*(2*a*d*(2*b*c - a*d) - b^2*(3*c^2 + d^2))*sqrt(a + b*tan(e + f*x))*sqrt(c + d*tan(e + f*x)))/(d^2*(c^2 + d^2)*f)]
    @test_int [(a + b*tan(e + f*x))^(5/2)/(c + d*tan(e + f*x))^(3/2), x, 13, -((im*(a - im*b)^(5/2)*atanh((sqrt(c - im*d)*sqrt(a + b*tan(e + f*x)))/(sqrt(a - im*b)*sqrt(c + d*tan(e + f*x)))))/((c - im*d)^(3/2)*f)) + (im*(a + im*b)^(5/2)*atanh((sqrt(c + im*d)*sqrt(a + b*tan(e + f*x)))/(sqrt(a + im*b)*sqrt(c + d*tan(e + f*x)))))/((c + im*d)^(3/2)*f) + (2*b^(5/2)*atanh((sqrt(d)*sqrt(a + b*tan(e + f*x)))/(sqrt(b)*sqrt(c + d*tan(e + f*x)))))/(d^(3/2)*f) - (2*(b*c - a*d)^2*sqrt(a + b*tan(e + f*x)))/(d*(c^2 + d^2)*f*sqrt(c + d*tan(e + f*x)))]
    @test_int [(a + b*tan(e + f*x))^(3/2)/(c + d*tan(e + f*x))^(3/2), x, 8, ((-im)*(a - im*b)^(3/2)*atanh((sqrt(c - im*d)*sqrt(a + b*tan(e + f*x)))/(sqrt(a - im*b)*sqrt(c + d*tan(e + f*x)))))/((c - im*d)^(3/2)*f) + (im*(a + im*b)^(3/2)*atanh((sqrt(c + im*d)*sqrt(a + b*tan(e + f*x)))/(sqrt(a + im*b)*sqrt(c + d*tan(e + f*x)))))/((c + im*d)^(3/2)*f) + (2*(b*c - a*d)*sqrt(a + b*tan(e + f*x)))/((c^2 + d^2)*f*sqrt(c + d*tan(e + f*x)))]
    @test_int [sqrt(a + b*tan(e + f*x))/(c + d*tan(e + f*x))^(3/2), x, 8, ((-im)*sqrt(a - im*b)*atanh((sqrt(c - im*d)*sqrt(a + b*tan(e + f*x)))/(sqrt(a - im*b)*sqrt(c + d*tan(e + f*x)))))/((c - im*d)^(3/2)*f) + (im*sqrt(a + im*b)*atanh((sqrt(c + im*d)*sqrt(a + b*tan(e + f*x)))/(sqrt(a + im*b)*sqrt(c + d*tan(e + f*x)))))/((c + im*d)^(3/2)*f) - (2*d*sqrt(a + b*tan(e + f*x)))/((c^2 + d^2)*f*sqrt(c + d*tan(e + f*x)))]
    @test_int [1/(sqrt(a + b*tan(e + f*x))*(c + d*tan(e + f*x))^(3/2)), x, 8, ((-im)*atanh((sqrt(c - im*d)*sqrt(a + b*tan(e + f*x)))/(sqrt(a - im*b)*sqrt(c + d*tan(e + f*x)))))/(sqrt(a - im*b)*(c - im*d)^(3/2)*f) + (im*atanh((sqrt(c + im*d)*sqrt(a + b*tan(e + f*x)))/(sqrt(a + im*b)*sqrt(c + d*tan(e + f*x)))))/(sqrt(a + im*b)*(c + im*d)^(3/2)*f) + (2*d^2*sqrt(a + b*tan(e + f*x)))/((b*c - a*d)*(c^2 + d^2)*f*sqrt(c + d*tan(e + f*x)))]
    @test_int [1/((a + b*tan(e + f*x))^(3/2)*(c + d*tan(e + f*x))^(3/2)), x, 9, ((-im)*atanh((sqrt(c - im*d)*sqrt(a + b*tan(e + f*x)))/(sqrt(a - im*b)*sqrt(c + d*tan(e + f*x)))))/((a - im*b)^(3/2)*(c - im*d)^(3/2)*f) + (im*atanh((sqrt(c + im*d)*sqrt(a + b*tan(e + f*x)))/(sqrt(a + im*b)*sqrt(c + d*tan(e + f*x)))))/((a + im*b)^(3/2)*(c + im*d)^(3/2)*f) - (2*b^2)/((a^2 + b^2)*(b*c - a*d)*f*sqrt(a + b*tan(e + f*x))*sqrt(c + d*tan(e + f*x))) - (2*d*(a^2*d^2 + b^2*(c^2 + 2*d^2))*sqrt(a + b*tan(e + f*x)))/((a^2 + b^2)*(b*c - a*d)^2*(c^2 + d^2)*f*sqrt(c + d*tan(e + f*x)))]
