@testset "4.5.1.2 (d sec)^n (a+b sec)^m" begin
    (a, b, c, d, e, f, m, n, x, ) = @variables a b c d e f m n x

    #= ::Package:: =#

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


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


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


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


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


    @test_int [sec(c + d*x)^4*(a + a*sec(c + d*x))^2, x, 7, (3*a^2*atanh(sin(c + d*x)))/(4*d) + (9*a^2*tan(c + d*x))/(5*d) + (3*a^2*sec(c + d*x)*tan(c + d*x))/(4*d) + (a^2*sec(c + d*x)^3*tan(c + d*x))/(2*d) + (a^2*sec(c + d*x)^4*tan(c + d*x))/(5*d) + (3*a^2*tan(c + d*x)^3)/(5*d)]
    @test_int [sec(c + d*x)^3*(a + a*sec(c + d*x))^2, x, 6, (7*a^2*atanh(sin(c + d*x)))/(8*d) + (2*a^2*tan(c + d*x))/d + (7*a^2*sec(c + d*x)*tan(c + d*x))/(8*d) + (a^2*sec(c + d*x)^3*tan(c + d*x))/(4*d) + (2*a^2*tan(c + d*x)^3)/(3*d)]
    @test_int [sec(c + d*x)^2*(a + a*sec(c + d*x))^2, x, 6, (a^2*atanh(sin(c + d*x)))/d + (5*a^2*tan(c + d*x))/(3*d) + (a^2*sec(c + d*x)*tan(c + d*x))/d + (a^2*sec(c + d*x)^2*tan(c + d*x))/(3*d)]
    @test_int [sec(c + d*x)^1*(a + a*sec(c + d*x))^2, x, 5, (3*a^2*atanh(sin(c + d*x)))/(2*d) + (2*a^2*tan(c + d*x))/d + (a^2*sec(c + d*x)*tan(c + d*x))/(2*d)]
    @test_int [sec(c + d*x)^0*(a + a*sec(c + d*x))^2, x, 4, a^2*x + (2*a^2*atanh(sin(c + d*x)))/d + (a^2*tan(c + d*x))/d]
    @test_int [cos(c + d*x)^1*(a + a*sec(c + d*x))^2, x, 4, 2*a^2*x + (a^2*atanh(sin(c + d*x)))/d + (a^2*sin(c + d*x))/d]
    @test_int [cos(c + d*x)^2*(a + a*sec(c + d*x))^2, x, 4, (3*a^2*x)/2 + (2*a^2*sin(c + d*x))/d + (a^2*cos(c + d*x)*sin(c + d*x))/(2*d)]
    @test_int [cos(c + d*x)^3*(a + a*sec(c + d*x))^2, x, 6, a^2*x + (2*a^2*sin(c + d*x))/d + (a^2*cos(c + d*x)*sin(c + d*x))/d - (a^2*sin(c + d*x)^3)/(3*d)]
    @test_int [cos(c + d*x)^4*(a + a*sec(c + d*x))^2, x, 6, (7*a^2*x)/8 + (2*a^2*sin(c + d*x))/d + (7*a^2*cos(c + d*x)*sin(c + d*x))/(8*d) + (a^2*cos(c + d*x)^3*sin(c + d*x))/(4*d) - (2*a^2*sin(c + d*x)^3)/(3*d)]
    @test_int [cos(c + d*x)^5*(a + a*sec(c + d*x))^2, x, 8, (3*a^2*x)/4 + (2*a^2*sin(c + d*x))/d + (3*a^2*cos(c + d*x)*sin(c + d*x))/(4*d) + (a^2*cos(c + d*x)^3*sin(c + d*x))/(2*d) - (a^2*sin(c + d*x)^3)/d + (a^2*sin(c + d*x)^5)/(5*d)]


    @test_int [sec(c + d*x)^3*(a + a*sec(c + d*x))^3, x, 11, (13*a^3*atanh(sin(c + d*x)))/(8*d) + (4*a^3*tan(c + d*x))/d + (13*a^3*sec(c + d*x)*tan(c + d*x))/(8*d) + (3*a^3*sec(c + d*x)^3*tan(c + d*x))/(4*d) + (5*a^3*tan(c + d*x)^3)/(3*d) + (a^3*tan(c + d*x)^5)/(5*d)]
    @test_int [sec(c + d*x)^2*(a + a*sec(c + d*x))^3, x, 11, (15*a^3*atanh(sin(c + d*x)))/(8*d) + (4*a^3*tan(c + d*x))/d + (15*a^3*sec(c + d*x)*tan(c + d*x))/(8*d) + (a^3*sec(c + d*x)^3*tan(c + d*x))/(4*d) + (a^3*tan(c + d*x)^3)/d]
    @test_int [sec(c + d*x)^1*(a + a*sec(c + d*x))^3, x, 9, (5*a^3*atanh(sin(c + d*x)))/(2*d) + (4*a^3*tan(c + d*x))/d + (3*a^3*sec(c + d*x)*tan(c + d*x))/(2*d) + (a^3*tan(c + d*x)^3)/(3*d)]
    @test_int [sec(c + d*x)^0*(a + a*sec(c + d*x))^3, x, 5, a^3*x + (7*a^3*atanh(sin(c + d*x)))/(2*d) + (5*a^3*tan(c + d*x))/(2*d) + ((a^3 + a^3*sec(c + d*x))*tan(c + d*x))/(2*d)]
    @test_int [cos(c + d*x)^1*(a + a*sec(c + d*x))^3, x, 6, 3*a^3*x + (3*a^3*atanh(sin(c + d*x)))/d + (a^3*sin(c + d*x))/d + (a^3*tan(c + d*x))/d]
    @test_int [cos(c + d*x)^2*(a + a*sec(c + d*x))^3, x, 6, (7*a^3*x)/2 + (a^3*atanh(sin(c + d*x)))/d + (3*a^3*sin(c + d*x))/d + (a^3*cos(c + d*x)*sin(c + d*x))/(2*d)]
    @test_int [cos(c + d*x)^3*(a + a*sec(c + d*x))^3, x, 7, (5*a^3*x)/2 + (4*a^3*sin(c + d*x))/d + (3*a^3*cos(c + d*x)*sin(c + d*x))/(2*d) - (a^3*sin(c + d*x)^3)/(3*d)]
    @test_int [cos(c + d*x)^4*(a + a*sec(c + d*x))^3, x, 10, (15*a^3*x)/8 + (4*a^3*sin(c + d*x))/d + (15*a^3*cos(c + d*x)*sin(c + d*x))/(8*d) + (a^3*cos(c + d*x)^3*sin(c + d*x))/(4*d) - (a^3*sin(c + d*x)^3)/d]
    @test_int [cos(c + d*x)^5*(a + a*sec(c + d*x))^3, x, 11, (13*a^3*x)/8 + (4*a^3*sin(c + d*x))/d + (13*a^3*cos(c + d*x)*sin(c + d*x))/(8*d) + (3*a^3*cos(c + d*x)^3*sin(c + d*x))/(4*d) - (5*a^3*sin(c + d*x)^3)/(3*d) + (a^3*sin(c + d*x)^5)/(5*d)]
    @test_int [cos(c + d*x)^6*(a + a*sec(c + d*x))^3, x, 13, (23*a^3*x)/16 + (4*a^3*sin(c + d*x))/d + (23*a^3*cos(c + d*x)*sin(c + d*x))/(16*d) + (23*a^3*cos(c + d*x)^3*sin(c + d*x))/(24*d) + (a^3*cos(c + d*x)^5*sin(c + d*x))/(6*d) - (7*a^3*sin(c + d*x)^3)/(3*d) + (3*a^3*sin(c + d*x)^5)/(5*d)]


    @test_int [sec(c + d*x)^3*(a + a*sec(c + d*x))^4, x, 15, (49*a^4*atanh(sin(c + d*x)))/(16*d) + (8*a^4*tan(c + d*x))/d + (49*a^4*sec(c + d*x)*tan(c + d*x))/(16*d) + (41*a^4*sec(c + d*x)^3*tan(c + d*x))/(24*d) + (a^4*sec(c + d*x)^5*tan(c + d*x))/(6*d) + (4*a^4*tan(c + d*x)^3)/d + (4*a^4*tan(c + d*x)^5)/(5*d)]
    @test_int [sec(c + d*x)^2*(a + a*sec(c + d*x))^4, x, 13, (7*a^4*atanh(sin(c + d*x)))/(2*d) + (8*a^4*tan(c + d*x))/d + (7*a^4*sec(c + d*x)*tan(c + d*x))/(2*d) + (a^4*sec(c + d*x)^3*tan(c + d*x))/d + (8*a^4*tan(c + d*x)^3)/(3*d) + (a^4*tan(c + d*x)^5)/(5*d)]
    @test_int [sec(c + d*x)^1*(a + a*sec(c + d*x))^4, x, 12, (35*a^4*atanh(sin(c + d*x)))/(8*d) + (8*a^4*tan(c + d*x))/d + (27*a^4*sec(c + d*x)*tan(c + d*x))/(8*d) + (a^4*sec(c + d*x)^3*tan(c + d*x))/(4*d) + (4*a^4*tan(c + d*x)^3)/(3*d)]
    @test_int [sec(c + d*x)^0*(a + a*sec(c + d*x))^4, x, 6, a^4*x + (6*a^4*atanh(sin(c + d*x)))/d + (5*a^4*tan(c + d*x))/d + ((a^2 + a^2*sec(c + d*x))^2*tan(c + d*x))/(3*d) + (4*(a^4 + a^4*sec(c + d*x))*tan(c + d*x))/(3*d)]
    @test_int [cos(c + d*x)^1*(a + a*sec(c + d*x))^4, x, 8, 4*a^4*x + (13*a^4*atanh(sin(c + d*x)))/(2*d) + (a^4*sin(c + d*x))/d + (4*a^4*tan(c + d*x))/d + (a^4*sec(c + d*x)*tan(c + d*x))/(2*d)]
    @test_int [cos(c + d*x)^2*(a + a*sec(c + d*x))^4, x, 8, (13*a^4*x)/2 + (4*a^4*atanh(sin(c + d*x)))/d + (4*a^4*sin(c + d*x))/d + (a^4*cos(c + d*x)*sin(c + d*x))/(2*d) + (a^4*tan(c + d*x))/d]
    @test_int [cos(c + d*x)^3*(a + a*sec(c + d*x))^4, x, 8, 6*a^4*x + (a^4*atanh(sin(c + d*x)))/d + (7*a^4*sin(c + d*x))/d + (2*a^4*cos(c + d*x)*sin(c + d*x))/d - (a^4*sin(c + d*x)^3)/(3*d)]
    @test_int [cos(c + d*x)^4*(a + a*sec(c + d*x))^4, x, 10, (35*a^4*x)/8 + (8*a^4*sin(c + d*x))/d + (27*a^4*cos(c + d*x)*sin(c + d*x))/(8*d) + (a^4*cos(c + d*x)^3*sin(c + d*x))/(4*d) - (4*a^4*sin(c + d*x)^3)/(3*d)]
    @test_int [cos(c + d*x)^5*(a + a*sec(c + d*x))^4, x, 12, (7*a^4*x)/2 + (8*a^4*sin(c + d*x))/d + (7*a^4*cos(c + d*x)*sin(c + d*x))/(2*d) + (a^4*cos(c + d*x)^3*sin(c + d*x))/d - (8*a^4*sin(c + d*x)^3)/(3*d) + (a^4*sin(c + d*x)^5)/(5*d)]
    @test_int [cos(c + d*x)^6*(a + a*sec(c + d*x))^4, x, 15, (49*a^4*x)/16 + (8*a^4*sin(c + d*x))/d + (49*a^4*cos(c + d*x)*sin(c + d*x))/(16*d) + (41*a^4*cos(c + d*x)^3*sin(c + d*x))/(24*d) + (a^4*cos(c + d*x)^5*sin(c + d*x))/(6*d) - (4*a^4*sin(c + d*x)^3)/d + (4*a^4*sin(c + d*x)^5)/(5*d)]
    @test_int [cos(c + d*x)^7*(a + a*sec(c + d*x))^4, x, 15, (11*a^4*x)/4 + (8*a^4*sin(c + d*x))/d + (11*a^4*cos(c + d*x)*sin(c + d*x))/(4*d) + (11*a^4*cos(c + d*x)^3*sin(c + d*x))/(6*d) + (2*a^4*cos(c + d*x)^5*sin(c + d*x))/(3*d) - (16*a^4*sin(c + d*x)^3)/(3*d) + (9*a^4*sin(c + d*x)^5)/(5*d) - (a^4*sin(c + d*x)^7)/(7*d)]


    @test_int [sec(c + d*x)^3*(a + a*sec(c + d*x))^5, x, 17, (93*a^5*atanh(sin(c + d*x)))/(16*d) + (16*a^5*tan(c + d*x))/d + (93*a^5*sec(c + d*x)*tan(c + d*x))/(16*d) + (85*a^5*sec(c + d*x)^3*tan(c + d*x))/(24*d) + (5*a^5*sec(c + d*x)^5*tan(c + d*x))/(6*d) + (28*a^5*tan(c + d*x)^3)/(3*d) + (13*a^5*tan(c + d*x)^5)/(5*d) + (a^5*tan(c + d*x)^7)/(7*d)]


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


    @test_int [sec(c + d*x)^5/(a + a*sec(c + d*x)), x, 6, -((3*atanh(sin(c + d*x)))/(2*a*d)) + (4*tan(c + d*x))/(a*d) - (3*sec(c + d*x)*tan(c + d*x))/(2*a*d) - (sec(c + d*x)^3*tan(c + d*x))/(d*(a + a*sec(c + d*x))) + (4*tan(c + d*x)^3)/(3*a*d)]
    @test_int [sec(c + d*x)^4/(a + a*sec(c + d*x)), x, 6, (3*atanh(sin(c + d*x)))/(2*a*d) - (2*tan(c + d*x))/(a*d) + (3*sec(c + d*x)*tan(c + d*x))/(2*a*d) - (sec(c + d*x)^2*tan(c + d*x))/(d*(a + a*sec(c + d*x)))]
    @test_int [sec(c + d*x)^3/(a + a*sec(c + d*x)), x, 4, -(atanh(sin(c + d*x))/(a*d)) + tan(c + d*x)/(d*(a + a*sec(c + d*x))) + tan(c + d*x)/(a*d)]
    @test_int [sec(c + d*x)^2/(a + a*sec(c + d*x)), x, 3, atanh(sin(c + d*x))/(a*d) - tan(c + d*x)/(d*(a + a*sec(c + d*x)))]
    @test_int [sec(c + d*x)^1/(a + a*sec(c + d*x)), x, 1, tan(c + d*x)/(d*(a + a*sec(c + d*x)))]
    @test_int [sec(c + d*x)^0/(a + a*sec(c + d*x)), x, 2, x/a - tan(c + d*x)/(d*(a + a*sec(c + d*x)))]
    @test_int [cos(c + d*x)^1/(a + a*sec(c + d*x)), x, 4, -(x/a) + (2*sin(c + d*x))/(a*d) - sin(c + d*x)/(d*(a + a*sec(c + d*x)))]
    @test_int [cos(c + d*x)^2/(a + a*sec(c + d*x)), x, 5, (3*x)/(2*a) - (2*sin(c + d*x))/(a*d) + (3*cos(c + d*x)*sin(c + d*x))/(2*a*d) - (cos(c + d*x)*sin(c + d*x))/(d*(a + a*sec(c + d*x)))]
    @test_int [cos(c + d*x)^3/(a + a*sec(c + d*x)), x, 6, -((3*x)/(2*a)) + (4*sin(c + d*x))/(a*d) - (3*cos(c + d*x)*sin(c + d*x))/(2*a*d) - (cos(c + d*x)^2*sin(c + d*x))/(d*(a + a*sec(c + d*x))) - (4*sin(c + d*x)^3)/(3*a*d)]
    @test_int [cos(c + d*x)^4/(a + a*sec(c + d*x)), x, 7, (15*x)/(8*a) - (4*sin(c + d*x))/(a*d) + (15*cos(c + d*x)*sin(c + d*x))/(8*a*d) + (5*cos(c + d*x)^3*sin(c + d*x))/(4*a*d) - (cos(c + d*x)^3*sin(c + d*x))/(d*(a + a*sec(c + d*x))) + (4*sin(c + d*x)^3)/(3*a*d)]


    @test_int [sec(c + d*x)^5/(a + a*sec(c + d*x))^2, x, 7, (7*atanh(sin(c + d*x)))/(2*a^2*d) - (16*tan(c + d*x))/(3*a^2*d) + (7*sec(c + d*x)*tan(c + d*x))/(2*a^2*d) - (8*sec(c + d*x)^2*tan(c + d*x))/(3*a^2*d*(1 + sec(c + d*x))) - (sec(c + d*x)^3*tan(c + d*x))/(3*d*(a + a*sec(c + d*x))^2)]
    @test_int [sec(c + d*x)^4/(a + a*sec(c + d*x))^2, x, 6, -((2*atanh(sin(c + d*x)))/(a^2*d)) + (4*tan(c + d*x))/(3*a^2*d) + (2*tan(c + d*x))/(a^2*d*(1 + sec(c + d*x))) - (sec(c + d*x)^2*tan(c + d*x))/(3*d*(a + a*sec(c + d*x))^2)]
    @test_int [sec(c + d*x)^3/(a + a*sec(c + d*x))^2, x, 4, atanh(sin(c + d*x))/(a^2*d) - (5*tan(c + d*x))/(3*a^2*d*(1 + sec(c + d*x))) + tan(c + d*x)/(3*d*(a + a*sec(c + d*x))^2)]
    @test_int [sec(c + d*x)^2/(a + a*sec(c + d*x))^2, x, 2, -(tan(c + d*x)/(3*d*(a + a*sec(c + d*x))^2)) + (2*tan(c + d*x))/(3*d*(a^2 + a^2*sec(c + d*x)))]
    @test_int [sec(c + d*x)^1/(a + a*sec(c + d*x))^2, x, 2, tan(c + d*x)/(3*d*(a + a*sec(c + d*x))^2) + tan(c + d*x)/(3*d*(a^2 + a^2*sec(c + d*x)))]
    @test_int [sec(c + d*x)^0/(a + a*sec(c + d*x))^2, x, 3, x/a^2 - (4*tan(c + d*x))/(3*a^2*d*(1 + sec(c + d*x))) - tan(c + d*x)/(3*d*(a + a*sec(c + d*x))^2)]
    @test_int [cos(c + d*x)^1/(a + a*sec(c + d*x))^2, x, 5, -((2*x)/a^2) + (10*sin(c + d*x))/(3*a^2*d) - (2*sin(c + d*x))/(a^2*d*(1 + sec(c + d*x))) - sin(c + d*x)/(3*d*(a + a*sec(c + d*x))^2)]
    @test_int [cos(c + d*x)^2/(a + a*sec(c + d*x))^2, x, 6, (7*x)/(2*a^2) - (16*sin(c + d*x))/(3*a^2*d) + (7*cos(c + d*x)*sin(c + d*x))/(2*a^2*d) - (8*cos(c + d*x)*sin(c + d*x))/(3*a^2*d*(1 + sec(c + d*x))) - (cos(c + d*x)*sin(c + d*x))/(3*d*(a + a*sec(c + d*x))^2)]
    @test_int [cos(c + d*x)^3/(a + a*sec(c + d*x))^2, x, 7, -((5*x)/a^2) + (12*sin(c + d*x))/(a^2*d) - (5*cos(c + d*x)*sin(c + d*x))/(a^2*d) - (10*cos(c + d*x)^2*sin(c + d*x))/(3*a^2*d*(1 + sec(c + d*x))) - (cos(c + d*x)^2*sin(c + d*x))/(3*d*(a + a*sec(c + d*x))^2) - (4*sin(c + d*x)^3)/(a^2*d)]