    @test_int [1/((a + b*tan(e + f*x))^(5/2)*(c + d*tan(e + f*x))^(3/2)), x, 10, ((-im)*atanh((sqrt(c - im*d)*sqrt(a + b*tan(e + f*x)))/(sqrt(a - im*b)*sqrt(c + d*tan(e + f*x)))))/((a - im*b)^(5/2)*(c - im*d)^(3/2)*f) + (im*atanh((sqrt(c + im*d)*sqrt(a + b*tan(e + f*x)))/(sqrt(a + im*b)*sqrt(c + d*tan(e + f*x)))))/((a + im*b)^(5/2)*(c + im*d)^(3/2)*f) - (2*b^2)/(3*(a^2 + b^2)*(b*c - a*d)*f*(a + b*tan(e + f*x))^(3/2)*sqrt(c + d*tan(e + f*x))) - (4*b^2*(3*a*b*c - 5*a^2*d - 2*b^2*d))/(3*(a^2 + b^2)^2*(b*c - a*d)^2*f*sqrt(a + b*tan(e + f*x))*sqrt(c + d*tan(e + f*x))) + (2*d*(3*a^4*d^3 - 6*a*b^3*c*(c^2 + d^2) + b^4*d*(5*c^2 + 8*d^2) + a^2*b^2*d*(11*c^2 + 17*d^2))*sqrt(a + b*tan(e + f*x)))/(3*(a^2 + b^2)^2*(b*c - a*d)^3*(c^2 + d^2)*f*sqrt(c + d*tan(e + f*x)))]


    @test_int [(a + b*tan(e + f*x))^(9/2)/(c + d*tan(e + f*x))^(5/2), x, 15, -((im*(a - im*b)^(9/2)*atanh((sqrt(c - im*d)*sqrt(a + b*tan(e + f*x)))/(sqrt(a - im*b)*sqrt(c + d*tan(e + f*x)))))/((c - im*d)^(5/2)*f)) + (im*(a + im*b)^(9/2)*atanh((sqrt(c + im*d)*sqrt(a + b*tan(e + f*x)))/(sqrt(a + im*b)*sqrt(c + d*tan(e + f*x)))))/((c + im*d)^(5/2)*f) - (b^(7/2)*(5*b*c - 9*a*d)*atanh((sqrt(d)*sqrt(a + b*tan(e + f*x)))/(sqrt(b)*sqrt(c + d*tan(e + f*x)))))/(d^(7/2)*f) - (2*(b*c - a*d)^2*(a + b*tan(e + f*x))^(5/2))/(3*d*(c^2 + d^2)*f*(c + d*tan(e + f*x))^(3/2)) - (2*(b*c - a*d)^2*(5*b*c^2 + 6*a*c*d + 11*b*d^2)*(a + b*tan(e + f*x))^(3/2))/(3*d^2*(c^2 + d^2)^2*f*sqrt(c + d*tan(e + f*x))) + (b*(4*a^3*c*d^3 - 4*a^2*b*d^2*(c^2 - 2*d^2) - 4*a*b^2*c*d*(c^2 + 4*d^2) + b^3*(5*c^4 + 10*c^2*d^2 + d^4))*sqrt(a + b*tan(e + f*x))*sqrt(c + d*tan(e + f*x)))/(d^3*(c^2 + d^2)^2*f)]
    @test_int [(a + b*tan(e + f*x))^(7/2)/(c + d*tan(e + f*x))^(5/2), x, 14, -((im*(a - im*b)^(7/2)*atanh((sqrt(c - im*d)*sqrt(a + b*tan(e + f*x)))/(sqrt(a - im*b)*sqrt(c + d*tan(e + f*x)))))/((c - im*d)^(5/2)*f)) + (im*(a + im*b)^(7/2)*atanh((sqrt(c + im*d)*sqrt(a + b*tan(e + f*x)))/(sqrt(a + im*b)*sqrt(c + d*tan(e + f*x)))))/((c + im*d)^(5/2)*f) + (2*b^(7/2)*atanh((sqrt(d)*sqrt(a + b*tan(e + f*x)))/(sqrt(b)*sqrt(c + d*tan(e + f*x)))))/(d^(5/2)*f) - (2*(b*c - a*d)^2*(a + b*tan(e + f*x))^(3/2))/(3*d*(c^2 + d^2)*f*(c + d*tan(e + f*x))^(3/2)) - (2*(b*c - a*d)^2*(2*a*c*d + b*(c^2 + 3*d^2))*sqrt(a + b*tan(e + f*x)))/(d^2*(c^2 + d^2)^2*f*sqrt(c + d*tan(e + f*x)))]
    @test_int [(a + b*tan(e + f*x))^(5/2)/(c + d*tan(e + f*x))^(5/2), x, 9, ((-im)*(a - im*b)^(5/2)*atanh((sqrt(c - im*d)*sqrt(a + b*tan(e + f*x)))/(sqrt(a - im*b)*sqrt(c + d*tan(e + f*x)))))/((c - im*d)^(5/2)*f) + (im*(a + im*b)^(5/2)*atanh((sqrt(c + im*d)*sqrt(a + b*tan(e + f*x)))/(sqrt(a + im*b)*sqrt(c + d*tan(e + f*x)))))/((c + im*d)^(5/2)*f) - (2*(b*c - a*d)^2*sqrt(a + b*tan(e + f*x)))/(3*d*(c^2 + d^2)*f*(c + d*tan(e + f*x))^(3/2)) + (2*(b*c - a*d)*(6*a*c*d + b*(c^2 + 7*d^2))*sqrt(a + b*tan(e + f*x)))/(3*d*(c^2 + d^2)^2*f*sqrt(c + d*tan(e + f*x)))]
    @test_int [(a + b*tan(e + f*x))^(3/2)/(c + d*tan(e + f*x))^(5/2), x, 9, -((im*(a - im*b)^(3/2)*atanh((sqrt(c - im*d)*sqrt(a + b*tan(e + f*x)))/(sqrt(a - im*b)*sqrt(c + d*tan(e + f*x)))))/((c - im*d)^(5/2)*f)) + (im*(a + im*b)^(3/2)*atanh((sqrt(c + im*d)*sqrt(a + b*tan(e + f*x)))/(sqrt(a + im*b)*sqrt(c + d*tan(e + f*x)))))/((c + im*d)^(5/2)*f) + (2*(b*c - a*d)*sqrt(a + b*tan(e + f*x)))/(3*(c^2 + d^2)*f*(c + d*tan(e + f*x))^(3/2)) + (4*(b*c^2 - 3*a*c*d - 2*b*d^2)*sqrt(a + b*tan(e + f*x)))/(3*(c^2 + d^2)^2*f*sqrt(c + d*tan(e + f*x)))]
    @test_int [sqrt(a + b*tan(e + f*x))/(c + d*tan(e + f*x))^(5/2), x, 9, ((-im)*sqrt(a - im*b)*atanh((sqrt(c - im*d)*sqrt(a + b*tan(e + f*x)))/(sqrt(a - im*b)*sqrt(c + d*tan(e + f*x)))))/((c - im*d)^(5/2)*f) + (im*sqrt(a + im*b)*atanh((sqrt(c + im*d)*sqrt(a + b*tan(e + f*x)))/(sqrt(a + im*b)*sqrt(c + d*tan(e + f*x)))))/((c + im*d)^(5/2)*f) - (2*d*sqrt(a + b*tan(e + f*x)))/(3*(c^2 + d^2)*f*(c + d*tan(e + f*x))^(3/2)) + (2*d*(6*a*c*d - b*(5*c^2 - d^2))*sqrt(a + b*tan(e + f*x)))/(3*(b*c - a*d)*(c^2 + d^2)^2*f*sqrt(c + d*tan(e + f*x)))]
    @test_int [1/(sqrt(a + b*tan(e + f*x))*(c + d*tan(e + f*x))^(5/2)), x, 9, ((-im)*atanh((sqrt(c - im*d)*sqrt(a + b*tan(e + f*x)))/(sqrt(a - im*b)*sqrt(c + d*tan(e + f*x)))))/(sqrt(a - im*b)*(c - im*d)^(5/2)*f) + (im*atanh((sqrt(c + im*d)*sqrt(a + b*tan(e + f*x)))/(sqrt(a + im*b)*sqrt(c + d*tan(e + f*x)))))/(sqrt(a + im*b)*(c + im*d)^(5/2)*f) + (2*d^2*sqrt(a + b*tan(e + f*x)))/(3*(b*c - a*d)*(c^2 + d^2)*f*(c + d*tan(e + f*x))^(3/2)) - (4*d^2*(3*a*c*d - b*(4*c^2 + d^2))*sqrt(a + b*tan(e + f*x)))/(3*(b*c - a*d)^2*(c^2 + d^2)^2*f*sqrt(c + d*tan(e + f*x)))]
    @test_int [1/((a + b*tan(e + f*x))^(3/2)*(c + d*tan(e + f*x))^(5/2)), x, 10, -((im*atanh((sqrt(c - im*d)*sqrt(a + b*tan(e + f*x)))/(sqrt(a - im*b)*sqrt(c + d*tan(e + f*x)))))/((a - im*b)^(3/2)*(c - im*d)^(5/2)*f)) + (im*atanh((sqrt(c + im*d)*sqrt(a + b*tan(e + f*x)))/(sqrt(a + im*b)*sqrt(c + d*tan(e + f*x)))))/((a + im*b)^(3/2)*(c + im*d)^(5/2)*f) - (2*b^2)/((a^2 + b^2)*(b*c - a*d)*f*sqrt(a + b*tan(e + f*x))*(c + d*tan(e + f*x))^(3/2)) - (2*d*(a^2*d^2 + b^2*(3*c^2 + 4*d^2))*sqrt(a + b*tan(e + f*x)))/(3*(a^2 + b^2)*(b*c - a*d)^2*(c^2 + d^2)*f*(c + d*tan(e + f*x))^(3/2)) + (2*(6*a^3*c*d^4 + 6*a*b^2*c*d^4 - a^2*b*d^3*(11*c^2 + 5*d^2) - b^3*(3*c^4*d + 17*c^2*d^3 + 8*d^5))*sqrt(a + b*tan(e + f*x)))/(3*(a^2 + b^2)*(b*c - a*d)^3*(c^2 + d^2)^2*f*sqrt(c + d*tan(e + f*x)))]
    @test_int [1/((a + b*tan(e + f*x))^(5/2)*(c + d*tan(e + f*x))^(5/2)), x, 11, ((-im)*atanh((sqrt(c - im*d)*sqrt(a + b*tan(e + f*x)))/(sqrt(a - im*b)*sqrt(c + d*tan(e + f*x)))))/((a - im*b)^(5/2)*(c - im*d)^(5/2)*f) + (im*atanh((sqrt(c + im*d)*sqrt(a + b*tan(e + f*x)))/(sqrt(a + im*b)*sqrt(c + d*tan(e + f*x)))))/((a + im*b)^(5/2)*(c + im*d)^(5/2)*f) - (2*b^2)/(3*(a^2 + b^2)*(b*c - a*d)*f*(a + b*tan(e + f*x))^(3/2)*(c + d*tan(e + f*x))^(3/2)) - (4*b^2*(a*b*c - 2*a^2*d - b^2*d))/((a^2 + b^2)^2*(b*c - a*d)^2*f*sqrt(a + b*tan(e + f*x))*(c + d*tan(e + f*x))^(3/2)) + (2*d*(a^4*d^3 - 