    @test_int [sec(c + d*x)^6/(a + a*sec(c + d*x))^3, x, 8, (13*atanh(sin(c + d*x)))/(2*a^3*d) - (152*tan(c + d*x))/(15*a^3*d) + (13*sec(c + d*x)*tan(c + d*x))/(2*a^3*d) - (sec(c + d*x)^4*tan(c + d*x))/(5*d*(a + a*sec(c + d*x))^3) - (11*sec(c + d*x)^3*tan(c + d*x))/(15*a*d*(a + a*sec(c + d*x))^2) - (76*sec(c + d*x)^2*tan(c + d*x))/(15*d*(a^3 + a^3*sec(c + d*x)))]
    @test_int [sec(c + d*x)^5/(a + a*sec(c + d*x))^3, x, 7, -((3*atanh(sin(c + d*x)))/(a^3*d)) + (9*tan(c + d*x))/(5*a^3*d) - (sec(c + d*x)^3*tan(c + d*x))/(5*d*(a + a*sec(c + d*x))^3) - (3*sec(c + d*x)^2*tan(c + d*x))/(5*a*d*(a + a*sec(c + d*x))^2) + (3*tan(c + d*x))/(d*(a^3 + a^3*sec(c + d*x)))]
    @test_int [sec(c + d*x)^4/(a + a*sec(c + d*x))^3, x, 5, atanh(sin(c + d*x))/(a^3*d) - (sec(c + d*x)^2*tan(c + d*x))/(5*d*(a + a*sec(c + d*x))^3) + (7*tan(c + d*x))/(15*a*d*(a + a*sec(c + d*x))^2) - (29*tan(c + d*x))/(15*d*(a^3 + a^3*sec(c + d*x)))]
    @test_int [sec(c + d*x)^3/(a + a*sec(c + d*x))^3, x, 3, tan(c + d*x)/(5*d*(a + a*sec(c + d*x))^3) - (8*tan(c + d*x))/(15*a*d*(a + a*sec(c + d*x))^2) + (7*tan(c + d*x))/(15*d*(a^3 + a^3*sec(c + d*x)))]
    @test_int [sec(c + d*x)^2/(a + a*sec(c + d*x))^3, x, 3, -(tan(c + d*x)/(5*d*(a + a*sec(c + d*x))^3)) + tan(c + d*x)/(5*a*d*(a + a*sec(c + d*x))^2) + tan(c + d*x)/(5*d*(a^3 + a^3*sec(c + d*x)))]
    @test_int [sec(c + d*x)^1/(a + a*sec(c + d*x))^3, x, 3, tan(c + d*x)/(5*d*(a + a*sec(c + d*x))^3) + (2*tan(c + d*x))/(15*a*d*(a + a*sec(c + d*x))^2) + (2*tan(c + d*x))/(15*d*(a^3 + a^3*sec(c + d*x)))]
    @test_int [sec(c + d*x)^0/(a + a*sec(c + d*x))^3, x, 4, x/a^3 - tan(c + d*x)/(5*d*(a + a*sec(c + d*x))^3) - (7*tan(c + d*x))/(15*a*d*(a + a*sec(c + d*x))^2) - (22*tan(c + d*x))/(15*d*(a^3 + a^3*sec(c + d*x)))]
    @test_int [cos(c + d*x)^1/(a + a*sec(c + d*x))^3, x, 6, -((3*x)/a^3) + (24*sin(c + d*x))/(5*a^3*d) - sin(c + d*x)/(5*d*(a + a*sec(c + d*x))^3) - (3*sin(c + d*x))/(5*a*d*(a + a*sec(c + d*x))^2) - (3*sin(c + d*x))/(d*(a^3 + a^3*sec(c + d*x)))]
    @test_int [cos(c + d*x)^2/(a + a*sec(c + d*x))^3, x, 7, (13*x)/(2*a^3) - (152*sin(c + d*x))/(15*a^3*d) + (13*cos(c + d*x)*sin(c + d*x))/(2*a^3*d) - (cos(c + d*x)*sin(c + d*x))/(5*d*(a + a*sec(c + d*x))^3) - (11*cos(c + d*x)*sin(c + d*x))/(15*a*d*(a + a*sec(c + d*x))^2) - (76*cos(c + d*x)*sin(c + d*x))/(15*d*(a^3 + a^3*sec(c + d*x)))]


    @test_int [sec(c + d*x)^7/(a + a*sec(c + d*x))^4, x, 9, (21*atanh(sin(c + d*x)))/(2*a^4*d) - (576*tan(c + d*x))/(35*a^4*d) + (21*sec(c + d*x)*tan(c + d*x))/(2*a^4*d) - (43*sec(c + d*x)^3*tan(c + d*x))/(35*a^4*d*(1 + sec(c + d*x))^2) - (288*sec(c + d*x)^2*tan(c + d*x))/(35*a^4*d*(1 + sec(c + d*x))) - (sec(c + d*x)^5*tan(c + d*x))/(7*d*(a + a*sec(c + d*x))^4) - (2*sec(c + d*x)^4*tan(c + d*x))/(5*a*d*(a + a*sec(c + d*x))^3)]
    @test_int [sec(c + d*x)^6/(a + a*sec(c + d*x))^4, x, 8, -((4*atanh(sin(c + d*x)))/(a^4*d)) + (244*tan(c + d*x))/(105*a^4*d) - (88*sec(c + d*x)^2*tan(c + d*x))/(105*a^4*d*(1 + sec(c + d*x))^2) + (4*tan(c + d*x))/(a^4*d*(1 + sec(c + d*x))) - (sec(c + d*x)^4*tan(c + d*x))/(7*d*(a + a*sec(c + d*x))^4) - (12*sec(c + d*x)^3*tan(c + d*x))/(35*a*d*(a + a*sec(c + d*x))^3)]
    @test_int [sec(c + d*x)^5/(a + a*sec(c + d*x))^4, x, 6, atanh(sin(c + d*x))/(a^4*d) + (11*tan(c + d*x))/(21*a^4*d*(1 + sec(c + d*x))^2) - (43*tan(c + d*x))/(21*a^4*d*(1 + sec(c + d*x))) - (sec(c + d*x)^3*tan(c + d*x))/(7*d*(a + a*sec(c + d*x))^4) - (2*sec(c + d*x)^2*tan(c + d*x))/(7*a*d*(a + a*sec(c + d*x))^3)]
    @test_int [sec(c + d*x)^4/(a + a*sec(c + d*x))^4, x, 4, (sec(c + d*x)^3*tan(c + d*x))/(7*d*(a + a*sec(c + d*x))^4) + (3*tan(c + d*x))/(35*a*d*(a + a*sec(c + d*x))^3) - (8*tan(c + d*x))/(35*d*(a^2 + a^2*sec(c + d*x))^2) + tan(c + d*x)/(5*d*(a^4 + a^4*sec(c + d*x)))]
    @test_int [sec(c + d*x)^3/(a + a*sec(c + d*x))^4, x, 4, tan(c + d*x)/(7*d*(a + a*sec(c + d*x))^4) - (11*tan(c + d*x))/(35*a*d*(a + a*sec(c + d*x))^3) + (13*tan(c + d*x))/(105*d*(a^2 + a^2*sec(c + d*x))^2) + (13*tan(c + d*x))/(105*d*(a^4 + a^4*sec(c + d*x)))]
    @test_int [sec(c + d*x)^2/(a + a*sec(c + d*x))^4, x, 4, -(tan(c + d*x)/(7*d*(a + a*sec(c + d*x))^4)) + (4*tan(c + d*x))/(35*a*d*(a + a*sec(c + d*x))^3) + (8*tan(c + d*x))/(105*d*(a^2 + a^2*sec(c + d*x))^2) + (8*tan(c + d*x))/(105*d*(a^4 + a^4*sec(c + d*x)))]
    @test_int [sec(c + d*x)^1/(a + a*sec(c + d*x))^4, x, 4, tan(c + d*x)/(7*d*(a + a*sec(c + d*x))^4) + (3*tan(c + d*x))/(35*a*d*(a + a*sec(c + d*x))^3) + (2*tan(c + d*x))/(35*d*(a^2 + a^2*sec(c + d*x))^2) + (2*tan(c + d*x))/(35*d*(a^4 + a^4*sec(c + d*x)))]
    @test_int [sec(c + d*x)^0/(a + a*sec(c + d*x))^4, x, 5, x/a^4 - (11*tan(c + d*x))/(21*a^4*d*(1 + sec(c + d*x))^2) - (32*tan(c + d*x))/(21*a^4*d*(1 + sec(c + d*x))) - tan(c + d*x)/(7*d*(a + a*sec(c + d*x))^4) - (2*tan(c + d*x))/(7*a*d*(a + a*sec(c + d*x))^3)]
    @test_int [cos(c + d*x)^1/(a + a*sec(c + d*x))^4, x, 7, -((4*x)/a^4) + (664*sin(c + d*x))/(105*a^4*d) - (88*sin(c + d*x))/(105*a^4*d*(1 + sec(c + d*x))^2) - (4*sin(c + d*x))/(a^4*d*(1 + sec(c + d*x))) - sin(c + d*x)/(7*d*(a + a*sec(c + d*x))^4) - (12*sin(c + d*x))/(35*a*d*(a + a*sec(c + d*x))^3)]
    @test_int [cos(c + d*x)^2/(a + a*sec(c + d*x))^4, x, 8, (21*x)/(2*a^4) - (576*sin(c + d*x))/(35*a^4*d) + (21*cos(c + d*x)*sin(c + d*x))/(2*a^4*d) - (43*cos(c + d*x)*sin(c + d*x))/(35*a^4*d*(1 + sec(c + d*x))^2) - (288*cos(c + d*x)*sin(c + d*x))/(35*a^4*d*(1 + sec(c + d*x))) - (cos(c + d*x)*sin(c + d*x))/(7*d*(a + a*sec(c + d*x))^4) - (2*cos(c + d*x)*sin(c + d*x))/(5*a*d*(a + a*sec(c + d*x))^3)]


    @test_int [sec(c + d*x)^7/(a + a*sec(c + d*x))^5, x, 9, -((5*atanh(sin(c + d*x)))/(a^5*d)) + (181*tan(c + d*x))/(63*a^5*d) - (sec(c + d*x)^5*tan(c + d*x))/(9*d*(a + a*sec(c + d*x))^5) - (5*sec(c + d*x)^4*tan(c + d*x))/(21*a*d*(a + a*sec(c + d*x))^4) - (29*sec(c + d*x)^3*tan(c + d*x))/(63*a^2*d*(a + a*sec(c + d*x))^3) - (67*sec(c + d*x)^2*tan(c + d*x))/(63*a^3*d*(a + a*sec(c + d*x))^2) + (5*tan(c + d*x))/(d*(a^5 + a^5*sec(c + d*x)))]
    @test_int [sec(c + d*x)^6/(a + a*sec(c + d*x))^5, x, 7, atanh(sin(c + d*x))/(a^5*d) - (sec(c + d*x)^4*tan(c + d*x))/(9*d*(a + a*sec(c + d*x))^5) - (13*sec(c + d*x)^3*tan(c + d*x))/(63*a*d*(a + a*sec(c + d*x))^4) - (34*sec(c + d*x)^2*tan(c + d*x))/(105*a^2*d*(a + a*sec(c + d*x))^3) + (173*tan(c + d*x))/(315*a^3*d*(a + a*sec(c + d*x))^2) - (661*tan(c + d*x))/(315*d*(a^5 + a^5*sec(c + d*x)))]
    @test_int [sec(c + d*x)^5/(a + a*sec(c + d*x))^5, x, 5, (sec(c + d*x)^4*tan(c + d*x))/(9*d*(a + a*sec(c + d*x))^5) + (4*sec(c + d*x)^3*tan(c + d*x))/(63*a*d*(a + a*sec(c + d*x))^4) + (4*tan(c + d*x))/(105*a^2*d*(a + a*sec(c + d*x))^3) - (32*tan(c + d*x))/(315*a*d*(a^2 + a^2*sec(c + d*x))^2) + (4*tan(c + d*x))/(45*d*(a^5 + a^5*sec(c + d*x)))]
    @test_int [sec(c + d*x)^4/(a + a*sec(c + d*x))^5, x, 5, -((sec(c + d*x)^4*tan(c + d*x))/(9*d*(a + a*sec(c + d*x))^5)) + (5*sec(c + d*x)^3*tan(c + d*x))/(63*a*d*(a + a*sec(c + d*x))^4) + tan(c + d*x)/(21*a^2*d*(a + a*sec(c + d*x))^3) - (8*tan(c + d*x))/(63*a*d*(a^2 + a^2*sec(c + d*x))^2) + tan(c + d*x)/(9*d*(a^5 + a^5*sec(c + d*x)))]
    @test_int [sec(c + d*x)^3/(a + a*sec(c + d*x))^5, x, 5, tan(c + d*x)/(9*d*(a + a*sec(c + d*x))^5) - (2*tan(c + d*x))/(9*a*d*(a + a*sec(c + d*x))^4) + tan(c + d*x)/(15*a^2*d*(a + a*sec(c + d*x))^3) + (2*tan(c + d*x))/(45*a^3*d*(a + a*sec(c + d*x))^2) + (2*tan(c + d*x))/(45*d*(a^5 + a^5*sec(c + d*x)))]
    @test_int [sec(c + d*x)^2/(a + a*sec(c + d*x))^5, x, 5, -(tan(c + d*x)/(9*d*(a + a*sec(c + d*x))^5)) + (5*tan(c + d*x))/(63*a*d*(a + a*sec(c + d*x))^4) + tan(c + d*x)/(21*a^2*d*(a + a*sec(c + d*x))^3) + (2*tan(c + d*x))/(63*a*d*(a^2 + a^2*sec(c + d*x))^2) + (2*tan(c + d*x))/(63*d*(a^5 + a^5*sec(c + d*x)))]
    @test_int [sec(c + d*x)^1/(a + a*sec(c + d*x))^5, x, 5, tan(c + d*x)/(9*d*(a + a*sec(c + d*x))^5) + (4*tan(c + d*x))/(63*a*d*(a + a*sec(c + d*x))^4) + (4*tan(c + d*x))/(105*a^2*d*(a + a*sec(c + d*x))^3) + (8*tan(c + d*x))/(315*a*d*(a^2 + a^2*sec(c + d*x))^2) + (8*tan(c + d*x))/(315*d*(a^5 + a^5*sec(c + d*x)))]
    @test_int [sec(c + d*x)^0/(a + a*sec(c + d*x))^5, x, 6, x/a^5 - tan(c + d*x)/(9*d*(a + a*sec(c + d*x))^5) - (13*tan(c + d*x))/(63*a*d*(a + a*sec(c + d*x))^4) - (34*tan(c + d*x))/(105*a^2*d*(a + a*sec(c + d*x))^3) - (173*tan(c + d*x))/(315*a^3*d*(a + a*sec(c + d*x))^2) - (488*tan(c + d*x))/(315*d*(a^5 + a^5*sec(c + d*x)))]
    @test_int [cos(c + d*x)^1/(a + a*sec(c + d*x))^5, x, 8, -((5*x)/a^5) + (496*sin(c + d*x))/(63*a^5*d) - sin(c + d*x)/(9*d*(a + a*sec(c + d*x))^5) - (5*sin(c + d*x))/(21*a*d*(a + a*sec(c + d*x))^4) - (29*sin(c + d*x))/(63*a^2*d*(a + a*sec(c + d*x))^3) - (67*sin(c + d*x))/(63*a^3*d*(a + a*sec(c + d*x))^2) - (5*sin(c + d*x))/(d*(a^5 + a^5*sec(c + d*x)))]
    @test_int [cos(c + d*x)^2/(a + a*sec(c + d*x))^5, x, 9, (31*x)/(2*a^5) - (7664*sin(c + d*x))/(315*a^5*d) + (31*cos(c + d*x)*sin(c + d*x))/(2*a^5*d) - (cos(c + d*x)*sin(c + d*x))/(9*d*(a + a*sec(c + d*x))^5) - (17*cos(c + d*x)*sin(c + d*x))/(63*a*d*(a + a*sec(c + d*x))^4) - (28*cos(c + d*x)*sin(c + d*x))/(45*a^2*d*(a + a*sec(c + d*x))^3) - (577*cos(c + d*x)*sin(c + d*x))/(315*a^3*d*(a + a*sec(c + d*x))^2) - (3832*cos(c + d*x)*sin(c + d*x))/(315*d*(a^5 + a^5*sec(c + d*x)))]