6*a*b^3*c*(c^2 + d^2) + b^4*d*(7*c^2 + 8*d^2) + a^2*b^2*d*(13*c^2 + 15*d^2))*sqrt(a + b*tan(e + f*x)))/(3*(a^2 + b^2)^2*(b*c - a*d)^3*(c^2 + d^2)*f*(c + d*tan(e + f*x))^(3/2)) - (4*d*(3*a^5*c*d^4 + 6*a^3*b^2*c*d^4 - a^4*b*d^3*(7*c^2 + 4*d^2) + 3*a*b^4*c*(c^4 + 2*c^2*d^2 + 2*d^4) - b^5*d*(4*c^4 + 15*c^2*d^2 + 8*d^4) - a^2*b^3*d*(7*c^4 + 28*c^2*d^2 + 15*d^4))*sqrt(a + b*tan(e + f*x)))/(3*(a^2 + b^2)^2*(b*c - a*d)^4*(c^2 + d^2)^2*f*sqrt(c + d*tan(e + f*x)))]


    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*(a+b*tan(e+f*x))^m*(c+d*tan(e+f*x))^n*with*m*and/or*n*symbolic=#


    @test_int [(a + b*tan(e + f*x))^m*(c + d*tan(e + f*x))^n, x, 7, (AppellF1(1 + m, -n, 1, 2 + m, -((d*(a + b*tan(e + f*x)))/(b*c - a*d)), (a + b*tan(e + f*x))/(a - im*b))*(a + b*tan(e + f*x))^(1 + m)*(c + d*tan(e + f*x))^n)/(((b*(c + d*tan(e + f*x)))/(b*c - a*d))^n*(2*(im*a + b)*f*(1 + m))) - (AppellF1(1 + m, -n, 1, 2 + m, -((d*(a + b*tan(e + f*x)))/(b*c - a*d)), (a + b*tan(e + f*x))/(a + im*b))*(a + b*tan(e + f*x))^(1 + m)*(c + d*tan(e + f*x))^n)/(((b*(c + d*tan(e + f*x)))/(b*c - a*d))^n*(2*(im*a - b)*f*(1 + m)))]


    @test_int [(a + b*tan(e + f*x))^m*(c + d*tan(e + f*x))^3, x, 7, (d^2*(3*b*c - a*d)*(a + b*tan(e + f*x))^(1 + m))/(b^2*f*(1 + m)) + ((im*c + d)^3*HypergeometricFunctions._₂F₁(1, 1 + m, 2 + m, (a + b*tan(e + f*x))/(a - im*b))*(a + b*tan(e + f*x))^(1 + m))/(2*(a - im*b)*f*(1 + m)) - ((im*c - d)^3*HypergeometricFunctions._₂F₁(1, 1 + m, 2 + m, (a + b*tan(e + f*x))/(a + im*b))*(a + b*tan(e + f*x))^(1 + m))/(2*(a + im*b)*f*(1 + m)) + (d^3*(a + b*tan(e + f*x))^(2 + m))/(b^2*f*(2 + m)), -((d^2*(a*d - b*c*(5 + 2*m))*(a + b*tan(e + f*x))^(1 + m))/(b^2*f*(1 + m)*(2 + m))) + ((c - im*d)^3*HypergeometricFunctions._₂F₁(1, 1 + m, 2 + m, (a + b*tan(e + f*x))/(a - im*b))*(a + b*tan(e + f*x))^(1 + m))/(2*(im*a + b)*f*(1 + m)) - ((im*c - d)^3*HypergeometricFunctions._₂F₁(1, 1 + m, 2 + m, (a + b*tan(e + f*x))/(a + im*b))*(a + b*tan(e + f*x))^(1 + m))/(2*(a + im*b)*f*(1 + m)) + (d^2*(a + b*tan(e + f*x))^(1 + m)*(c + d*tan(e + f*x)))/(b*f*(2 + m))]
    @test_int [(a + b*tan(e + f*x))^m*(c + d*tan(e + f*x))^2, x, 6, (d^2*(a + b*tan(e + f*x))^(1 + m))/(b*f*(1 + m)) + ((c - im*d)^2*HypergeometricFunctions._₂F₁(1, 1 + m, 2 + m, (a + b*tan(e + f*x))/(a - im*b))*(a + b*tan(e + f*x))^(1 + m))/(2*(im*a + b)*f*(1 + m)) - ((c + im*d)^2*HypergeometricFunctions._₂F₁(1, 1 + m, 2 + m, (a + b*tan(e + f*x))/(a + im*b))*(a + b*tan(e + f*x))^(1 + m))/(2*(im*a - b)*f*(1 + m))]
    @test_int [(a + b*tan(e + f*x))^m*(c + d*tan(e + f*x))^1, x, 5, ((c - im*d)*HypergeometricFunctions._₂F₁(1, 1 + m, 2 + m, (a + b*tan(e + f*x))/(a - im*b))*(a + b*tan(e + f*x))^(1 + m))/(2*(im*a + b)*f*(1 + m)) + ((im*c - d)*HypergeometricFunctions._₂F₁(1, 1 + m, 2 + m, (a + b*tan(e + f*x))/(a + im*b))*(a + b*tan(e + f*x))^(1 + m))/(2*(a + im*b)*f*(1 + m))]