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


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


    @test_int [sec(c + d*x)^4*sqrt(a + a*sec(c + d*x)), x, 4, (4*a*tan(c + d*x))/(5*d*sqrt(a + a*sec(c + d*x))) + (2*a*sec(c + d*x)^3*tan(c + d*x))/(7*d*sqrt(a + a*sec(c + d*x))) - (8*sqrt(a + a*sec(c + d*x))*tan(c + d*x))/(35*d) + (12*(a + a*sec(c + d*x))^(3/2)*tan(c + d*x))/(35*a*d)]
    @test_int [sec(c + d*x)^3*sqrt(a + a*sec(c + d*x)), x, 3, (14*a*tan(c + d*x))/(15*d*sqrt(a + a*sec(c + d*x))) - (4*sqrt(a + a*sec(c + d*x))*tan(c + d*x))/(15*d) + (2*(a + a*sec(c + d*x))^(3/2)*tan(c + d*x))/(5*a*d)]
    @test_int [sec(c + d*x)^2*sqrt(a + a*sec(c + d*x)), x, 2, (2*a*tan(c + d*x))/(3*d*sqrt(a + a*sec(c + d*x))) + (2*sqrt(a + a*sec(c + d*x))*tan(c + d*x))/(3*d)]
    @test_int [sec(c + d*x)^1*sqrt(a + a*sec(c + d*x)), x, 1, (2*a*tan(c + d*x))/(d*sqrt(a + a*sec(c + d*x)))]
    @test_int [sec(c + d*x)^0*sqrt(a + a*sec(c + d*x)), x, 2, (2*sqrt(a)*atan((sqrt(a)*tan(c + d*x))/sqrt(a + a*sec(c + d*x))))/d]
    @test_int [cos(c + d*x)^1*sqrt(a + a*sec(c + d*x)), x, 3, (sqrt(a)*atan((sqrt(a)*tan(c + d*x))/sqrt(a + a*sec(c + d*x))))/d + (a*sin(c + d*x))/(d*sqrt(a + a*sec(c + d*x)))]
    @test_int [cos(c + d*x)^2*sqrt(a + a*sec(c + d*x)), x, 4, (3*sqrt(a)*atan((sqrt(a)*tan(c + d*x))/sqrt(a + a*sec(c + d*x))))/(4*d) + (3*a*sin(c + d*x))/(4*d*sqrt(a + a*sec(c + d*x))) + (a*cos(c + d*x)*sin(c + d*x))/(2*d*sqrt(a + a*sec(c + d*x)))]
    @test_int [cos(c + d*x)^3*sqrt(a + a*sec(c + d*x)), x, 5, (5*sqrt(a)*atan((sqrt(a)*tan(c + d*x))/sqrt(a + a*sec(c + d*x))))/(8*d) + (5*a*sin(c + d*x))/(8*d*sqrt(a + a*sec(c + d*x))) + (5*a*cos(c + d*x)*sin(c + d*x))/(12*d*sqrt(a + a*sec(c + d*x))) + (a*cos(c + d*x)^2*sin(c + d*x))/(3*d*sqrt(a + a*sec(c + d*x)))]
    @test_int [cos(c + d*x)^4*sqrt(a + a*sec(c + d*x)), x, 6, (35*sqrt(a)*atan((sqrt(a)*tan(c + d*x))/sqrt(a + a*sec(c + d*x))))/(64*d) + (35*a*sin(c + d*x))/(64*d*sqrt(a + a*sec(c + d*x))) + (35*a*cos(c + d*x)*sin(c + d*x))/(96*d*sqrt(a + a*sec(c + d*x))) + (7*a*cos(c + d*x)^2*sin(c + d*x))/(24*d*sqrt(a + a*sec(c + d*x))) + (a*cos(c + d*x)^3*sin(c + d*x))/(4*d*sqrt(a + a*sec(c + d*x)))]


    @test_int [sec(c + d*x)^4*(a + a*sec(c + d*x))^(3/2), x, 6, (68*a^2*tan(c + d*x))/(45*d*sqrt(a + a*sec(c + d*x))) + (34*a^2*sec(c + d*x)^3*tan(c + d*x))/(63*d*sqrt(a + a*sec(c + d*x))) + (2*a^2*sec(c + d*x)^4*tan(c + d*x))/(9*d*sqrt(a + a*sec(c + d*x))) - (136*a*sqrt(a + a*sec(c + d*x))*tan(c + d*x))/(315*d) + (68*(a + a*sec(c + d*x))^(3/2)*tan(c + d*x))/(105*d)]
    @test_int [sec(c + d*x)^3*(a + a*sec(c + d*x))^(3/2), x, 4, (152*a^2*tan(c + d*x))/(105*d*sqrt(a + a*sec(c + d*x))) + (38*a*sqrt(a + a*sec(c + d*x))*tan(c + d*x))/(105*d) - (4*(a + a*sec(c + d*x))^(3/2)*tan(c + d*x))/(35*d) + (2*(a + a*sec(c + d*x))^(5/2)*tan(c + d*x))/(7*a*d)]
    @test_int [sec(c + d*x)^2*(a + a*sec(c + d*x))^(3/2), x, 3, (8*a^2*tan(c + d*x))/(5*d*sqrt(a + a*sec(c + d*x))) + (2*a*sqrt(a + a*sec(c + d*x))*tan(c + d*x))/(5*d) + (2*(a + a*sec(c + d*x))^(3/2)*tan(c + d*x))/(5*d)]
    @test_int [sec(c + d*x)^1*(a + a*sec(c + d*x))^(3/2), x, 2, (8*a^2*tan(c + d*x))/(3*d*sqrt(a + a*sec(c + d*x))) + (2*a*sqrt(a + a*sec(c + d*x))*tan(c + d*x))/(3*d)]
    @test_int [sec(c + d*x)^0*(a + a*sec(c + d*x))^(3/2), x, 4, (2*a^(3/2)*atan((sqrt(a)*tan(c + d*x))/sqrt(a + a*sec(c + d*x))))/d + (2*a^2*tan(c + d*x))/(d*sqrt(a + a*sec(c + d*x)))]
    @test_int [cos(c + d*x)^1*(a + a*sec(c + d*x))^(3/2), x, 5, (3*a^(3/2)*atan((sqrt(a)*tan(c + d*x))/sqrt(a + a*sec(c + d*x))))/d + (a^2*sin(c + d*x))/(d*sqrt(a + a*sec(c + d*x)))]
    @test_int [cos(c + d*x)^2*(a + a*sec(c + d*x))^(3/2), x, 5, (7*a^(3/2)*atan((sqrt(a)*tan(c + d*x))/sqrt(a + a*sec(c + d*x))))/(4*d) + (7*a^2*sin(c + d*x))/(4*d*sqrt(a + a*sec(c + d*x))) + (a^2*cos(c + d*x)*sin(c + d*x))/(2*d*sqrt(a + a*sec(c + d*x)))]
    @test_int [cos(c + d*x)^3*(a + a*sec(c + d*x))^(3/2), x, 6, (11*a^(3/2)*atan((sqrt(a)*tan(c + d*x))/sqrt(a + a*sec(c + d*x))))/(8*d) + (11*a^2*sin(c + d*x))/(8*d*sqrt(a + a*sec(c + d*x))) + (11*a^2*cos(c + d*x)*sin(c + d*x))/(12*d*sqrt(a + a*sec(c + d*x))) + (a^2*cos(c + d*x)^2*sin(c + d*x))/(3*d*sqrt(a + a*sec(c + d*x)))]


    @test_int [sec(c + d*x)^4*(a + a*sec(c + d*x))^(5/2), x, 6, (284*a^3*tan(c + d*x))/(99*d*sqrt(a + a*sec(c + d*x))) + (710*a^3*sec(c + d*x)^3*tan(c + d*x))/(693*d*sqrt(a + a*sec(c + d*x))) + (46*a^3*sec(c + d*x)^4*tan(c + d*x))/(99*d*sqrt(a + a*sec(c + d*x))) - (568*a^2*sqrt(a + a*sec(c + d*x))*tan(c + d*x))/(693*d) + (2*a^2*sec(c + d*x)^4*sqrt(a + a*sec(c + d*x))*tan(c + d*x))/(11*d) + (284*a*(a + a*sec(c + d*x))^(3/2)*tan(c + d*x))/(231*d)]
    @test_int [sec(c + d*x)^3*(a + a*sec(c + d*x))^(5/2), x, 5, (832*a^3*tan(c + d*x))/(315*d*sqrt(a + a*sec(c + d*x))) + (208*a^2*sqrt(a + a*sec(c + d*x))*tan(c + d*x))/(315*d) + (26*a*(a + a*sec(c + d*x))^(3/2)*tan(c + d*x))/(105*d) - (4*(a + a*sec(c + d*x))^(5/2)*tan(c + d*x))/(63*d) + (2*(a + a*sec(c + d*x))^(7/2)*tan(c + d*x))/(9*a*d)]
    @test_int [sec(c + d*x)^2*(a + a*sec(c + d*x))^(5/2), x, 4, (64*a^3*tan(c + d*x))/(21*d*sqrt(a + a*sec(c + d*x))) + (16*a^2*sqrt(a + a*sec(c + d*x))*tan(c + d*x))/(21*d) + (2*a*(a + a*sec(c + d*x))^(3/2)*tan(c + d*x))/(7*d) + (2*(a + a*sec(c + d*x))^(5/2)*tan(c + d*x))/(7*d)]
    @test_int [sec(c + d*x)^1*(a + a*sec(c + d*x))^(5/2), x, 3, (64*a^3*tan(c + d*x))/(15*d*sqrt(a + a*sec(c + d*x))) + (16*a^2*sqrt(a + a*sec(c + d*x))*tan(c + d*x))/(15*d) + (2*a*(a + a*sec(c + d*x))^(3/2)*tan(c + d*x))/(5*d)]
    @test_int [sec(c + d*x)^0*(a + a*sec(c + d*x))^(5/2), x, 5, (2*a^(5/2)*atan((sqrt(a)*tan(c + d*x))/sqrt(a + a*sec(c + d*x))))/d + (14*a^3*tan(c + d*x))/(3*d*sqrt(a + a*sec(c + d*x))) + (2*a^2*sqrt(a + a*sec(c + d*x))*tan(c + d*x))/(3*d)]
    @test_int [cos(c + d*x)^1*(a + a*sec(c + d*x))^(5/2), x, 4, (5*a^(5/2)*atan((sqrt(a)*tan(c + d*x))/sqrt(a + a*sec(c + d*x))))/d - (a^3*sin(c + d*x))/(d*sqrt(a + a*sec(c + d*x))) + (2*a^2*sqrt(a + a*sec(c + d*x))*sin(c + d*x))/d]
    @test_int [cos(c + d*x)^2*(a + a*sec(c + d*x))^(5/2), x, 4, (19*a^(5/2)*atan((sqrt(a)*tan(c + d*x))/sqrt(a + a*sec(c + d*x))))/(4*d) + (9*a^3*sin(c + d*x))/(4*d*sqrt(a + a*sec(c + d*x))) + (a^2*cos(c + d*x)*sqrt(a + a*sec(c + d*x))*sin(c + d*x))/(2*d)]
    @test_int [cos(c + d*x)^3*(a + a*sec(c + d*x))^(5/2), x, 5, (25*a^(5/2)*atan((sqrt(a)*tan(c + d*x))/sqrt(a + a*sec(c + d*x))))/(8*d) + (25*a^3*sin(c + d*x))/(8*d*sqrt(a + a*sec(c + d*x))) + (13*a^3*cos(c + d*x)*sin(c + d*x))/(12*d*sqrt(a + a*sec(c + d*x))) + (a^2*cos(c + d*x)^2*sqrt(a + a*sec(c + d*x))*sin(c + d*x))/(3*d)]
    @test_int [cos(c + d*x)^4*(a + a*sec(c + d*x))^(5/2), x, 6, (163*a^(5/2)*atan((sqrt(a)*tan(c + d*x))/sqrt(a + a*sec(c + d*x))))/(64*d) + (163*a^3*sin(c + d*x))/(64*d*sqrt(a + a*sec(c + d*x))) + (163*a^3*cos(c + d*x)*sin(c + d*x))/(96*d*sqrt(a + a*sec(c + d*x))) + (17*a^3*cos(c + d*x)^2*sin(c + d*x))/(24*d*sqrt(a + a*sec(c + d*x))) + (a^2*cos(c + d*x)^3*sqrt(a + a*sec(c + d*x))*sin(c + d*x))/(4*d)]


    @test_int [sec(c + d*x)^1*sqrt(a - a*sec(c + d*x)), x, 1, -((2*a*tan(c + d*x))/(d*sqrt(a - a*sec(c + d*x))))]
    @test_int [sec(c + d*x)^0*sqrt(a - a*sec(c + d*x)), x, 2, (2*sqrt(a)*atan((sqrt(a)*tan(c + d*x))/sqrt(a - a*sec(c + d*x))))/d]
    @test_int [cos(c + d*x)^1*sqrt(a - a*sec(c + d*x)), x, 3, -((sqrt(a)*atan((sqrt(a)*tan(c + d*x))/sqrt(a - a*sec(c + d*x))))/d) + (a*sin(c + d*x))/(d*sqrt(a - a*sec(c + d*x)))]


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


    @test_int [sec(c + d*x)^4/sqrt(a + a*sec(c + d*x)), x, 5, -((sqrt(2)*atan((sqrt(a)*tan(c + d*x))/(sqrt(2)*sqrt(a + a*sec(c + d*x)))))/(sqrt(a)*d)) + (28*tan(c + d*x))/(15*d*sqrt(a + a*sec(c + d*x))) + (2*sec(c + d*x)^2*tan(c + d*x))/(5*d*sqrt(a + a*sec(c + d*x))) - (2*sqrt(a + a*sec(c + d*x))*tan(c + d*x))/(15*a*d)]
    @test_int [sec(c + d*x)^3/sqrt(a + a*sec(c + d*x)), x, 4, (sqrt(2)*atan((sqrt(a)*tan(c + d*x))/(sqrt(2)*sqrt(a + a*sec(c + d*x)))))/(sqrt(a)*d) - (4*tan(c + d*x))/(3*d*sqrt(a + a*sec(c + d*x))) + (2*sqrt(a + a*sec(c + d*x))*tan(c + d*x))/(3*a*d)]
    @test_int [sec(c + d*x)^2/sqrt(a + a*sec(c + d*x)), x, 3, -((sqrt(2)*atan((sqrt(a)*tan(c + d*x))/(sqrt(2)*sqrt(a + a*sec(c + d*x)))))/(sqrt(a)*d)) + (2*tan(c + d*x))/(d*sqrt(a + a*sec(c + d*x)))]
    @test_int [sec(c + d*x)^1/sqrt(a + a*sec(c + d*x)), x, 2, (sqrt(2)*atan((sqrt(a)*tan(c + d*x))/(sqrt(2)*sqrt(a + a*sec(c + d*x)))))/(sqrt(a)*d)]
    @test_int [sec(c + d*x)^0/sqrt(a + a*sec(c + d*x)), x, 5, (2*atan((sqrt(a)*tan(c + d*x))/sqrt(a + a*sec(c + d*x))))/(sqrt(a)*d) - (sqrt(2)*atan((sqrt(a)*tan(c + d*x))/(sqrt(2)*sqrt(a + a*sec(c + d*x)))))/(sqrt(a)*d)]
    @test_int [cos(c + d*x)^1/sqrt(a + a*sec(c + d*x)), x, 6, -(atan((sqrt(a)*tan(c + d*x))/sqrt(a + a*sec(c + d*x)))/(sqrt(a)*d)) + (sqrt(2)*atan((sqrt(a)*tan(c + d*x))/(sqrt(2)*sqrt(a + a*sec(c + d*x)))))/(sqrt(a)*d) + sin(c + d*x)/(d*sqrt(a + a*sec(c + d*x)))]
    @test_int [cos(c + d*x)^2/sqrt(a + a*sec(c + d*x)), x, 7, (7*atan((sqrt(a)*tan(c + d*x))/sqrt(a + a*sec(c + d*x))))/(4*sqrt(a)*d) - (sqrt(2)*atan((sqrt(a)*tan(c + d*x))/(sqrt(2)*sqrt(a + a*sec(c + d*x)))))/(sqrt(a)*d) - sin(c + d*x)/(4*d*sqrt(a + a*sec(c + d*x))) + (cos(c + d*x)*sin(c + d*x))/(2*d*sqrt(a + a*sec(c + d*x)))]