    @test_int [(a + b*tan(e + f*x))^m*(c + d*tan(e + f*x))^0, x, 5, (b*HypergeometricFunctions._₂F₁(1, 1 + m, 2 + m, (a + b*tan(e + f*x))/(a - sqrt(-b^2)))*(a + b*tan(e + f*x))^(1 + m))/(2*sqrt(-b^2)*(a - sqrt(-b^2))*f*(1 + m)) - (b*HypergeometricFunctions._₂F₁(1, 1 + m, 2 + m, (a + b*tan(e + f*x))/(a + sqrt(-b^2)))*(a + b*tan(e + f*x))^(1 + m))/(2*sqrt(-b^2)*(a + sqrt(-b^2))*f*(1 + m))]
    @test_int [(a + b*tan(e + f*x))^m/(c + d*tan(e + f*x))^1, x, 8, If(13>=8, (HypergeometricFunctions._₂F₁(1, 1 + m, 2 + m, (a + b*tan(e + f*x))/(a - im*b))*(a + b*tan(e + f*x))^(1 + m))/(2*(im*a + b)*(c - im*d)*f*(1 + m)) - (HypergeometricFunctions._₂F₁(1, 1 + m, 2 + m, (a + b*tan(e + f*x))/(a + im*b))*(a + b*tan(e + f*x))^(1 + m))/(2*(a + im*b)*(im*c - d)*f*(1 + m)) + (d^2*HypergeometricFunctions._₂F₁(1, 1 + m, 2 + m, -((d*(a + b*tan(e + f*x)))/(b*c - a*d)))*(a + b*tan(e + f*x))^(1 + m))/((b*c - a*d)*(c^2 + d^2)*f*(1 + m)), (HypergeometricFunctions._₂F₁(1, 1 + m, 2 + m, (a + b*tan(e + f*x))/(a - im*b))*(a + b*tan(e + f*x))^(1 + m))/(2*(im*a + b)*(c - im*d)*f*(1 + m)) - (HypergeometricFunctions._₂F₁(1, 1 + m, 2 + m, (a + b*tan(e + f*x))/(a + im*b))*(a + b*tan(e + f*x))^(1 + m))/(2*(im*a - b)*(c + im*d)*f*(1 + m)) + (d^2*HypergeometricFunctions._₂F₁(1, 1 + m, 2 + m, -((d*(a + b*tan(e + f*x)))/(b*c - a*d)))*(a + b*tan(e + f*x))^(1 + m))/((b*c - a*d)*(c^2 + d^2)*f*(1 + m)))]
    @test_int [(a + b*tan(e + f*x))^m/(c + d*tan(e + f*x))^2, x, 9, (HypergeometricFunctions._₂F₁(1, 1 + m, 2 + m, (a + b*tan(e + f*x))/(a - im*b))*(a + b*tan(e + f*x))^(1 + m))/(2*(im*a + b)*(c - im*d)^2*f*(1 + m)) - (HypergeometricFunctions._₂F₁(1, 1 + m, 2 + m, (a + b*tan(e + f*x))/(a + im*b))*(a + b*tan(e + f*x))^(1 + m))/(2*(im*a - b)*(c + im*d)^2*f*(1 + m)) - (d^2*(2*a*c*d - b*(c^2*(2 - m) - d^2*m))*HypergeometricFunctions._₂F₁(1, 1 + m, 2 + m, -((d*(a + b*tan(e + f*x)))/(b*c - a*d)))*(a + b*tan(e + f*x))^(1 + m))/((b*c - a*d)^2*(c^2 + d^2)^2*f*(1 + m)) + (d^2*(a + b*tan(e + f*x))^(1 + m))/((b*c - a*d)*(c^2 + d^2)*f*(c + d*tan(e + f*x)))]
    @test_int [(a + b*tan(e + f*x))^m/(c + d*tan(e + f*x))^3, x, 10, (HypergeometricFunctions._₂F₁(1, 1 + m, 2 + m, (a + b*tan(e + f*x))/(a - im*b))*(a + b*tan(e + f*x))^(1 + m))/(2*(im*a + b)*(c - im*d)^3*f*(1 + m)) + (HypergeometricFunctions._₂F₁(1, 1 + m, 2 + m, (a + b*tan(e + f*x))/(a + im*b))*(a + b*tan(e + f*x))^(1 + m))/(2*(a + im*b)*(im*c - d)^3*f*(1 + m)) + (d^2*(2*a^2*d^2*(3*c^2 - d^2) - 4*a*b*c*d*(c^2*(3 - m) - d^2*(1 + m)) - b^2*(d^4*(1 - m)*m + 2*c^2*d^2*(1 + 3*m - m^2) - c^4*(6 - 5*m + m^2)))*HypergeometricFunctions._₂F₁(1, 1 + m, 2 + m, -((d*(a + b*tan(e + f*x)))/(b*c - a*d)))*(a + b*tan(e + f*x))^(1 + m))/(2*(b*c - a*d)^3*(c^2 + d^2)^3*f*(1 + m)) + (d^2*(a + b*tan(e + f*x))^(1 + m))/(2*(b*c - a*d)*(c^2 + d^2)*f*(c + d*tan(e + f*x))^2) - (d^2*(4*a*c*d - b*(d^2*(1 - m) + c^2*(5 - m)))*(a + b*tan(e + f*x))^(1 + m))/(2*(b*c - a*d)^2*(c^2 + d^2)^2*f*(c + d*tan(e + f*x)))]