    @test_int [sec(c + d*x)^5/(a + a*sec(c + d*x))^(3/2), x, 6, -((15*atan((sqrt(a)*tan(c + d*x))/(sqrt(2)*sqrt(a + a*sec(c + d*x)))))/(2*sqrt(2)*a^(3/2)*d)) - (sec(c + d*x)^3*tan(c + d*x))/(2*d*(a + a*sec(c + d*x))^(3/2)) + (31*tan(c + d*x))/(5*a*d*sqrt(a + a*sec(c + d*x))) + (9*sec(c + d*x)^2*tan(c + d*x))/(10*a*d*sqrt(a + a*sec(c + d*x))) - (13*sqrt(a + a*sec(c + d*x))*tan(c + d*x))/(10*a^2*d)]
    @test_int [sec(c + d*x)^4/(a + a*sec(c + d*x))^(3/2), x, 5, (11*atan((sqrt(a)*tan(c + d*x))/(sqrt(2)*sqrt(a + a*sec(c + d*x)))))/(2*sqrt(2)*a^(3/2)*d) - (sec(c + d*x)^2*tan(c + d*x))/(2*d*(a + a*sec(c + d*x))^(3/2)) - (13*tan(c + d*x))/(3*a*d*sqrt(a + a*sec(c + d*x))) + (7*sqrt(a + a*sec(c + d*x))*tan(c + d*x))/(6*a^2*d)]
    @test_int [sec(c + d*x)^3/(a + a*sec(c + d*x))^(3/2), x, 4, -((7*atan((sqrt(a)*tan(c + d*x))/(sqrt(2)*sqrt(a + a*sec(c + d*x)))))/(2*sqrt(2)*a^(3/2)*d)) + tan(c + d*x)/(2*d*(a + a*sec(c + d*x))^(3/2)) + (2*tan(c + d*x))/(a*d*sqrt(a + a*sec(c + d*x)))]
    @test_int [sec(c + d*x)^2/(a + a*sec(c + d*x))^(3/2), x, 3, (3*atan((sqrt(a)*tan(c + d*x))/(sqrt(2)*sqrt(a + a*sec(c + d*x)))))/(2*sqrt(2)*a^(3/2)*d) - tan(c + d*x)/(2*d*(a + a*sec(c + d*x))^(3/2))]
    @test_int [sec(c + d*x)^1/(a + a*sec(c + d*x))^(3/2), x, 3, atan((sqrt(a)*tan(c + d*x))/(sqrt(2)*sqrt(a + a*sec(c + d*x))))/(2*sqrt(2)*a^(3/2)*d) + tan(c + d*x)/(2*d*(a + a*sec(c + d*x))^(3/2))]
    @test_int [sec(c + d*x)^0/(a + a*sec(c + d*x))^(3/2), x, 6, (2*atan((sqrt(a)*tan(c + d*x))/sqrt(a + a*sec(c + d*x))))/(a^(3/2)*d) - (5*atan((sqrt(a)*tan(c + d*x))/(sqrt(2)*sqrt(a + a*sec(c + d*x)))))/(2*sqrt(2)*a^(3/2)*d) - tan(c + d*x)/(2*d*(a + a*sec(c + d*x))^(3/2))]
    @test_int [cos(c + d*x)^1/(a + a*sec(c + d*x))^(3/2), x, 7, -((3*atan((sqrt(a)*tan(c + d*x))/sqrt(a + a*sec(c + d*x))))/(a^(3/2)*d)) + (9*atan((sqrt(a)*tan(c + d*x))/(sqrt(2)*sqrt(a + a*sec(c + d*x)))))/(2*sqrt(2)*a^(3/2)*d) - sin(c + d*x)/(2*d*(a + a*sec(c + d*x))^(3/2)) + (3*sin(c + d*x))/(2*a*d*sqrt(a + a*sec(c + d*x)))]
    @test_int [cos(c + d*x)^2/(a + a*sec(c + d*x))^(3/2), x, 8, (19*atan((sqrt(a)*tan(c + d*x))/sqrt(a + a*sec(c + d*x))))/(4*a^(3/2)*d) - (13*atan((sqrt(a)*tan(c + d*x))/(sqrt(2)*sqrt(a + a*sec(c + d*x)))))/(2*sqrt(2)*a^(3/2)*d) - (cos(c + d*x)*sin(c + d*x))/(2*d*(a + a*sec(c + d*x))^(3/2)) - (7*sin(c + d*x))/(4*a*d*sqrt(a + a*sec(c + d*x))) + (cos(c + d*x)*sin(c + d*x))/(a*d*sqrt(a + a*sec(c + d*x)))]


    @test_int [sec(c + d*x)^5/(a + a*sec(c + d*x))^(5/2), x, 6, (163*atan((sqrt(a)*tan(c + d*x))/(sqrt(2)*sqrt(a + a*sec(c + d*x)))))/(16*sqrt(2)*a^(5/2)*d) - (sec(c + d*x)^3*tan(c + d*x))/(4*d*(a + a*sec(c + d*x))^(5/2)) - (17*sec(c + d*x)^2*tan(c + d*x))/(16*a*d*(a + a*sec(c + d*x))^(3/2)) - (197*tan(c + d*x))/(24*a^2*d*sqrt(a + a*sec(c + d*x))) + (95*sqrt(a + a*sec(c + d*x))*tan(c + d*x))/(48*a^3*d)]
    @test_int [sec(c + d*x)^4/(a + a*sec(c + d*x))^(5/2), x, 5, -((75*atan((sqrt(a)*tan(c + d*x))/(sqrt(2)*sqrt(a + a*sec(c + d*x)))))/(16*sqrt(2)*a^(5/2)*d)) - (sec(c + d*x)^2*tan(c + d*x))/(4*d*(a + a*sec(c + d*x))^(5/2)) + (13*tan(c + d*x))/(16*a*d*(a + a*sec(c + d*x))^(3/2)) + (9*tan(c + d*x))/(4*a^2*d*sqrt(a + a*sec(c + d*x)))]
    @test_int [sec(c + d*x)^3/(a + a*sec(c + d*x))^(5/2), x, 4, (19*atan((sqrt(a)*tan(c + d*x))/(sqrt(2)*sqrt(a + a*sec(c + d*x)))))/(16*sqrt(2)*a^(5/2)*d) + tan(c + d*x)/(4*d*(a + a*sec(c + d*x))^(5/2)) - (13*tan(c + d*x))/(16*a*d*(a + a*sec(c + d*x))^(3/2))]
    @test_int [sec(c + d*x)^2/(a + a*sec(c + d*x))^(5/2), x, 4, (5*atan((sqrt(a)*tan(c + d*x))/(sqrt(2)*sqrt(a + a*sec(c + d*x)))))/(16*sqrt(2)*a^(5/2)*d) - tan(c + d*x)/(4*d*(a + a*sec(c + d*x))^(5/2)) + (5*tan(c + d*x))/(16*a*d*(a + a*sec(c + d*x))^(3/2))]
    @test_int [sec(c + d*x)^1/(a + a*sec(c + d*x))^(5/2), x, 4, (3*atan((sqrt(a)*tan(c + d*x))/(sqrt(2)*sqrt(a + a*sec(c + d*x)))))/(16*sqrt(2)*a^(5/2)*d) + tan(c + d*x)/(4*d*(a + a*sec(c + d*x))^(5/2)) + (3*tan(c + d*x))/(16*a*d*(a + a*sec(c + d*x))^(3/2))]
    @test_int [sec(c + d*x)^0/(a + a*sec(c + d*x))^(5/2), x, 7, (2*atan((sqrt(a)*tan(c + d*x))/sqrt(a + a*sec(c + d*x))))/(a^(5/2)*d) - (43*atan((sqrt(a)*tan(c + d*x))/(sqrt(2)*sqrt(a + a*sec(c + d*x)))))/(16*sqrt(2)*a^(5/2)*d) - tan(c + d*x)/(4*d*(a + a*sec(c + d*x))^(5/2)) - (11*tan(c + d*x))/(16*a*d*(a + a*sec(c + d*x))^(3/2))]
    @test_int [cos(c + d*x)^1/(a + a*sec(c + d*x))^(5/2), x, 8, -((5*atan((sqrt(a)*tan(c + d*x))/sqrt(a + a*sec(c + d*x))))/(a^(5/2)*d)) + (115*atan((sqrt(a)*tan(c + d*x))/(sqrt(2)*sqrt(a + a*sec(c + d*x)))))/(16*sqrt(2)*a^(5/2)*d) - sin(c + d*x)/(4*d*(a + a*sec(c + d*x))^(5/2)) - (15*sin(c + d*x))/(16*a*d*(a + a*sec(c + d*x))^(3/2)) + (35*sin(c + d*x))/(16*a^2*d*sqrt(a + a*sec(c + d*x)))]


    @test_int [sec(c + d*x)/sqrt(a - a*sec(c + d*x)), x, 2, -((sqrt(2)*atan((sqrt(a)*tan(c + d*x))/(sqrt(2)*sqrt(a - a*sec(c + d*x)))))/(sqrt(a)*d))]
    @test_int [1/sqrt(a - a*sec(c + d*x)), x, 5, (2*atan((sqrt(a)*tan(c + d*x))/sqrt(a - a*sec(c + d*x))))/(sqrt(a)*d) - (sqrt(2)*atan((sqrt(a)*tan(c + d*x))/(sqrt(2)*sqrt(a - a*sec(c + d*x)))))/(sqrt(a)*d)]


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


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


    @test_int [sec(c + d*x)^3*(a + a*sec(c + d*x))^(2/3), x, 7, -((9*(a + a*sec(c + d*x))^(2/3)*tan(c + d*x))/(40*d)) + (57*(a + a*sec(c + d*x))^(2/3)*tan(c + d*x))/(80*d*(1 + sec(c + d*x))) + (3*(a + a*sec(c + d*x))^(5/3)*tan(c + d*x))/(8*a*d) - (19*3^(3/4)*Elliptic.F(acos((2^(1/3) - (1 - sqrt(3))*(1 + sec(c + d*x))^(1/3))/(2^(1/3) - (1 + sqrt(3))*(1 + sec(c + d*x))^(1/3))), (1/4)*(2 + sqrt(3)))*(a + a*sec(c + d*x))^(2/3)*(2^(1/3) - (1 + sec(c + d*x))^(1/3))*sqrt((2^(2/3) + 2^(1/3)*(1 + sec(c + d*x))^(1/3) + (1 + sec(c + d*x))^(2/3))/(2^(1/3) - (1 + sqrt(3))*(1 + sec(c + d*x))^(1/3))^2)*tan(c + d*x))/(80*2^(1/3)*d*(1 - sec(c + d*x))*(1 + sec(c + d*x))*sqrt(-(((1 + sec(c + d*x))^(1/3)*(2^(1/3) - (1 + sec(c + d*x))^(1/3)))/(2^(1/3) - (1 + sqrt(3))*(1 + sec(c + d*x))^(1/3))^2)))]
    @test_int [sec(c + d*x)^2*(a + a*sec(c + d*x))^(2/3), x, 6, (3*(a + a*sec(c + d*x))^(2/3)*tan(c + d*x))/(5*d) + (3*(a + a*sec(c + d*x))^(2/3)*tan(c + d*x))/(5*d*(1 + sec(c + d*x))) - (3^(3/4)*Elliptic.F(acos((2^(1/3) - (1 - sqrt(3))*(1 + sec(c + d*x))^(1/3))/(2^(1/3) - (1 + sqrt(3))*(1 + sec(c + d*x))^(1/3))), (1/4)*(2 + sqrt(3)))*(a + a*sec(c + d*x))^(2/3)*(2^(1/3) - (1 + sec(c + d*x))^(1/3))*sqrt((2^(2/3) + 2^(1/3)*(1 + sec(c + d*x))^(1/3) + (1 + sec(c + d*x))^(2/3))/(2^(1/3) - (1 + sqrt(3))*(1 + sec(c + d*x))^(1/3))^2)*tan(c + d*x))/(5*2^(1/3)*d*(1 - sec(c + d*x))*(1 + sec(c + d*x))*sqrt(-(((1 + sec(c + d*x))^(1/3)*(2^(1/3) - (1 + sec(c + d*x))^(1/3)))/(2^(1/3) - (1 + sqrt(3))*(1 + sec(c + d*x))^(1/3))^2)))]
    @test_int [sec(c + d*x)^1*(a + a*sec(c + d*x))^(2/3), x, 5, (3*(a + a*sec(c + d*x))^(2/3)*tan(c + d*x))/(2*d*(1 + sec(c + d*x))) - (3^(3/4)*Elliptic.F(acos((2^(1/3) - (1 - sqrt(3))*(1 + sec(c + d*x))^(1/3))/(2^(1/3) - (1 + sqrt(3))*(1 + sec(c + d*x))^(1/3))), (1/4)*(2 + sqrt(3)))*(a + a*sec(c + d*x))^(2/3)*(2^(1/3) - (1 + sec(c + d*x))^(1/3))*sqrt((2^(2/3) + 2^(1/3)*(1 + sec(c + d*x))^(1/3) + (1 + sec(c + d*x))^(2/3))/(2^(1/3) - (1 + sqrt(3))*(1 + sec(c + d*x))^(1/3))^2)*tan(c + d*x))/(2*2^(1/3)*d*(1 - sec(c + d*x))*(1 + sec(c + d*x))*sqrt(-(((1 + sec(c + d*x))^(1/3)*(2^(1/3) - (1 + sec(c + d*x))^(1/3)))/(2^(1/3) - (1 + sqrt(3))*(1 + sec(c + d*x))^(1/3))^2)))]
    @test_int [sec(c + d*x)^0*(a + a*sec(c + d*x))^(2/3), x, 3, (3*sqrt(2)*AppellF1(7/6, 1/2, 1, 13/6, (1/2)*(1 + sec(c + d*x)), 1 + sec(c + d*x))*(a + a*sec(c + d*x))^(2/3)*tan(c + d*x))/(7*d*sqrt(1 - sec(c + d*x)))]
    @test_int [cos(c + d*x)^1*(a + a*sec(c + d*x))^(2/3), x, 3, -((3*sqrt(2)*AppellF1(7/6, 1/2, 2, 13/6, (1/2)*(1 + sec(c + d*x)), 1 + sec(c + d*x))*(a + a*sec(c + d*x))^(2/3)*tan(c + d*x))/(7*d*sqrt(1 - sec(c + d*x))))]


    @test_int [sec(c + d*x)^3*(a + a*sec(c + d*x))^(5/3), x, 8, (147*a*(a + a*sec(c + d*x))^(2/3)*tan(c + d*x))/(440*d) + (1029*a*(a + a*sec(c + d*x))^(2/3)*tan(c + d*x))/(880*d*(1 + sec(c + d*x))) - (9*(a + a*sec(c + d*x))^(5/3)*tan(c + d*x))/(88*d) + (3*(a + a*sec(c + d*x))^(8/3)*tan(c + d*x))/(11*a*d) - (343*3^(3/4)*a*Elliptic.F(acos((2^(1/3) - (1 - sqrt(3))*(1 + sec(c + d*x))^(1/3))/(2^(1/3) - (1 + sqrt(3))*(1 + sec(c + d*x))^(1/3))), (1/4)*(2 + sqrt(3)))*(a + a*sec(c + d*x))^(2/3)*(2^(1/3) - (1 + sec(c + d*x))^(1/3))*sqrt((2^(2/3) + 2^(1/3)*(1 + sec(c + d*x))^(1/3) + (1 + sec(c + d*x))^(2/3))/(2^(1/3) - (1 + sqrt(3))*(1 + sec(c + d*x))^(1/3))^2)*tan(c + d*x))/(880*2^(1/3)*d*(1 - sec(c + d*x))*(1 + sec(c + d*x))*sqrt(-(((1 + sec(c + d*x))^(1/3)*(2^(1/3) - (1 + sec(c + d*x))^(1/3)))/(2^(1/3) - (1 + sqrt(3))*(1 + sec(c + d*x))^(1/3))^2)))]
    @test_int [sec(c + d*x)^2*(a + a*sec(c + d*x))^(5/3), x, 7, (3*a*(a + a*sec(c + d*x))^(2/3)*tan(c + d*x))/(8*d) + (21*a*(a + a*sec(c + d*x))^(2/3)*tan(c + d*x))/(16*d*(1 + sec(c + d*x))) + (3*(a + a*sec(c + d*x))^(5/3)*tan(c + d*x))/(8*d) - (7*3^(3/4)*a*Elliptic.F(acos((2^(1/3) - (1 - sqrt(3))*(1 + sec(c + d*x))^(1/3))/(2^(1/3) - (1 + sqrt(3))*(1 + sec(c + d*x))^(1/3))), (1/4)*(2 + sqrt(3)))*(a + a*sec(c + d*x))^(2/3)*(2^(1/3) - (1 + sec(c + d*x))^(1/3))*sqrt((2^(2/3) + 2^(1/3)*(1 + sec(c + d*x))^(1/3) + (1 + sec(c + d*x))^(2/3))/(2^(1/3) - (1 + sqrt(3))*(1 + sec(c + d*x))^(1/3))^2)*tan(c + d*x))/(16*2^(1/3)*d*(1 - sec(c + d*x))*(1 + sec(c + d*x))*sqrt(-(((1 + sec(c + d*x))^(1/3)*(2^(1/3) - (1 + sec(c + d*x))^(1/3)))/(2^(1/3) - (1 + sqrt(3))*(1 + sec(c + d*x))^(1/3))^2)))]
    @test_int [sec(c + d*x)^1*(a + a*sec(c + d*x))^(5/3), x, 6, (3*a*(a + a*sec(c + d*x))^(2/3)*tan(c + d*x))/(5*d) + (21*a*(a + a*sec(c + d*x))^(2/3)*tan(c + d*x))/(10*d*(1 + sec(c + d*x))) - (7*3^(3/4)*a*Elliptic.F(acos((2^(1/3) - (1 - sqrt(3))*(1 + sec(c + d*x))^(1/3))/(2^(1/3) - (1 + sqrt(3))*(1 + sec(c + d*x))^(1/3))), (1/4)*(2 + sqrt(3)))*(a + a*sec(c + d*x))^(2/3)*(2^(1/3) - (1 + sec(c + d*x))^(1/3))*sqrt((2^(2/3) + 2^(1/3)*(1 + sec(c + d*x))^(1/3) + (1 + sec(c + d*x))^(2/3))/(2^(1/3) - (1 + sqrt(3))*(1 + sec(c + d*x))^(1/3))^2)*tan(c + d*x))/(10*2^(1/3)*d*(1 - sec(c + d*x))*(1 + sec(c + d*x))*sqrt(-(((1 + sec(c + d*x))^(1/3)*(2^(1/3) - (1 + sec(c + d*x))^(1/3)))/(2^(1/3) - (1 + sqrt(3))*(1 + sec(c + d*x))^(1/3))^2)))]
    @test_int [sec(c + d*x)^0*(a + a*sec(c + d*x))^(5/3), x, 3, (3*sqrt(2)*a*AppellF1(13/6, 1/2, 1, 19/6, (1/2)*(1 + sec(c + d*x)), 1 + sec(c + d*x))*(1 + sec(c + d*x))*(a + a*sec(c + d*x))^(2/3)*tan(c + d*x))/(13*d*sqrt(1 - sec(c + d*x)))]
    @test_int [cos(c + d*x)^1*(a + a*sec(c + d*x))^(5/3), x, 3, -((3*sqrt(2)*a*AppellF1(13/6, 1/2, 2, 19/6, (1/2)*(1 + sec(c + d*x)), 1 + sec(c + d*x))*(1 + sec(c + d*x))*(a + a*sec(c + d*x))^(2/3)*tan(c + d*x))/(13*d*sqrt(1 - sec(c + d*x))))]


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


    @test_int [sec(c + d*x)^4/(a + a*sec(c + d*x))^(1/3), x, 7, (99*tan(c + d*x))/(80*d*(a + a*sec(c + d*x))^(1/3)) + (3*sec(c + d*x)^2*tan(c + d*x))/(8*d*(a + a*sec(c + d*x))^(1/3)) - (3*(a + a*sec(c + d*x))^(2/3)*tan(c + d*x))/(40*a*d) + (37*3^(3/4)*Elliptic.F(acos((2^(1/3) - (1 - sqrt(3))*(1 + sec(c + d*x))^(1/3))/(2^(1/3) - (1 + sqrt(3))*(1 + sec(c + d*x))^(1/3))), (1/4)*(2 + sqrt(3)))*(2^(1/3) - (1 + sec(c + d*x))^(1/3))*sqrt((2^(2/3) + 2^(1/3)*(1 + sec(c + d*x))^(1/3) + (1 + sec(c + d*x))^(2/3))/(2^(1/3) - (1 + sqrt(3))*(1 + sec(c + d*x))^(1/3))^2)*tan(c + d*x))/(80*2^(1/3)*d*(1 - sec(c + d*x))*(a + a*sec(c + d*x))^(1/3)*sqrt(-(((1 + sec(c + d*x))^(1/3)*(2^(1/3) - (1 + sec(c + d*x))^(1/3)))/(2^(1/3) - (1 + sqrt(3))*(1 + sec(c + d*x))^(1/3))^2)))]
    @test_int [sec(c + d*x)^3/(a + a*sec(c + d*x))^(1/3), x, 6, -((9*tan(c + d*x))/(10*d*(a + a*sec(c + d*x))^(1/3))) + (3*(a + a*sec(c + d*x))^(2/3)*tan(c + d*x))/(5*a*d) - (7*3^(3/4)*Elliptic.F(acos((2^(1/3) - (1 - sqrt(3))*(1 + sec(c + d*x))^(1/3))/(2^(1/3) - (1 + sqrt(3))*(1 + sec(c + d*x))^(1/3))), (1/4)*(2 + sqrt(3)))*(2^(1/3) - (1 + sec(c + d*x))^(1/3))*sqrt((2^(2/3) + 2^(1/3)*(1 + sec(c + d*x))^(1/3) + (1 + sec(c + d*x))^(2/3))/(2^(1/3) - (1 + sqrt(3))*(1 + sec(c + d*x))^(1/3))^2)*tan(c + d*x))/(10*2^(1/3)*d*(1 - sec(c + d*x))*(a + a*sec(c + d*x))^(1/3)*sqrt(-(((1 + sec(c + d*x))^(1/3)*(2^(1/3) - (1 + sec(c + d*x))^(1/3)))/(2^(1/3) - (1 + sqrt(3))*(1 + sec(c + d*x))^(1/3))^2)))]
    @test_int [sec(c + d*x)^2/(a + a*sec(c + d*x))^(1/3), x, 5, (3*tan(c + d*x))/(2*d*(a + a*sec(c + d*x))^(1/3)) + (3^(3/4)*Elliptic.F(acos((2^(1/3) - (1 - sqrt(3))*(1 + sec(c + d*x))^(1/3))/(2^(1/3) - (1 + sqrt(3))*(1 + sec(c + d*x))^(1/3))), (1/4)*(2 + sqrt(3)))*(2^(1/3) - (1 + sec(c + d*x))^(1/3))*sqrt((2^(2/3) + 2^(1/3)*(1 + sec(c + d*x))^(1/3) + (1 + sec(c + d*x))^(2/3))/(2^(1/3) - (1 + sqrt(3))*(1 + sec(c + d*x))^(1/3))^2)*tan(c + d*x))/(2*2^(1/3)*d*(1 - sec(c + d*x))*(a + a*sec(c + d*x))^(1/3)*sqrt(-(((1 + sec(c + d*x))^(1/3)*(2^(1/3) - (1 + sec(c + d*x))^(1/3)))/(2^(1/3) - (1 + sqrt(3))*(1 + sec(c + d*x))^(1/3))^2)))]
    @test_int [sec(c + d*x)^1/(a + a*sec(c + d*x))^(1/3), x, 4, -((3^(3/4)*Elliptic.F(acos((2^(1/3) - (1 - sqrt(3))*(1 + sec(c + d*x))^(1/3))/(2^(1/3) - (1 + sqrt(3))*(1 + sec(c + d*x))^(1/3))), (1/4)*(2 + sqrt(3)))*(2^(1/3) - (1 + sec(c + d*x))^(1/3))*sqrt((2^(2/3) + 2^(1/3)*(1 + sec(c + d*x))^(1/3) + (1 + sec(c + d*x))^(2/3))/(2^(1/3) - (1 + sqrt(3))*(1 + sec(c + d*x))^(1/3))^2)*tan(c + d*x))/(2^(1/3)*d*(1 - sec(c + d*x))*(a + a*sec(c + d*x))^(1/3)*sqrt(-(((1 + sec(c + d*x))^(1/3)*(2^(1/3) - (1 + sec(c + d*x))^(1/3)))/(2^(1/3) - (1 + sqrt(3))*(1 + sec(c + d*x))^(1/3))^2))))]
    @test_int [sec(c + d*x)^0/(a + a*sec(c + d*x))^(1/3), x, 3, (3*sqrt(2)*AppellF1(1/6, 1/2, 1, 7/6, (1/2)*(1 + sec(c + d*x)), 1 + sec(c + d*x))*tan(c + d*x))/(d*sqrt(1 - sec(c + d*x))*(a + a*sec(c + d*x))^(1/3))]
    @test_int [cos(c + d*x)^1/(a + a*sec(c + d*x))^(1/3), x, 3, -((3*sqrt(2)*AppellF1(1/6, 1/2, 2, 7/6, (1/2)*(1 + sec(c + d*x)), 1 + sec(c + d*x))*tan(c + d*x))/(d*sqrt(1 - sec(c + d*x))*(a + a*sec(c + d*x))^(1/3)))]


    @test_int [sec(c + d*x)^4/(a + a*sec(c + d*x))^(5/3), x, 9, -((33*tan(c + d*x))/(28*d*(a + a*sec(c + d*x))^(5/3))) + (3*sec(c + d*x)^2*tan(c + d*x))/(4*d*(a + a*sec(c + d*x))^(5/3)) + (135*tan(c + d*x))/(14*a*d*(a + a*sec(c + d*x))^(2/3)) + (375*(1 + sqrt(3))*(a + a*sec(c + d*x))^(1/3)*tan(c + d*x))/(28*a^2*d*(1 + sec(c + d*x))^(2/3)*(2^(1/3) - (1 + sqrt(3))*(1 + sec(c + d*x))^(1/3))) - (375*3^(1/4)*Elliptic.E(acos((2^(1/3) - (1 - sqrt(3))*(1 + sec(c + d*x))^(1/3))/(2^(1/3) - (1 + sqrt(3))*(1 + sec(c + d*x))^(1/3))), (1/4)*(2 + sqrt(3)))*(a + a*sec(c + d*x))^(1/3)*(2^(1/3) - (1 + sec(c + d*x))^(1/3))*sqrt((2^(2/3) + 2^(1/3)*(1 + sec(c + d*x))^(1/3) + (1 + sec(c + d*x))^(2/3))/(2^(1/3) - (1 + sqrt(3))*(1 + sec(c + d*x))^(1/3))^2)*tan(c + d*x))/(14*2^(2/3)*a^2*d*(1 - sec(c + d*x))*(1 + sec(c + d*x))^(2/3)*sqrt(-(((1 + sec(c + d*x))^(1/3)*(2^(1/3) - (1 + sec(c + d*x))^(1/3)))/(2^(1/3) - (1 + sqrt(3))*(1 + sec(c + d*x))^(1/3))^2))) - (125*3^(3/4)*(1 - sqrt(3))*Elliptic.F(acos((2^(1/3) - (1 - sqrt(3))*(1 + sec(c + d*x))^(1/3))/(2^(1/3) - (1 + sqrt(3))*(1 + sec(c + d*x))^(1/3))), (1/4)*(2 + sqrt(3)))*(a + a*sec(c + d*x))^(1/3)*(2^(1/3) - (1 + sec(c + d*x))^(1/3))*sqrt((2^(2/3) + 2^(1/3)*(1 + sec(c + d*x))^(1/3) + (1 + sec(c + d*x))^(2/3))/(2^(1/3) - (1 + sqrt(3))*(1 + sec(c + d*x))^(1/3))^2)*tan(c + d*x))/(28*2^(2/3)*a^2*d*(1 - sec(c + d*x))*(1 + sec(c + d*x))^(2/3)*sqrt(-(((1 + sec(c + d*x))^(1/3)*(2^(1/3) - (1 + sec(c + d*x))^(1/3)))/(2^(1/3) - (1 + sqrt(3))*(1 + sec(c + d*x))^(1/3))^2)))]
    @test_int [sec(c + d*x)^3/(a + a*sec(c + d*x))^(5/3), x, 8, (3*tan(c + d*x))/(7*d*(a + a*sec(c + d*x))^(5/3)) - (36*tan(c + d*x))/(7*a*d*(a + a*sec(c + d*x))^(2/3)) - (57*(1 + sqrt(3))*(a + a*sec(c + d*x))^(1/3)*tan(c + d*x))/(7*a^2*d*(1 + sec(c + d*x))^(2/3)*(2^(1/3) - (1 + sqrt(3))*(1 + sec(c + d*x))^(1/3))) + (57*2^(1/3)*3^(1/4)*Elliptic.E(acos((2^(1/3) - (1 - sqrt(3))*(1 + sec(c + d*x))^(1/3))/(2^(1/3) - (1 + sqrt(3))*(1 + sec(c + d*x))^(1/3))), (1/4)*(2 + sqrt(3)))*(a + a*sec(c + d*x))^(1/3)*(2^(1/3) - (1 + sec(c + d*x))^(1/3))*sqrt((2^(2/3) + 2^(1/3)*(1 + sec(c + d*x))^(1/3) + (1 + sec(c + d*x))^(2/3))/(2^(1/3) - (1 + sqrt(3))*(1 + sec(c + d*x))^(1/3))^2)*tan(c + d*x))/(7*a^2*d*(1 - sec(c + d*x))*(1 + sec(c + d*x))^(2/3)*sqrt(-(((1 + sec(c + d*x))^(1/3)*(2^(1/3) - (1 + sec(c + d*x))^(1/3)))/(2^(1/3) - (1 + sqrt(3))*(1 + sec(c + d*x))^(1/3))^2))) + (19*3^(3/4)*(1 - sqrt(3))*Elliptic.F(acos((2^(1/3) - (1 - sqrt(3))*(1 + sec(c + d*x))^(1/3))/(2^(1/3) - (1 + sqrt(3))*(1 + sec(c + d*x))^(1/3))), (1/4)*(2 + sqrt(3)))*(a + a*sec(c + d*x))^(1/3)*(2^(1/3) - (1 + sec(c + d*x))^(1/3))*sqrt((2^(2/3) + 2^(1/3)*(1 + sec(c + d*x))^(1/3) + (1 + sec(c + d*x))^(2/3))/(2^(1/3) - (1 + sqrt(3))*(1 + sec(c + d*x))^(1/3))^2)*tan(c + d*x))/(7*2^(2/3)*a^2*d*(1 - sec(c + d*x))*(1 + sec(c + d*x))^(2/3)*sqrt(-(((1 + sec(c + d*x))^(1/3)*(2^(1/3) - (1 + sec(c + d*x))^(1/3)))/(2^(1/3) - (1 + sqrt(3))*(1 + sec(c + d*x))^(1/3))^2)))]
    @test_int [sec(c + d*x)^2/(a + a*sec(c + d*x))^(5/3), x, 8, -((3*tan(c + d*x))/(7*d*(a + a*sec(c + d*x))^(5/3))) + (15*tan(c + d*x))/(7*a*d*(a + a*sec(c + d*x))^(2/3)) + (15*(1 + sqrt(3))*(1 + sec(c + d*x))^(1/3)*tan(c + d*x))/(7*a*d*(a + a*sec(c + d*x))^(2/3)*(2^(1/3) - (1 + sqrt(3))*(1 + sec(c + d*x))^(1/3))) - (15*2^(1/3)*3^(1/4)*Elliptic.E(acos((2^(1/3) - (1 - sqrt(3))*(1 + sec(c + d*x))^(1/3))/(2^(1/3) - (1 + sqrt(3))*(1 + sec(c + d*x))^(1/3))), (1/4)*(2 + sqrt(3)))*(1 + sec(c + d*x))^(1/3)*(2^(1/3) - (1 + sec(c + d*x))^(1/3))*sqrt((2^(2/3) + 2^(1/3)*(1 + sec(c + d*x))^(1/3) + (1 + sec(c + d*x))^(2/3))/(2^(1/3) - (1 + sqrt(3))*(1 + sec(c + d*x))^(1/3))^2)*tan(c + d*x))/(7*a*d*(1 - sec(c + d*x))*(a + a*sec(c + d*x))^(2/3)*sqrt(-(((1 + sec(c + d*x))^(1/3)*(2^(1/3) - (1 + sec(c + d*x))^(1/3)))/(2^(1/3) - (1 + sqrt(3))*(1 + sec(c + d*x))^(1/3))^2))) - (5*3^(3/4)*(1 - sqrt(3))*Elliptic.F(acos((2^(1/3) - (1 - sqrt(3))*(1 + sec(c + d*x))^(1/3))/(2^(1/3) - (1 + sqrt(3))*(1 + sec(c + d*x))^(1/3))), (1/4)*(2 + sqrt(3)))*(1 + sec(c + d*x))^(1/3)*(2^(1/3) - (1 + sec(c + d*x))^(1/3))*sqrt((2^(2/3) + 2^(1/3)*(1 + sec(c + d*x))^(1/3) + (1 + sec(c + d*x))^(2/3))/(2^(1/3) - (1 + sqrt(3))*(1 + sec(c + d*x))^(1/3))^2)*tan(c + d*x))/(7*2^(2/3)*a*d*(1 - sec(c + d*x))*(a + a*sec(c + d*x))^(2/3)*sqrt(-(((1 + sec(c + d*x))^(1/3)*(2^(1/3) - (1 + sec(c + d*x))^(1/3)))/(2^(1/3) - (1 + sqrt(3))*(1 + sec(c + d*x))^(1/3))^2)))]
    @test_int [sec(c + d*x)^1/(a + a*sec(c + d*x))^(5/3), x, 8, (6*tan(c + d*x))/(7*a*d*(a + a*sec(c + d*x))^(2/3)) + (3*tan(c + d*x))/(7*a*d*(1 + sec(c + d*x))*(a + a*sec(c + d*x))^(2/3)) + (6*(1 + sqrt(3))*(1 + sec(c + d*x))^(1/3)*tan(c + d*x))/(7*a*d*(a + a*sec(c + d*x))^(2/3)*(2^(1/3) - (1 + sqrt(3))*(1 + sec(c + d*x))^(1/3))) - (6*2^(1/3)*3^(1/4)*Elliptic.E(acos((2^(1/3) - (1 - sqrt(3))*(1 + sec(c + d*x))^(1/3))/(2^(1/3) - (1 + sqrt(3))*(1 + sec(c + d*x))^(1/3))), (1/4)*(2 + sqrt(3)))*(1 + sec(c + d*x))^(1/3)*(2^(1/3) - (1 + sec(c + d*x))^(1/3))*sqrt((2^(2/3) + 2^(1/3)*(1 + sec(c + d*x))^(1/3) + (1 + sec(c + d*x))^(2/3))/(2^(1/3) - (1 + sqrt(3))*(1 + sec(c + d*x))^(1/3))^2)*tan(c + d*x))/(7*a*d*(1 - sec(c + d*x))*(a + a*sec(c + d*x))^(2/3)*sqrt(-(((1 + sec(c + d*x))^(1/3)*(2^(1/3) - (1 + sec(c + d*x))^(1/3)))/(2^(1/3) - (1 + sqrt(3))*(1 + sec(c + d*x))^(1/3))^2))) - (2^(1/3)*3^(3/4)*(1 - sqrt(3))*Elliptic.F(acos((2^(1/3) - (1 - sqrt(3))*(1 + sec(c + d*x))^(1/3))/(2^(1/3) - (1 + sqrt(3))*(1 + sec(c + d*x))^(1/3))), (1/4)*(2 + sqrt(3)))*(1 + sec(c + d*x))^(1/3)*(2^(1/3) - (1 + sec(c + d*x))^(1/3))*sqrt((2^(2/3) + 2^(1/3)*(1 + sec(c + d*x))^(1/3) + (1 + sec(c + d*x))^(2/3))/(2^(1/3) - (1 + sqrt(3))*(1 + sec(c + d*x))^(1/3))^2)*tan(c + d*x))/(7*a*d*(1 - sec(c + d*x))*(a + a*sec(c + d*x))^(2/3)*sqrt(-(((1 + sec(c + d*x))^(1/3)*(2^(1/3) - (1 + sec(c + d*x))^(1/3)))/(2^(1/3) - (1 + sqrt(3))*(1 + sec(c + d*x))^(1/3))^2)))]
    @test_int [sec(c + d*x)^0/(a + a*sec(c + d*x))^(5/3), x, 3, -((3*sqrt(2)*AppellF1(-(7/6), 1/2, 1, -(1/6), (1/2)*(1 + sec(c + d*x)), 1 + sec(c + d*x))*tan(c + d*x))/(7*a*d*sqrt(1 - sec(c + d*x))*(1 + sec(c + d*x))*(a + a*sec(c + d*x))^(2/3)))]
    @test_int [cos(c + d*x)^1/(a + a*sec(c + d*x))^(5/3), x, 3, (3*sqrt(2)*AppellF1(-(7/6), 1/2, 2, -(1/6), (1/2)*(1 + sec(c + d*x)), 1 + sec(c + d*x))*tan(c + d*x))/(7*a*d*sqrt(1 - sec(c + d*x))*(1 + sec(c + d*x))*(a + a*sec(c + d*x))^(2/3))]