    @test_int [(a + b*tan(e + f*x))^m*(c + d*tan(e + f*x))^(3/2), x, 7, ((b*c - a*d)*AppellF1(1 + m, -(3/2), 1, 2 + m, -((d*(a + b*tan(e + f*x)))/(b*c - a*d)), (a + b*tan(e + f*x))/(a - im*b))*(a + b*tan(e + f*x))^(1 + m)*sqrt(c + d*tan(e + f*x)))/(2*b*(im*a + b)*f*(1 + m)*sqrt((b*(c + d*tan(e + f*x)))/(b*c - a*d))) - ((b*c - a*d)*AppellF1(1 + m, -(3/2), 1, 2 + m, -((d*(a + b*tan(e + f*x)))/(b*c - a*d)), (a + b*tan(e + f*x))/(a + im*b))*(a + b*tan(e + f*x))^(1 + m)*sqrt(c + d*tan(e + f*x)))/(2*(im*a - b)*b*f*(1 + m)*sqrt((b*(c + d*tan(e + f*x)))/(b*c - a*d)))]
    @test_int [(a + b*tan(e + f*x))^m*sqrt(c + d*tan(e + f*x)), x, 7, (AppellF1(1 + m, -(1/2), 1, 2 + m, -((d*(a + b*tan(e + f*x)))/(b*c - a*d)), (a + b*tan(e + f*x))/(a - im*b))*(a + b*tan(e + f*x))^(1 + m)*sqrt(c + d*tan(e + f*x)))/(2*(im*a + b)*f*(1 + m)*sqrt((b*(c + d*tan(e + f*x)))/(b*c - a*d))) - (AppellF1(1 + m, -(1/2), 1, 2 + m, -((d*(a + b*tan(e + f*x)))/(b*c - a*d)), (a + b*tan(e + f*x))/(a + im*b))*(a + b*tan(e + f*x))^(1 + m)*sqrt(c + d*tan(e + f*x)))/(2*(im*a - b)*f*(1 + m)*sqrt((b*(c + d*tan(e + f*x)))/(b*c - a*d)))]
    @test_int [(a + b*tan(e + f*x))^m/sqrt(c + d*tan(e + f*x)), x, 7, (AppellF1(1 + m, 1/2, 1, 2 + m, -((d*(a + b*tan(e + f*x)))/(b*c - a*d)), (a + b*tan(e + f*x))/(a - im*b))*(a + b*tan(e + f*x))^(1 + m)*sqrt((b*(c + d*tan(e + f*x)))/(b*c - a*d)))/(2*(im*a + b)*f*(1 + m)*sqrt(c + d*tan(e + f*x))) - (AppellF1(1 + m, 1/2, 1, 2 + m, -((d*(a + b*tan(e + f*x)))/(b*c - a*d)), (a + b*tan(e + f*x))/(a + im*b))*(a + b*tan(e + f*x))^(1 + m)*sqrt((b*(c + d*tan(e + f*x)))/(b*c - a*d)))/(2*(im*a - b)*f*(1 + m)*sqrt(c + d*tan(e + f*x)))]
    @test_int [(a + b*tan(e + f*x))^m/(c + d*tan(e + f*x))^(3/2), x, 7, (b*AppellF1(1 + m, 3/2, 1, 2 + m, -((d*(a + b*tan(e + f*x)))/(b*c - a*d)), (a + b*tan(e + f*x))/(a - im*b))*(a + b*tan(e + f*x))^(1 + m)*sqrt((b*(c + d*tan(e + f*x)))/(b*c - a*d)))/(2*(im*a + b)*(b*c - a*d)*f*(1 + m)*sqrt(c + d*tan(e + f*x))) - (b*AppellF1(1 + m, 3/2, 1, 2 + m, -((d*(a + b*tan(e + f*x)))/(b*c - a*d)), (a + b*tan(e + f*x))/(a + im*b))*(a + b*tan(e + f*x))^(1 + m)*sqrt((b*(c + d*tan(e + f*x)))/(b*c - a*d)))/(2*(im*a - b)*(b*c - a*d)*f*(1 + m)*sqrt(c + d*tan(e + f*x)))]
    @test_int [(a + b*tan(e + f*x))^m/(c + d*tan(e + f*x))^(5/2), x, 7, (b^2*AppellF1(1 + m, 5/2, 1, 2 + m, -((d*(a + b*tan(e + f*x)))/(b*c - a*d)), (a + b*tan(e + f*x))/(a - im*b))*(a + b*tan(e + f*x))^(1 + m)*sqrt((b*(c + d*tan(e + f*x)))/(b*c - a*d)))/(2*(im*a + b)*(b*c - a*d)^2*f*(1 + m)*sqrt(c + d*tan(e + f*x))) - (b^2*AppellF1(1 + m, 5/2, 1, 2 + m, -((d*(a + b*tan(e + f*x)))/(b*c - a*d)), (a + b*tan(e + f*x))/(a + im*b))*(a + b*tan(e + f*x))^(1 + m)*sqrt((b*(c + d*tan(e + f*x)))/(b*c - a*d)))/(2*(im*a - b)*(b*c - a*d)^2*f*(1 + m)*sqrt(c + d*tan(e + f*x)))]