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


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


    @test_int [(a + a*sec(c + d*x))*sec(c + d*x)^(5/2), x, 8, (-6*a*sqrt(cos(c + d*x))*Elliptic.E((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(5*d) + (2*a*sqrt(cos(c + d*x))*Elliptic.F((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(3*d) + (6*a*sqrt(sec(c + d*x))*sin(c + d*x))/(5*d) + (2*a*sec(c + d*x)^(3/2)*sin(c + d*x))/(3*d) + (2*a*sec(c + d*x)^(5/2)*sin(c + d*x))/(5*d)]
    @test_int [(a + a*sec(c + d*x))*sec(c + d*x)^(3/2), x, 7, (-2*a*sqrt(cos(c + d*x))*Elliptic.E((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/d + (2*a*sqrt(cos(c + d*x))*Elliptic.F((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(3*d) + (2*a*sqrt(sec(c + d*x))*sin(c + d*x))/d + (2*a*sec(c + d*x)^(3/2)*sin(c + d*x))/(3*d)]
    @test_int [(a + a*sec(c + d*x))*sec(c + d*x)^(1/2), x, 6, (-2*a*sqrt(cos(c + d*x))*Elliptic.E((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/d + (2*a*sqrt(cos(c + d*x))*Elliptic.F((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/d + (2*a*sqrt(sec(c + d*x))*sin(c + d*x))/d]
    @test_int [(a + a*sec(c + d*x))/sec(c + d*x)^(1/2), x, 5, (2*a*sqrt(cos(c + d*x))*Elliptic.E((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/d + (2*a*sqrt(cos(c + d*x))*Elliptic.F((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/d]
    @test_int [(a + a*sec(c + d*x))/sec(c + d*x)^(3/2), x, 6, (2*a*sqrt(cos(c + d*x))*Elliptic.E((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/d + (2*a*sqrt(cos(c + d*x))*Elliptic.F((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(3*d) + (2*a*sin(c + d*x))/(3*d*sqrt(sec(c + d*x)))]
    @test_int [(a + a*sec(c + d*x))/sec(c + d*x)^(5/2), x, 7, (6*a*sqrt(cos(c + d*x))*Elliptic.E((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(5*d) + (2*a*sqrt(cos(c + d*x))*Elliptic.F((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(3*d) + (2*a*sin(c + d*x))/(5*d*sec(c + d*x)^(3/2)) + (2*a*sin(c + d*x))/(3*d*sqrt(sec(c + d*x)))]
    @test_int [(a + a*sec(c + d*x))/sec(c + d*x)^(7/2), x, 8, (6*a*sqrt(cos(c + d*x))*Elliptic.E((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(5*d) + (10*a*sqrt(cos(c + d*x))*Elliptic.F((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(21*d) + (2*a*sin(c + d*x))/(7*d*sec(c + d*x)^(5/2)) + (2*a*sin(c + d*x))/(5*d*sec(c + d*x)^(3/2)) + (10*a*sin(c + d*x))/(21*d*sqrt(sec(c + d*x)))]


    @test_int [(a + a*sec(c + d*x))^2*sec(c + d*x)^(5/2), x, 9, (-12*a^2*sqrt(cos(c + d*x))*Elliptic.E((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(5*d) + (8*a^2*sqrt(cos(c + d*x))*Elliptic.F((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(7*d) + (12*a^2*sqrt(sec(c + d*x))*sin(c + d*x))/(5*d) + (8*a^2*sec(c + d*x)^(3/2)*sin(c + d*x))/(7*d) + (4*a^2*sec(c + d*x)^(5/2)*sin(c + d*x))/(5*d) + (2*a^2*sec(c + d*x)^(7/2)*sin(c + d*x))/(7*d)]
    @test_int [(a + a*sec(c + d*x))^2*sec(c + d*x)^(3/2), x, 8, (-16*a^2*sqrt(cos(c + d*x))*Elliptic.E((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(5*d) + (4*a^2*sqrt(cos(c + d*x))*Elliptic.F((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(3*d) + (16*a^2*sqrt(sec(c + d*x))*sin(c + d*x))/(5*d) + (4*a^2*sec(c + d*x)^(3/2)*sin(c + d*x))/(3*d) + (2*a^2*sec(c + d*x)^(5/2)*sin(c + d*x))/(5*d)]
    @test_int [(a + a*sec(c + d*x))^2*sec(c + d*x)^(1/2), x, 7, (-4*a^2*sqrt(cos(c + d*x))*Elliptic.E((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/d + (8*a^2*sqrt(cos(c + d*x))*Elliptic.F((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(3*d) + (4*a^2*sqrt(sec(c + d*x))*sin(c + d*x))/d + (2*a^2*sec(c + d*x)^(3/2)*sin(c + d*x))/(3*d)]
    @test_int [(a + a*sec(c + d*x))^2/sec(c + d*x)^(1/2), x, 4, (4*a^2*sqrt(cos(c + d*x))*Elliptic.F((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/d + (2*a^2*sqrt(sec(c + d*x))*sin(c + d*x))/d]
    @test_int [(a + a*sec(c + d*x))^2/sec(c + d*x)^(3/2), x, 6, (4*a^2*sqrt(cos(c + d*x))*Elliptic.E((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/d + (8*a^2*sqrt(cos(c + d*x))*Elliptic.F((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(3*d) + (2*a^2*sin(c + d*x))/(3*d*sqrt(sec(c + d*x)))]
    @test_int [(a + a*sec(c + d*x))^2/sec(c + d*x)^(5/2), x, 7, (16*a^2*sqrt(cos(c + d*x))*Elliptic.E((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(5*d) + (4*a^2*sqrt(cos(c + d*x))*Elliptic.F((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(3*d) + (2*a^2*sin(c + d*x))/(5*d*sec(c + d*x)^(3/2)) + (4*a^2*sin(c + d*x))/(3*d*sqrt(sec(c + d*x)))]
    @test_int [(a + a*sec(c + d*x))^2/sec(c + d*x)^(7/2), x, 8, (12*a^2*sqrt(cos(c + d*x))*Elliptic.E((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(5*d) + (8*a^2*sqrt(cos(c + d*x))*Elliptic.F((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(7*d) + (2*a^2*sin(c + d*x))/(7*d*sec(c + d*x)^(5/2)) + (4*a^2*sin(c + d*x))/(5*d*sec(c + d*x)^(3/2)) + (8*a^2*sin(c + d*x))/(7*d*sqrt(sec(c + d*x)))]


    @test_int [(a + a*sec(c + d*x))^3*sec(c + d*x)^(3/2), x, 16, -((28*a^3*sqrt(cos(c + d*x))*Elliptic.E((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(5*d)) + (52*a^3*sqrt(cos(c + d*x))*Elliptic.F((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(21*d) + (28*a^3*sqrt(sec(c + d*x))*sin(c + d*x))/(5*d) + (52*a^3*sec(c + d*x)^(3/2)*sin(c + d*x))/(21*d) + (6*a^3*sec(c + d*x)^(5/2)*sin(c + d*x))/(5*d) + (2*a^3*sec(c + d*x)^(7/2)*sin(c + d*x))/(7*d)]
    @test_int [(a + a*sec(c + d*x))^3*sec(c + d*x)^(1/2), x, 14, -((36*a^3*sqrt(cos(c + d*x))*Elliptic.E((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(5*d)) + (4*a^3*sqrt(cos(c + d*x))*Elliptic.F((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/d + (36*a^3*sqrt(sec(c + d*x))*sin(c + d*x))/(5*d) + (2*a^3*sec(c + d*x)^(3/2)*sin(c + d*x))/d + (2*a^3*sec(c + d*x)^(5/2)*sin(c + d*x))/(5*d)]
    @test_int [(a + a*sec(c + d*x))^3/sec(c + d*x)^(1/2), x, 12, -((4*a^3*sqrt(cos(c + d*x))*Elliptic.E((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/d) + (20*a^3*sqrt(cos(c + d*x))*Elliptic.F((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(3*d) + (6*a^3*sqrt(sec(c + d*x))*sin(c + d*x))/d + (2*a^3*sec(c + d*x)^(3/2)*sin(c + d*x))/(3*d)]
    @test_int [(a + a*sec(c + d*x))^3/sec(c + d*x)^(3/2), x, 12, (4*a^3*sqrt(cos(c + d*x))*Elliptic.E((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/d + (20*a^3*sqrt(cos(c + d*x))*Elliptic.F((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(3*d) + (2*a^3*sin(c + d*x))/(3*d*sqrt(sec(c + d*x))) + (2*a^3*sqrt(sec(c + d*x))*sin(c + d*x))/d]
    @test_int [(a + a*sec(c + d*x))^3/sec(c + d*x)^(5/2), x, 12, (36*a^3*sqrt(cos(c + d*x))*Elliptic.E((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(5*d) + (4*a^3*sqrt(cos(c + d*x))*Elliptic.F((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/d + (2*a^3*sin(c + d*x))/(5*d*sec(c + d*x)^(3/2)) + (2*a^3*sin(c + d*x))/(d*sqrt(sec(c + d*x)))]
    @test_int [(a + a*sec(c + d*x))^3/sec(c + d*x)^(7/2), x, 14, (28*a^3*sqrt(cos(c + d*x))*Elliptic.E((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(5*d) + (52*a^3*sqrt(cos(c + d*x))*Elliptic.F((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(21*d) + (2*a^3*sin(c + d*x))/(7*d*sec(c + d*x)^(5/2)) + (6*a^3*sin(c + d*x))/(5*d*sec(c + d*x)^(3/2)) + (52*a^3*sin(c + d*x))/(21*d*sqrt(sec(c + d*x)))]
    @test_int [(a + a*sec(c + d*x))^3/sec(c + d*x)^(9/2), x, 16, (68*a^3*sqrt(cos(c + d*x))*Elliptic.E((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(15*d) + (44*a^3*sqrt(cos(c + d*x))*Elliptic.F((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(21*d) + (2*a^3*sin(c + d*x))/(9*d*sec(c + d*x)^(7/2)) + (6*a^3*sin(c + d*x))/(7*d*sec(c + d*x)^(5/2)) + (68*a^3*sin(c + d*x))/(45*d*sec(c + d*x)^(3/2)) + (44*a^3*sin(c + d*x))/(21*d*sqrt(sec(c + d*x)))]


    @test_int [(a + a*sec(c + d*x))^4*sec(c + d*x)^(3/2), x, 21, -((152*a^4*sqrt(cos(c + d*x))*Elliptic.E((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(15*d)) + (32*a^4*sqrt(cos(c + d*x))*Elliptic.F((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(7*d) + (152*a^4*sqrt(sec(c + d*x))*sin(c + d*x))/(15*d) + (32*a^4*sec(c + d*x)^(3/2)*sin(c + d*x))/(7*d) + (122*a^4*sec(c + d*x)^(5/2)*sin(c + d*x))/(45*d) + (8*a^4*sec(c + d*x)^(7/2)*sin(c + d*x))/(7*d) + (2*a^4*sec(c + d*x)^(9/2)*sin(c + d*x))/(9*d)]
    @test_int [(a + a*sec(c + d*x))^4*sec(c + d*x)^(1/2), x, 18, -((64*a^4*sqrt(cos(c + d*x))*Elliptic.E((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(5*d)) + (136*a^4*sqrt(cos(c + d*x))*Elliptic.F((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(21*d) + (64*a^4*sqrt(sec(c + d*x))*sin(c + d*x))/(5*d) + (94*a^4*sec(c + d*x)^(3/2)*sin(c + d*x))/(21*d) + (8*a^4*sec(c + d*x)^(5/2)*sin(c + d*x))/(5*d) + (2*a^4*sec(c + d*x)^(7/2)*sin(c + d*x))/(7*d)]
    @test_int [(a + a*sec(c + d*x))^4/sec(c + d*x)^(1/2), x, 16, -((56*a^4*sqrt(cos(c + d*x))*Elliptic.E((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(5*d)) + (32*a^4*sqrt(cos(c + d*x))*Elliptic.F((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(3*d) + (66*a^4*sqrt(sec(c + d*x))*sin(c + d*x))/(5*d) + (8*a^4*sec(c + d*x)^(3/2)*sin(c + d*x))/(3*d) + (2*a^4*sec(c + d*x)^(5/2)*sin(c + d*x))/(5*d)]
    @test_int [(a + a*sec(c + d*x))^4/sec(c + d*x)^(3/2), x, 15, (40*a^4*sqrt(cos(c + d*x))*Elliptic.F((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(3*d) + (2*a^4*sin(c + d*x))/(3*d*sqrt(sec(c + d*x))) + (8*a^4*sqrt(sec(c + d*x))*sin(c + d*x))/d + (2*a^4*sec(c + d*x)^(3/2)*sin(c + d*x))/(3*d)]
    @test_int [(a + a*sec(c + d*x))^4/sec(c + d*x)^(5/2), x, 15, (56*a^4*sqrt(cos(c + d*x))*Elliptic.E((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(5*d) + (32*a^4*sqrt(cos(c + d*x))*Elliptic.F((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(3*d) + (2*a^4*sin(c + d*x))/(5*d*sec(c + d*x)^(3/2)) + (8*a^4*sin(c + d*x))/(3*d*sqrt(sec(c + d*x))) + (2*a^4*sqrt(sec(c + d*x))*sin(c + d*x))/d]
    @test_int [(a + a*sec(c + d*x))^4/sec(c + d*x)^(7/2), x, 16, (64*a^4*sqrt(cos(c + d*x))*Elliptic.E((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(5*d) + (136*a^4*sqrt(cos(c + d*x))*Elliptic.F((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(21*d) + (2*a^4*sin(c + d*x))/(7*d*sec(c + d*x)^(5/2)) + (8*a^4*sin(c + d*x))/(5*d*sec(c + d*x)^(3/2)) + (94*a^4*sin(c + d*x))/(21*d*sqrt(sec(c + d*x)))]
    @test_int [(a + a*sec(c + d*x))^4/sec(c + d*x)^(9/2), x, 18, (152*a^4*sqrt(cos(c + d*x))*Elliptic.E((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(15*d) + (32*a^4*sqrt(cos(c + d*x))*Elliptic.F((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(7*d) + (2*a^4*sin(c + d*x))/(9*d*sec(c + d*x)^(7/2)) + (8*a^4*sin(c + d*x))/(7*d*sec(c + d*x)^(5/2)) + (122*a^4*sin(c + d*x))/(45*d*sec(c + d*x)^(3/2)) + (32*a^4*sin(c + d*x))/(7*d*sqrt(sec(c + d*x)))]
    @test_int [(a + a*sec(c + d*x))^4/sec(c + d*x)^(11/2), x, 21, (128*a^4*sqrt(cos(c + d*x))*Elliptic.E((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(15*d) + (904*a^4*sqrt(cos(c + d*x))*Elliptic.F((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(231*d) + (2*a^4*sin(c + d*x))/(11*d*sec(c + d*x)^(9/2)) + (8*a^4*sin(c + d*x))/(9*d*sec(c + d*x)^(7/2)) + (150*a^4*sin(c + d*x))/(77*d*sec(c + d*x)^(5/2)) + (128*a^4*sin(c + d*x))/(45*d*sec(c + d*x)^(3/2)) + (904*a^4*sin(c + d*x))/(231*d*sqrt(sec(c + d*x)))]


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


    @test_int [sec(c + d*x)^(7/2)/(a + a*sec(c + d*x)), x, 8, (3*sqrt(cos(c + d*x))*Elliptic.E((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(a*d) + (5*sqrt(cos(c + d*x))*Elliptic.F((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(3*a*d) - (3*sqrt(sec(c + d*x))*sin(c + d*x))/(a*d) + (5*sec(c + d*x)^(3/2)*sin(c + d*x))/(3*a*d) - (sec(c + d*x)^(5/2)*sin(c + d*x))/(d*(a + a*sec(c + d*x)))]
    @test_int [sec(c + d*x)^(5/2)/(a + a*sec(c + d*x)), x, 7, -((3*sqrt(cos(c + d*x))*Elliptic.E((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(a*d)) - (sqrt(cos(c + d*x))*Elliptic.F((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(a*d) + (3*sqrt(sec(c + d*x))*sin(c + d*x))/(a*d) - (sec(c + d*x)^(3/2)*sin(c + d*x))/(d*(a + a*sec(c + d*x)))]
    @test_int [sec(c + d*x)^(3/2)/(a + a*sec(c + d*x)), x, 6, (sqrt(cos(c + d*x))*Elliptic.E((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(a*d) + (sqrt(cos(c + d*x))*Elliptic.F((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(a*d) - (sqrt(sec(c + d*x))*sin(c + d*x))/(d*(a + a*sec(c + d*x)))]
    @test_int [sec(c + d*x)^(1/2)/(a + a*sec(c + d*x)), x, 6, -((sqrt(cos(c + d*x))*Elliptic.E((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(a*d)) + (sqrt(cos(c + d*x))*Elliptic.F((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(a*d) + (sqrt(sec(c + d*x))*sin(c + d*x))/(d*(a + a*sec(c + d*x)))]
    @test_int [1/(sec(c + d*x)^(1/2)*(a + a*sec(c + d*x))), x, 6, (3*sqrt(cos(c + d*x))*Elliptic.E((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(a*d) - (sqrt(cos(c + d*x))*Elliptic.F((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(a*d) - (sqrt(sec(c + d*x))*sin(c + d*x))/(d*(a + a*sec(c + d*x)))]
    @test_int [1/(sec(c + d*x)^(3/2)*(a + a*sec(c + d*x))), x, 7, -((3*sqrt(cos(c + d*x))*Elliptic.E((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(a*d)) + (5*sqrt(cos(c + d*x))*Elliptic.F((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(3*a*d) + (5*sin(c + d*x))/(3*a*d*sqrt(sec(c + d*x))) - sin(c + d*x)/(d*sqrt(sec(c + d*x))*(a + a*sec(c + d*x)))]
    @test_int [1/(sec(c + d*x)^(5/2)*(a + a*sec(c + d*x))), x, 8, (21*sqrt(cos(c + d*x))*Elliptic.E((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(5*a*d) - (5*sqrt(cos(c + d*x))*Elliptic.F((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(3*a*d) + (7*sin(c + d*x))/(5*a*d*sec(c + d*x)^(3/2)) - (5*sin(c + d*x))/(3*a*d*sqrt(sec(c + d*x))) - sin(c + d*x)/(d*sec(c + d*x)^(3/2)*(a + a*sec(c + d*x)))]


    @test_int [sec(c + d*x)^(9/2)/(a + a*sec(c + d*x))^2, x, 9, (7*sqrt(cos(c + d*x))*Elliptic.E((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(a^2*d) + (10*sqrt(cos(c + d*x))*Elliptic.F((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(3*a^2*d) - (7*sqrt(sec(c + d*x))*sin(c + d*x))/(a^2*d) + (10*sec(c + d*x)^(3/2)*sin(c + d*x))/(3*a^2*d) - (7*sec(c + d*x)^(5/2)*sin(c + d*x))/(3*a^2*d*(1 + sec(c + d*x))) - (sec(c + d*x)^(7/2)*sin(c + d*x))/(3*d*(a + a*sec(c + d*x))^2)]
    @test_int [sec(c + d*x)^(7/2)/(a + a*sec(c + d*x))^2, x, 8, -((4*sqrt(cos(c + d*x))*Elliptic.E((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(a^2*d)) - (5*sqrt(cos(c + d*x))*Elliptic.F((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(3*a^2*d) + (4*sqrt(sec(c + d*x))*sin(c + d*x))/(a^2*d) - (5*sec(c + d*x)^(3/2)*sin(c + d*x))/(3*a^2*d*(1 + sec(c + d*x))) - (sec(c + d*x)^(5/2)*sin(c + d*x))/(3*d*(a + a*sec(c + d*x))^2)]
    @test_int [sec(c + d*x)^(5/2)/(a + a*sec(c + d*x))^2, x, 7, (sqrt(cos(c + d*x))*Elliptic.E((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(a^2*d) + (2*sqrt(cos(c + d*x))*Elliptic.F((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(3*a^2*d) - (sqrt(sec(c + d*x))*sin(c + d*x))/(a^2*d*(1 + sec(c + d*x))) - (sec(c + d*x)^(3/2)*sin(c + d*x))/(3*d*(a + a*sec(c + d*x))^2)]
    @test_int [sec(c + d*x)^(3/2)/(a + a*sec(c + d*x))^2, x, 4, (sqrt(cos(c + d*x))*Elliptic.F((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(3*a^2*d) + (sec(c + d*x)^(3/2)*sin(c + d*x))/(3*d*(a + a*sec(c + d*x))^2)]
    @test_int [sec(c + d*x)^(1/2)/(a + a*sec(c + d*x))^2, x, 7, -((sqrt(cos(c + d*x))*Elliptic.E((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(a^2*d)) + (2*sqrt(cos(c + d*x))*Elliptic.F((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(3*a^2*d) + (sqrt(sec(c + d*x))*sin(c + d*x))/(a^2*d*(1 + sec(c + d*x))) - (sec(c + d*x)^(3/2)*sin(c + d*x))/(3*d*(a + a*sec(c + d*x))^2)]
    @test_int [1/(sec(c + d*x)^(1/2)*(a + a*sec(c + d*x))^2), x, 7, (4*sqrt(cos(c + d*x))*Elliptic.E((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(a^2*d) - (5*sqrt(cos(c + d*x))*Elliptic.F((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(3*a^2*d) - (5*sqrt(sec(c + d*x))*sin(c + d*x))/(3*a^2*d*(1 + sec(c + d*x))) - (sqrt(sec(c + d*x))*sin(c + d*x))/(3*d*(a + a*sec(c + d*x))^2)]
    @test_int [1/(sec(c + d*x)^(3/2)*(a + a*sec(c + d*x))^2), x, 8, -((7*sqrt(cos(c + d*x))*Elliptic.E((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(a^2*d)) + (10*sqrt(cos(c + d*x))*Elliptic.F((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(3*a^2*d) + (10*sin(c + d*x))/(3*a^2*d*sqrt(sec(c + d*x))) - (7*sin(c + d*x))/(3*a^2*d*sqrt(sec(c + d*x))*(1 + sec(c + d*x))) - sin(c + d*x)/(3*d*sqrt(sec(c + d*x))*(a + a*sec(c + d*x))^2)]
    @test_int [1/(sec(c + d*x)^(5/2)*(a + a*sec(c + d*x))^2), x, 9, (56*sqrt(cos(c + d*x))*Elliptic.E((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(5*a^2*d) - (5*sqrt(cos(c + d*x))*Elliptic.F((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(a^2*d) + (56*sin(c + d*x))/(15*a^2*d*sec(c + d*x)^(3/2)) - (5*sin(c + d*x))/(a^2*d*sqrt(sec(c + d*x))) - (3*sin(c + d*x))/(a^2*d*sec(c + d*x)^(3/2)*(1 + sec(c + d*x))) - sin(c + d*x)/(3*d*sec(c + d*x)^(3/2)*(a + a*sec(c + d*x))^2)]


    @test_int [sec(c + d*x)^(11/2)/(a + a*sec(c + d*x))^3, x, 10, (119*sqrt(cos(c + d*x))*Elliptic.E((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(10*a^3*d) + (11*sqrt(cos(c + d*x))*Elliptic.F((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(2*a^3*d) - (119*sqrt(sec(c + d*x))*sin(c + d*x))/(10*a^3*d) + (11*sec(c + d*x)^(3/2)*sin(c + d*x))/(2*a^3*d) - (sec(c + d*x)^(9/2)*sin(c + d*x))/(5*d*(a + a*sec(c + d*x))^3) - (2*sec(c + d*x)^(7/2)*sin(c + d*x))/(3*a*d*(a + a*sec(c + d*x))^2) - (119*sec(c + d*x)^(5/2)*sin(c + d*x))/(30*d*(a^3 + a^3*sec(c + d*x)))]
    @test_int [sec(c + d*x)^(9/2)/(a + a*sec(c + d*x))^3, x, 9, -((49*sqrt(cos(c + d*x))*Elliptic.E((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(10*a^3*d)) - (13*sqrt(cos(c + d*x))*Elliptic.F((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(6*a^3*d) + (49*sqrt(sec(c + d*x))*sin(c + d*x))/(10*a^3*d) - (sec(c + d*x)^(7/2)*sin(c + d*x))/(5*d*(a + a*sec(c + d*x))^3) - (8*sec(c + d*x)^(5/2)*sin(c + d*x))/(15*a*d*(a + a*sec(c + d*x))^2) - (13*sec(c + d*x)^(3/2)*sin(c + d*x))/(6*d*(a^3 + a^3*sec(c + d*x)))]
    @test_int [sec(c + d*x)^(7/2)/(a + a*sec(c + d*x))^3, x, 8, (9*sqrt(cos(c + d*x))*Elliptic.E((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(10*a^3*d) + (sqrt(cos(c + d*x))*Elliptic.F((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(2*a^3*d) - (sec(c + d*x)^(5/2)*sin(c + d*x))/(5*d*(a + a*sec(c + d*x))^3) - (2*sec(c + d*x)^(3/2)*sin(c + d*x))/(5*a*d*(a + a*sec(c + d*x))^2) - (9*sqrt(sec(c + d*x))*sin(c + d*x))/(10*d*(a^3 + a^3*sec(c + d*x)))]
    @test_int [sec(c + d*x)^(5/2)/(a + a*sec(c + d*x))^3, x, 8, (sqrt(cos(c + d*x))*Elliptic.E((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(10*a^3*d) + (sqrt(cos(c + d*x))*Elliptic.F((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(6*a^3*d) - (sec(c + d*x)^(3/2)*sin(c + d*x))/(5*d*(a + a*sec(c + d*x))^3) - (4*sqrt(sec(c + d*x))*sin(c + d*x))/(15*a*d*(a + a*sec(c + d*x))^2) + (sqrt(sec(c + d*x))*sin(c + d*x))/(6*d*(a^3 + a^3*sec(c + d*x)))]
    @test_int [sec(c + d*x)^(3/2)/(a + a*sec(c + d*x))^3, x, 8, -((sqrt(cos(c + d*x))*Elliptic.E((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(10*a^3*d)) + (sqrt(cos(c + d*x))*Elliptic.F((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(6*a^3*d) + (sec(c + d*x)^(3/2)*sin(c + d*x))/(5*d*(a + a*sec(c + d*x))^3) - (sqrt(sec(c + d*x))*sin(c + d*x))/(15*a*d*(a + a*sec(c + d*x))^2) + (sqrt(sec(c + d*x))*sin(c + d*x))/(6*d*(a^3 + a^3*sec(c + d*x)))]
    @test_int [sec(c + d*x)^(1/2)/(a + a*sec(c + d*x))^3, x, 8, -((9*sqrt(cos(c + d*x))*Elliptic.E((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(10*a^3*d)) + (sqrt(cos(c + d*x))*Elliptic.F((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(2*a^3*d) - (sec(c + d*x)^(3/2)*sin(c + d*x))/(5*d*(a + a*sec(c + d*x))^3) + (2*sqrt(sec(c + d*x))*sin(c + d*x))/(5*a*d*(a + a*sec(c + d*x))^2) + (sqrt(sec(c + d*x))*sin(c + d*x))/(2*d*(a^3 + a^3*sec(c + d*x)))]
    @test_int [1/(sec(c + d*x)^(1/2)*(a + a*sec(c + d*x))^3), x, 8, (49*sqrt(cos(c + d*x))*Elliptic.E((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(10*a^3*d) - (13*sqrt(cos(c + d*x))*Elliptic.F((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(6*a^3*d) - (sqrt(sec(c + d*x))*sin(c + d*x))/(5*d*(a + a*sec(c + d*x))^3) - (8*sqrt(sec(c + d*x))*sin(c + d*x))/(15*a*d*(a + a*sec(c + d*x))^2) - (13*sqrt(sec(c + d*x))*sin(c + d*x))/(6*d*(a^3 + a^3*sec(c + d*x)))]
    @test_int [1/(sec(c + d*x)^(3/2)*(a + a*sec(c + d*x))^3), x, 9, -((119*sqrt(cos(c + d*x))*Elliptic.E((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(10*a^3*d)) + (11*sqrt(cos(c + d*x))*Elliptic.F((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(2*a^3*d) + (11*sin(c + d*x))/(2*a^3*d*sqrt(sec(c + d*x))) - sin(c + d*x)/(5*d*sqrt(sec(c + d*x))*(a + a*sec(c + d*x))^3) - (2*sin(c + d*x))/(3*a*d*sqrt(sec(c + d*x))*(a + a*sec(c + d*x))^2) - (119*sin(c + d*x))/(30*d*sqrt(sec(c + d*x))*(a^3 + a^3*sec(c + d*x)))]
    @test_int [1/(sec(c + d*x)^(5/2)*(a + a*sec(c + d*x))^3), x, 10, (231*sqrt(cos(c + d*x))*Elliptic.E((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(10*a^3*d) - (21*sqrt(cos(c + d*x))*Elliptic.F((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(2*a^3*d) + (77*sin(c + d*x))/(10*a^3*d*sec(c + d*x)^(3/2)) - (21*sin(c + d*x))/(2*a^3*d*sqrt(sec(c + d*x))) - sin(c + d*x)/(5*d*sec(c + d*x)^(3/2)*(a + a*sec(c + d*x))^3) - (4*sin(c + d*x))/(5*a*d*sec(c + d*x)^(3/2)*(a + a*sec(c + d*x))^2) - (63*sin(c + d*x))/(10*d*sec(c + d*x)^(3/2)*(a^3 + a^3*sec(c + d*x)))]


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


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


    @test_int [sec(c + d*x)^(5/2)*sqrt(a + a*sec(c + d*x)), x, 4, (3*sqrt(a)*asinh((sqrt(a)*tan(c + d*x))/sqrt(a + a*sec(c + d*x))))/(4*d) + (3*a*sec(c + d*x)^(3/2)*sin(c + d*x))/(4*d*sqrt(a + a*sec(c + d*x))) + (a*sec(c + d*x)^(5/2)*sin(c + d*x))/(2*d*sqrt(a + a*sec(c + d*x)))]
    @test_int [sec(c + d*x)^(3/2)*sqrt(a + a*sec(c + d*x)), x, 3, (sqrt(a)*asinh((sqrt(a)*tan(c + d*x))/sqrt(a + a*sec(c + d*x))))/d + (a*sec(c + d*x)^(3/2)*sin(c + d*x))/(d*sqrt(a + a*sec(c + d*x)))]
    @test_int [sec(c + d*x)^(1/2)*sqrt(a + a*sec(c + d*x)), x, 2, (2*sqrt(a)*asinh((sqrt(a)*tan(c + d*x))/sqrt(a + a*sec(c + d*x))))/d]
    @test_int [sqrt(a + a*sec(c + d*x))/sec(c + d*x)^(1/2), x, 1, (2*a*sqrt(sec(c + d*x))*sin(c + d*x))/(d*sqrt(a + a*sec(c + d*x)))]
    @test_int [sqrt(a + a*sec(c + d*x))/sec(c + d*x)^(3/2), x, 2, (2*a*sin(c + d*x))/(3*d*sqrt(sec(c + d*x))*sqrt(a + a*sec(c + d*x))) + (4*a*sqrt(sec(c + d*x))*sin(c + d*x))/(3*d*sqrt(a + a*sec(c + d*x)))]
    @test_int [sqrt(a + a*sec(c + d*x))/sec(c + d*x)^(5/2), x, 3, (2*a*sin(c + d*x))/(5*d*sec(c + d*x)^(3/2)*sqrt(a + a*sec(c + d*x))) + (8*a*sin(c + d*x))/(15*d*sqrt(sec(c + d*x))*sqrt(a + a*sec(c + d*x))) + (16*a*sqrt(sec(c + d*x))*sin(c + d*x))/(15*d*sqrt(a + a*sec(c + d*x)))]
    @test_int [sqrt(a + a*sec(c + d*x))/sec(c + d*x)^(7/2), x, 4, (2*a*sin(c + d*x))/(7*d*sec(c + d*x)^(5/2)*sqrt(a + a*sec(c + d*x))) + (12*a*sin(c + d*x))/(35*d*sec(c + d*x)^(3/2)*sqrt(a + a*sec(c + d*x))) + (16*a*sin(c + d*x))/(35*d*sqrt(sec(c + d*x))*sqrt(a + a*sec(c + d*x))) + (32*a*sqrt(sec(c + d*x))*sin(c + d*x))/(35*d*sqrt(a + a*sec(c + d*x)))]


    @test_int [sec(c + d*x)^(5/2)*(a + a*sec(c + d*x))^(3/2), x, 6, (11*a^(3/2)*asinh((sqrt(a)*tan(c + d*x))/sqrt(a + a*sec(c + d*x))))/(8*d) + (11*a^2*sec(c + d*x)^(3/2)*sin(c + d*x))/(8*d*sqrt(a + a*sec(c + d*x))) + (11*a^2*sec(c + d*x)^(5/2)*sin(c + d*x))/(12*d*sqrt(a + a*sec(c + d*x))) + (a^2*sec(c + d*x)^(7/2)*sin(c + d*x))/(3*d*sqrt(a + a*sec(c + d*x)))]
    @test_int [sec(c + d*x)^(3/2)*(a + a*sec(c + d*x))^(3/2), x, 5, (7*a^(3/2)*asinh((sqrt(a)*tan(c + d*x))/sqrt(a + a*sec(c + d*x))))/(4*d) + (7*a^2*sec(c + d*x)^(3/2)*sin(c + d*x))/(4*d*sqrt(a + a*sec(c + d*x))) + (a^2*sec(c + d*x)^(5/2)*sin(c + d*x))/(2*d*sqrt(a + a*sec(c + d*x)))]
    @test_int [sec(c + d*x)^(1/2)*(a + a*sec(c + d*x))^(3/2), x, 4, (3*a^(3/2)*asinh((sqrt(a)*tan(c + d*x))/sqrt(a + a*sec(c + d*x))))/d + (a^2*sec(c + d*x)^(3/2)*sin(c + d*x))/(d*sqrt(a + a*sec(c + d*x)))]
    @test_int [(a + a*sec(c + d*x))^(3/2)/sec(c + d*x)^(1/2), x, 4, (2*a^(3/2)*asinh((sqrt(a)*tan(c + d*x))/sqrt(a + a*sec(c + d*x))))/d + (2*a^2*sqrt(sec(c + d*x))*sin(c + d*x))/(d*sqrt(a + a*sec(c + d*x)))]
    @test_int [(a + a*sec(c + d*x))^(3/2)/sec(c + d*x)^(3/2), x, 2, (8*a^2*sqrt(sec(c + d*x))*sin(c + d*x))/(3*d*sqrt(a + a*sec(c + d*x))) + (2*a*sqrt(a + a*sec(c + d*x))*sin(c + d*x))/(3*d*sqrt(sec(c + d*x)))]
    @test_int [(a + a*sec(c + d*x))^(3/2)/sec(c + d*x)^(5/2), x, 3, (8*a^2*sqrt(sec(c + d*x))*sin(c + d*x))/(5*d*sqrt(a + a*sec(c + d*x))) + (2*a*sqrt(a + a*sec(c + d*x))*sin(c + d*x))/(5*d*sqrt(sec(c + d*x))) + (2*(a + a*sec(c + d*x))^(3/2)*sin(c + d*x))/(5*d*sec(c + d*x)^(3/2))]
    @test_int [(a + a*sec(c + d*x))^(3/2)/sec(c + d*x)^(7/2), x, 5, (2*a^2*sin(c + d*x))/(7*d*sec(c + d*x)^(5/2)*sqrt(a + a*sec(c + d*x))) + (26*a^2*sin(c + d*x))/(35*d*sec(c + d*x)^(3/2)*sqrt(a + a*sec(c + d*x))) + (104*a^2*sin(c + d*x))/(105*d*sqrt(sec(c + d*x))*sqrt(a + a*sec(c + d*x))) + (208*a^2*sqrt(sec(c + d*x))*sin(c + d*x))/(105*d*sqrt(a + a*sec(c + d*x)))]
    @test_int [(a + a*sec(c + d*x))^(3/2)/sec(c + d*x)^(9/2), x, 6, (2*a^2*sin(c + d*x))/(9*d*sec(c + d*x)^(7/2)*sqrt(a + a*sec(c + d*x))) + (34*a^2*sin(c + d*x))/(63*d*sec(c + d*x)^(5/2)*sqrt(a + a*sec(c + d*x))) + (68*a^2*sin(c + d*x))/(105*d*sec(c + d*x)^(3/2)*sqrt(a + a*sec(c + d*x))) + (272*a^2*sin(c + d*x))/(315*d*sqrt(sec(c + d*x))*sqrt(a + a*sec(c + d*x))) + (544*a^2*sqrt(sec(c + d*x))*sin(c + d*x))/(315*d*sqrt(a + a*sec(c + d*x)))]