    #= ::Title::Closed:: =#
    #=Integrands*of*the*form*(a+b*tan(e+f*x))^m*(c*(d*tan(e+f*x))^p)^n=#


    #= ::Section::Closed:: =#
    #=Integrands*of*the*form*(a+im*a*tan(e+f*x))^m*(c*(d*tan(e+f*x))^p)^n=#


    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*(a+im*a*tan(e+f*x))^m*(c*(d*tan(e+f*x))^p)^n*with*n*and*p*symbolic=#


    @test_int [(a + im*a*tan(e + f*x))^m*(c*(d*tan(e + f*x))^p)^n, x, 4, (AppellF1(1 + n*p, 1 - m, 1, 2 + n*p, (-im)*tan(e + f*x), im*tan(e + f*x))*tan(e + f*x)*(c*(d*tan(e + f*x))^p)^n*(a + im*a*tan(e + f*x))^m)/((1 + im*tan(e + f*x))^m*(f*(1 + n*p)))]


    @test_int [(c*(d*tan(e + f*x))^p)^n*(a + im*a*tan(e + f*x))^3, x, 8, -((3*a^3*tan(e + f*x)*(c*(d*tan(e + f*x))^p)^n)/(f*(1 + n*p))) + (4*a^3*HypergeometricFunctions._₂F₁(1, 1 + n*p, 2 + n*p, im*tan(e + f*x))*tan(e + f*x)*(c*(d*tan(e + f*x))^p)^n)/(f*(1 + n*p)) - (im*a^3*tan(e + f*x)^2*(c*(d*tan(e + f*x))^p)^n)/(f*(2 + n*p))]
    @test_int [(c*(d*tan(e + f*x))^p)^n*(a + im*a*tan(e + f*x))^2, x, 5, -((a^2*tan(e + f*x)*(c*(d*tan(e + f*x))^p)^n)/(f*(1 + n*p))) + (2*a^2*HypergeometricFunctions._₂F₁(1, 1 + n*p, 2 + n*p, im*tan(e + f*x))*tan(e + f*x)*(c*(d*tan(e + f*x))^p)^n)/(f*(1 + n*p))]
    @test_int [(c*(d*tan(e + f*x))^p)^n*(a + im*a*tan(e + f*x))^1, x, 4, (a*HypergeometricFunctions._₂F₁(1, 1 + n*p, 2 + n*p, im*tan(e + f*x))*tan(e + f*x)*(c*(d*tan(e + f*x))^p)^n)/(f*(1 + n*p))]
    @test_int [(c*(d*tan(e + f*x))^p)^n/(a + im*a*tan(e + f*x))^1, x, 8, (HypergeometricFunctions._₂F₁(2, (1/2)*(1 + n*p), (1/2)*(3 + n*p), -tan(e + f*x)^2)*tan(e + f*x)*(c*(d*tan(e + f*x))^p)^n)/(a*f*(1 + n*p)) - (im*HypergeometricFunctions._₂F₁(2, (1/2)*(2 + n*p), (1/2)*(4 + n*p), -tan(e + f*x)^2)*tan(e + f*x)^2*(c*(d*tan(e + f*x))^p)^n)/(a*f*(2 + n*p))]
    @test_int [(c*(d*tan(e + f*x))^p)^n/(a + im*a*tan(e + f*x))^2, x, 8, ((1 - 4*n*p + 2*n^2*p^2)*HypergeometricFunctions._₂F₁(1, 1 + n*p, 2 + n*p, (-im)*tan(e + f*x))*tan(e + f*x)*(c*(d*tan(e + f*x))^p)^n)/(8*a^2*f*(1 + n*p)) + (HypergeometricFunctions._₂F₁(1, 1 + n*p, 2 + n*p, im*tan(e + f*x))*tan(e + f*x)*(c*(d*tan(e + f*x))^p)^n)/(8*a^2*f*(1 + n*p)) + (tan(e + f*x)*(c*(d*tan(e + f*x))^p)^n)/(4*a^2*f*(1 + im*tan(e + f*x))^2) + ((2 - n*p)*tan(e + f*x)*(c*(d*tan(e + f*x))^p)^n)/(4*a^2*f*(1 + im*tan(e + f*x)))]


    #= ::Section::Closed:: =#
    #=Integrands*of*the*form*(a+b*tan(e+f*x))^m*(c*(d*tan(e+f*x))^p)^n=#


    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*(a+b*tan(e+f*x))^m*(c*(d*tan(e+f*x))^p)^n*with*n*and*p*symbolic=#


    @test_int [(a + b*tan(e + f*x))^m*(c*(d*tan(e + f*x))^p)^n, x, 8, (AppellF1(1 + n*p, -m, 1, 2 + n*p, -((b*tan(e + f*x))/a), (-im)*tan(e + f*x))*tan(e + f*x)*(c*(d*tan(e + f*x))^p)^n*(a + b*tan(e + f*x))^m)/((1 + (b*tan(e + f*x))/a)^m*(2*f*(1 + n*p))) + (AppellF1(1 + n*p, -m, 1, 2 + n*p, -((b*tan(e + f*x))/a), im*tan(e + f*x))*tan(e + f*x)*(c*(d*tan(e + f*x))^p)^n*(a + b*tan(e + f*x))^m)/((1 + (b*tan(e + f*x))/a)^m*(2*f*(1 + n*p)))]


    @test_int [(c*(d*tan(e + f*x))^p)^n*(a + b*tan(e + f*x))^3, x, 7, (3*a*b^2*tan(e + f*x)*(c*(d*tan(e + f*x))^p)^n)/(f*(1 + n*p)) + (a*(a^2 - 3*b^2)*HypergeometricFunctions._₂F₁(1, (1/2)*(1 + n*p), (1/2)*(3 + n*p), -tan(e + f*x)^2)*tan(e + f*x)*(c*(d*tan(e + f*x))^p)^n)/(f*(1 + n*p)) + (b^3*tan(e + f*x)^2*(c*(d*tan(e + f*x))^p)^n)/(f*(2 + n*p)) + (b*(3*a^2 - b^2)*HypergeometricFunctions._₂F₁(1, (1/2)*(2 + n*p), (1/2)*(4 + n*p), -tan(e + f*x)^2)*tan(e + f*x)^2*(c*(d*tan(e + f*x))^p)^n)/(f*(2 + n*p))]
    @test_int [(c*(d*tan(e + f*x))^p)^n*(a + b*tan(e + f*x))^2, x, 7, (b^2*tan(e + f*x)*(c*(d*tan(e + f*x))^p)^n)/(f*(1 + n*p)) + ((a^2 - b^2)*HypergeometricFunctions._₂F₁(1, (1/2)*(1 + n*p), (1/2)*(3 + n*p), -tan(e + f*x)^2)*tan(e + f*x)*(c*(d*tan(e + f*x))^p)^n)/(f*(1 + n*p)) + (2*a*b*HypergeometricFunctions._₂F₁(1, (1/2)*(2 + n*p), (1/2)*(4 + n*p), -tan(e + f*x)^2)*tan(e + f*x)^2*(c*(d*tan(e + f*x))^p)^n)/(f*(2 + n*p))]
    @test_int [(c*(d*tan(e + f*x))^p)^n*(a + b*tan(e + f*x))^1, x, 5, (a*HypergeometricFunctions._₂F₁(1, (1/2)*(1 + n*p), (1/2)*(3 + n*p), -tan(e + f*x)^2)*tan(e + f*x)*(c*(d*tan(e + f*x))^p)^n)/(f*(1 + n*p)) + (b*HypergeometricFunctions._₂F₁(1, (1/2)*(2 + n*p), (1/2)*(4 + n*p), -tan(e + f*x)^2)*tan(e + f*x)^2*(c*(d*tan(e + f*x))^p)^n)/(f*(2 + n*p))]
    @test_int [(c*(d*tan(e + f*x))^p)^n/(a + b*tan(e + f*x))^1, x, 8, (a*HypergeometricFunctions._₂F₁(1, (1/2)*(1 + n*p), (1/2)*(3 + n*p), -tan(e + f*x)^2)*tan(e + f*x)*(c*(d*tan(e + f*x))^p)^n)/((a^2 + b^2)*f*(1 + n*p)) + (b^2*HypergeometricFunctions._₂F₁(1, 1 + n*p, 2 + n*p, -((b*tan(e + f*x))/a))*tan(e + f*x)*(c*(d*tan(e + f*x))^p)^n)/(a*(a^2 + b^2)*f*(1 + n*p)) - (b*HypergeometricFunctions._₂F₁(1, (1/2)*(2 + n*p), (1/2)*(4 + n*p), -tan(e + f*x)^2)*tan(e + f*x)^2*(c*(d*tan(e + f*x))^p)^n)/((a^2 + b^2)*f*(2 + n*p))]
    @test_int [(c*(d*tan(e + f*x))^p)^n/(a + b*tan(e + f*x))^2, x, 9, ((a^2 - b^2)*HypergeometricFunctions._₂F₁(1, (1/2)*(1 + n*p), (1/2)*(3 + n*p), -tan(e + f*x)^2)*tan(e + f*x)*(c*(d*tan(e + f*x))^p)^n)/((a^2 + b^2)^2*f*(1 + n*p)) + (2*b^2*HypergeometricFunctions._₂F₁(1, 1 + n*p, 2 + n*p, -((b*tan(e + f*x))/a))*tan(e + f*x)*(c*(d*tan(e + f*x))^p)^n)/((a^2 + b^2)^2*f*(1 + n*p)) + (b^2*HypergeometricFunctions._₂F₁(2, 1 + n*p, 2 + n*p, -((b*tan(e + f*x))/a))*tan(e + f*x)*(c*(d*tan(e + f*x))^p)^n)/(a^2*(a^2 + b^2)*f*(1 + n*p)) - (2*a*b*HypergeometricFunctions._₂F₁(1, (1/2)*(2 + n*p), (1/2)*(4 + n*p), -tan(e + f*x)^2)*tan(e + f*x)^2*(c*(d*tan(e + f*x))^p)^n)/((a^2 + b^2)^2*f*(2 + n*p))]
    end