    @test_int [sec(c + d*x)^(5/2)*(a + a*sec(c + d*x))^(5/2), x, 6, (163*a^(5/2)*asinh((sqrt(a)*tan(c + d*x))/sqrt(a + a*sec(c + d*x))))/(64*d) + (163*a^3*sec(c + d*x)^(3/2)*sin(c + d*x))/(64*d*sqrt(a + a*sec(c + d*x))) + (163*a^3*sec(c + d*x)^(5/2)*sin(c + d*x))/(96*d*sqrt(a + a*sec(c + d*x))) + (17*a^3*sec(c + d*x)^(7/2)*sin(c + d*x))/(24*d*sqrt(a + a*sec(c + d*x))) + (a^2*sec(c + d*x)^(7/2)*sqrt(a + a*sec(c + d*x))*sin(c + d*x))/(4*d)]
    @test_int [sec(c + d*x)^(3/2)*(a + a*sec(c + d*x))^(5/2), x, 5, (25*a^(5/2)*asinh((sqrt(a)*tan(c + d*x))/sqrt(a + a*sec(c + d*x))))/(8*d) + (25*a^3*sec(c + d*x)^(3/2)*sin(c + d*x))/(8*d*sqrt(a + a*sec(c + d*x))) + (13*a^3*sec(c + d*x)^(5/2)*sin(c + d*x))/(12*d*sqrt(a + a*sec(c + d*x))) + (a^2*sec(c + d*x)^(5/2)*sqrt(a + a*sec(c + d*x))*sin(c + d*x))/(3*d)]
    @test_int [sec(c + d*x)^(1/2)*(a + a*sec(c + d*x))^(5/2), x, 4, (19*a^(5/2)*asinh((sqrt(a)*tan(c + d*x))/sqrt(a + a*sec(c + d*x))))/(4*d) + (9*a^3*sec(c + d*x)^(3/2)*sin(c + d*x))/(4*d*sqrt(a + a*sec(c + d*x))) + (a^2*sec(c + d*x)^(3/2)*sqrt(a + a*sec(c + d*x))*sin(c + d*x))/(2*d)]
    @test_int [(a + a*sec(c + d*x))^(5/2)/sec(c + d*x)^(1/2), x, 4, (5*a^(5/2)*asinh((sqrt(a)*tan(c + d*x))/sqrt(a + a*sec(c + d*x))))/d + (a^3*sqrt(sec(c + d*x))*sin(c + d*x))/(d*sqrt(a + a*sec(c + d*x))) + (a^2*sqrt(sec(c + d*x))*sqrt(a + a*sec(c + d*x))*sin(c + d*x))/d]
    @test_int [(a + a*sec(c + d*x))^(5/2)/sec(c + d*x)^(3/2), x, 4, (2*a^(5/2)*asinh((sqrt(a)*tan(c + d*x))/sqrt(a + a*sec(c + d*x))))/d + (14*a^3*sqrt(sec(c + d*x))*sin(c + d*x))/(3*d*sqrt(a + a*sec(c + d*x))) + (2*a^2*sqrt(a + a*sec(c + d*x))*sin(c + d*x))/(3*d*sqrt(sec(c + d*x)))]
    @test_int [(a + a*sec(c + d*x))^(5/2)/sec(c + d*x)^(5/2), x, 3, (64*a^3*sqrt(sec(c + d*x))*sin(c + d*x))/(15*d*sqrt(a + a*sec(c + d*x))) + (16*a^2*sqrt(a + a*sec(c + d*x))*sin(c + d*x))/(15*d*sqrt(sec(c + d*x))) + (2*a*(a + a*sec(c + d*x))^(3/2)*sin(c + d*x))/(5*d*sec(c + d*x)^(3/2))]
    @test_int [(a + a*sec(c + d*x))^(5/2)/sec(c + d*x)^(7/2), x, 4, (64*a^3*sqrt(sec(c + d*x))*sin(c + d*x))/(21*d*sqrt(a + a*sec(c + d*x))) + (16*a^2*sqrt(a + a*sec(c + d*x))*sin(c + d*x))/(21*d*sqrt(sec(c + d*x))) + (2*a*(a + a*sec(c + d*x))^(3/2)*sin(c + d*x))/(7*d*sec(c + d*x)^(3/2)) + (2*(a + a*sec(c + d*x))^(5/2)*sin(c + d*x))/(7*d*sec(c + d*x)^(5/2))]
    @test_int [(a + a*sec(c + d*x))^(5/2)/sec(c + d*x)^(9/2), x, 5, (38*a^3*sin(c + d*x))/(63*d*sec(c + d*x)^(5/2)*sqrt(a + a*sec(c + d*x))) + (146*a^3*sin(c + d*x))/(105*d*sec(c + d*x)^(3/2)*sqrt(a + a*sec(c + d*x))) + (584*a^3*sin(c + d*x))/(315*d*sqrt(sec(c + d*x))*sqrt(a + a*sec(c + d*x))) + (1168*a^3*sqrt(sec(c + d*x))*sin(c + d*x))/(315*d*sqrt(a + a*sec(c + d*x))) + (2*a^2*sqrt(a + a*sec(c + d*x))*sin(c + d*x))/(9*d*sec(c + d*x)^(7/2))]
    @test_int [(a + a*sec(c + d*x))^(5/2)/sec(c + d*x)^(11/2), x, 6, (46*a^3*sin(c + d*x))/(99*d*sec(c + d*x)^(7/2)*sqrt(a + a*sec(c + d*x))) + (710*a^3*sin(c + d*x))/(693*d*sec(c + d*x)^(5/2)*sqrt(a + a*sec(c + d*x))) + (284*a^3*sin(c + d*x))/(231*d*sec(c + d*x)^(3/2)*sqrt(a + a*sec(c + d*x))) + (1136*a^3*sin(c + d*x))/(693*d*sqrt(sec(c + d*x))*sqrt(a + a*sec(c + d*x))) + (2272*a^3*sqrt(sec(c + d*x))*sin(c + d*x))/(693*d*sqrt(a + a*sec(c + d*x))) + (2*a^2*sqrt(a + a*sec(c + d*x))*sin(c + d*x))/(11*d*sec(c + d*x)^(9/2))]


    @test_int [sec(c + d*x)^(-1/4)*(a + a*sec(c + d*x))^(3/2), x, 2, (4*a^2*sec(c + d*x)^(3/4)*sin(c + d*x))/(d*sqrt(a + a*sec(c + d*x)))]


    @test_int [sqrt(sec(e + f*x))*sqrt(a + a*sec(e + f*x)), x, 2, (2*sqrt(a)*asinh((sqrt(a)*tan(e + f*x))/sqrt(a + a*sec(e + f*x))))/f]
    @test_int [sqrt(-sec(e + f*x))*sqrt(a - a*sec(e + f*x)), x, 2, (2*sqrt(a)*asinh((sqrt(a)*tan(e + f*x))/sqrt(a - a*sec(e + f*x))))/f]


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


    @test_int [sec(c + d*x)^(5/2)/sqrt(a + a*sec(c + d*x)), x, 6, -(asinh((sqrt(a)*tan(c + d*x))/sqrt(a + a*sec(c + d*x)))/(sqrt(a)*d)) + (sqrt(2)*atanh((sqrt(a)*sqrt(sec(c + d*x))*sin(c + d*x))/(sqrt(2)*sqrt(a + a*sec(c + d*x)))))/(sqrt(a)*d) + (sec(c + d*x)^(3/2)*sin(c + d*x))/(d*sqrt(a + a*sec(c + d*x)))]
    @test_int [sec(c + d*x)^(3/2)/sqrt(a + a*sec(c + d*x)), x, 5, (2*asinh((sqrt(a)*tan(c + d*x))/sqrt(a + a*sec(c + d*x))))/(sqrt(a)*d) - (sqrt(2)*atanh((sqrt(a)*sqrt(sec(c + d*x))*sin(c + d*x))/(sqrt(2)*sqrt(a + a*sec(c + d*x)))))/(sqrt(a)*d)]
    @test_int [sqrt(sec(c + d*x))/sqrt(a + a*sec(c + d*x)), x, 2, (sqrt(2)*atanh((sqrt(a)*sqrt(sec(c + d*x))*sin(c + d*x))/(sqrt(2)*sqrt(a + a*sec(c + d*x)))))/(sqrt(a)*d)]
    @test_int [1/(sqrt(sec(c + d*x))*sqrt(a + a*sec(c + d*x))), x, 3, -((sqrt(2)*atanh((sqrt(a)*sqrt(sec(c + d*x))*sin(c + d*x))/(sqrt(2)*sqrt(a + a*sec(c + d*x)))))/(sqrt(a)*d)) + (2*sqrt(sec(c + d*x))*sin(c + d*x))/(d*sqrt(a + a*sec(c + d*x)))]
    @test_int [1/(sec(c + d*x)^(3/2)*sqrt(a + a*sec(c + d*x))), x, 4, (sqrt(2)*atanh((sqrt(a)*sqrt(sec(c + d*x))*sin(c + d*x))/(sqrt(2)*sqrt(a + a*sec(c + d*x)))))/(sqrt(a)*d) + (2*sin(c + d*x))/(3*d*sqrt(sec(c + d*x))*sqrt(a + a*sec(c + d*x))) - (2*sqrt(sec(c + d*x))*sin(c + d*x))/(3*d*sqrt(a + a*sec(c + d*x)))]
    @test_int [1/(sec(c + d*x)^(5/2)*sqrt(a + a*sec(c + d*x))), x, 5, -((sqrt(2)*atanh((sqrt(a)*sqrt(sec(c + d*x))*sin(c + d*x))/(sqrt(2)*sqrt(a + a*sec(c + d*x)))))/(sqrt(a)*d)) + (2*sin(c + d*x))/(5*d*sec(c + d*x)^(3/2)*sqrt(a + a*sec(c + d*x))) - (2*sin(c + d*x))/(15*d*sqrt(sec(c + d*x))*sqrt(a + a*sec(c + d*x))) + (26*sqrt(sec(c + d*x))*sin(c + d*x))/(15*d*sqrt(a + a*sec(c + d*x)))]


    @test_int [sec(c + d*x)^(7/2)/(a + a*sec(c + d*x))^(3/2), x, 7, -((3*asinh((sqrt(a)*tan(c + d*x))/sqrt(a + a*sec(c + d*x))))/(a^(3/2)*d)) + (9*atanh((sqrt(a)*sqrt(sec(c + d*x))*sin(c + d*x))/(sqrt(2)*sqrt(a + a*sec(c + d*x)))))/(2*sqrt(2)*a^(3/2)*d) - (sec(c + d*x)^(5/2)*sin(c + d*x))/(2*d*(a + a*sec(c + d*x))^(3/2)) + (3*sec(c + d*x)^(3/2)*sin(c + d*x))/(2*a*d*sqrt(a + a*sec(c + d*x)))]
    @test_int [sec(c + d*x)^(5/2)/(a + a*sec(c + d*x))^(3/2), x, 6, (2*asinh((sqrt(a)*tan(c + d*x))/sqrt(a + a*sec(c + d*x))))/(a^(3/2)*d) - (5*atanh((sqrt(a)*sqrt(sec(c + d*x))*sin(c + d*x))/(sqrt(2)*sqrt(a + a*sec(c + d*x)))))/(2*sqrt(2)*a^(3/2)*d) - (sec(c + d*x)^(3/2)*sin(c + d*x))/(2*d*(a + a*sec(c + d*x))^(3/2))]
    @test_int [sec(c + d*x)^(3/2)/(a + a*sec(c + d*x))^(3/2), x, 3, atanh((sqrt(a)*sqrt(sec(c + d*x))*sin(c + d*x))/(sqrt(2)*sqrt(a + a*sec(c + d*x))))/(2*sqrt(2)*a^(3/2)*d) + (sec(c + d*x)^(3/2)*sin(c + d*x))/(2*d*(a + a*sec(c + d*x))^(3/2))]
    @test_int [sqrt(sec(c + d*x))/(a + a*sec(c + d*x))^(3/2), x, 3, (3*atanh((sqrt(a)*sqrt(sec(c + d*x))*sin(c + d*x))/(sqrt(2)*sqrt(a + a*sec(c + d*x)))))/(2*sqrt(2)*a^(3/2)*d) - (sec(c + d*x)^(3/2)*sin(c + d*x))/(2*d*(a + a*sec(c + d*x))^(3/2))]
    @test_int [1/(sqrt(sec(c + d*x))*(a + a*sec(c + d*x))^(3/2)), x, 4, -((7*atanh((sqrt(a)*sqrt(sec(c + d*x))*sin(c + d*x))/(sqrt(2)*sqrt(a + a*sec(c + d*x)))))/(2*sqrt(2)*a^(3/2)*d)) - (sqrt(sec(c + d*x))*sin(c + d*x))/(2*d*(a + a*sec(c + d*x))^(3/2)) + (5*sqrt(sec(c + d*x))*sin(c + d*x))/(2*a*d*sqrt(a + a*sec(c + d*x)))]
    @test_int [1/(sec(c + d*x)^(3/2)*(a + a*sec(c + d*x))^(3/2)), x, 5, (11*atanh((sqrt(a)*sqrt(sec(c + d*x))*sin(c + d*x))/(sqrt(2)*sqrt(a + a*sec(c + d*x)))))/(2*sqrt(2)*a^(3/2)*d) - sin(c + d*x)/(2*d*sqrt(sec(c + d*x))*(a + a*sec(c + d*x))^(3/2)) + (7*sin(c + d*x))/(6*a*d*sqrt(sec(c + d*x))*sqrt(a + a*sec(c + d*x))) - (19*sqrt(sec(c + d*x))*sin(c + d*x))/(6*a*d*sqrt(a + a*sec(c + d*x)))]
    @test_int [1/(sec(c + d*x)^(5/2)*(a + a*sec(c + d*x))^(3/2)), x, 6, -((15*atanh((sqrt(a)*sqrt(sec(c + d*x))*sin(c + d*x))/(sqrt(2)*sqrt(a + a*sec(c + d*x)))))/(2*sqrt(2)*a^(3/2)*d)) - sin(c + d*x)/(2*d*sec(c + d*x)^(3/2)*(a + a*sec(c + d*x))^(3/2)) + (9*sin(c + d*x))/(10*a*d*sec(c + d*x)^(3/2)*sqrt(a + a*sec(c + d*x))) - (13*sin(c + d*x))/(10*a*d*sqrt(sec(c + d*x))*sqrt(a + a*sec(c + d*x))) + (49*sqrt(sec(c + d*x))*sin(c + d*x))/(10*a*d*sqrt(a + a*sec(c + d*x)))]


    @test_int [sec(c + d*x)^(9/2)/(a + a*sec(c + d*x))^(5/2), x, 8, -((5*asinh((sqrt(a)*tan(c + d*x))/sqrt(a + a*sec(c + d*x))))/(a^(5/2)*d)) + (115*atanh((sqrt(a)*sqrt(sec(c + d*x))*sin(c + d*x))/(sqrt(2)*sqrt(a + a*sec(c + d*x)))))/(16*sqrt(2)*a^(5/2)*d) - (sec(c + d*x)^(7/2)*sin(c + d*x))/(4*d*(a + a*sec(c + d*x))^(5/2)) - (15*sec(c + d*x)^(5/2)*sin(c + d*x))/(16*a*d*(a + a*sec(c + d*x))^(3/2)) + (35*sec(c + d*x)^(3/2)*sin(c + d*x))/(16*a^2*d*sqrt(a + a*sec(c + d*x)))]
    @test_int [sec(c + d*x)^(7/2)/(a + a*sec(c + d*x))^(5/2), x, 7, (2*asinh((sqrt(a)*tan(c + d*x))/sqrt(a + a*sec(c + d*x))))/(a^(5/2)*d) - (43*atanh((sqrt(a)*sqrt(sec(c + d*x))*sin(c + d*x))/(sqrt(2)*sqrt(a + a*sec(c + d*x)))))/(16*sqrt(2)*a^(5/2)*d) - (sec(c + d*x)^(5/2)*sin(c + d*x))/(4*d*(a + a*sec(c + d*x))^(5/2)) - (11*sec(c + d*x)^(3/2)*sin(c + d*x))/(16*a*d*(a + a*sec(c + d*x))^(3/2))]
    @test_int [sec(c + d*x)^(5/2)/(a + a*sec(c + d*x))^(5/2), x, 4, (3*atanh((sqrt(a)*sqrt(sec(c + d*x))*sin(c + d*x))/(sqrt(2)*sqrt(a + a*sec(c + d*x)))))/(16*sqrt(2)*a^(5/2)*d) + (sec(c + d*x)^(5/2)*sin(c + d*x))/(4*d*(a + a*sec(c + d*x))^(5/2)) + (3*sec(c + d*x)^(3/2)*sin(c + d*x))/(16*a*d*(a + a*sec(c + d*x))^(3/2))]
    @test_int [sec(c + d*x)^(3/2)/(a + a*sec(c + d*x))^(5/2), x, 4, (5*atanh((sqrt(a)*sqrt(sec(c + d*x))*sin(c + d*x))/(sqrt(2)*sqrt(a + a*sec(c + d*x)))))/(16*sqrt(2)*a^(5/2)*d) - (sec(c + d*x)^(5/2)*sin(c + d*x))/(4*d*(a + a*sec(c + d*x))^(5/2)) + (5*sec(c + d*x)^(3/2)*sin(c + d*x))/(16*a*d*(a + a*sec(c + d*x))^(3/2))]
    @test_int [sqrt(sec(c + d*x))/(a + a*sec(c + d*x))^(5/2), x, 4, (19*atanh((sqrt(a)*sqrt(sec(c + d*x))*sin(c + d*x))/(sqrt(2)*sqrt(a + a*sec(c + d*x)))))/(16*sqrt(2)*a^(5/2)*d) - (sec(c + d*x)^(3/2)*sin(c + d*x))/(4*d*(a + a*sec(c + d*x))^(5/2)) - (9*sec(c + d*x)^(3/2)*sin(c + d*x))/(16*a*d*(a + a*sec(c + d*x))^(3/2))]
    @test_int [1/(sqrt(sec(c + d*x))*(a + a*sec(c + d*x))^(5/2)), x, 5, -((75*atanh((sqrt(a)*sqrt(sec(c + d*x))*sin(c + d*x))/(sqrt(2)*sqrt(a + a*sec(c + d*x)))))/(16*sqrt(2)*a^(5/2)*d)) - (sqrt(sec(c + d*x))*sin(c + d*x))/(4*d*(a + a*sec(c + d*x))^(5/2)) - (13*sqrt(sec(c + d*x))*sin(c + d*x))/(16*a*d*(a + a*sec(c + d*x))^(3/2)) + (49*sqrt(sec(c + d*x))*sin(c + d*x))/(16*a^2*d*sqrt(a + a*sec(c + d*x)))]
    @test_int [1/(sec(c + d*x)^(3/2)*(a + a*sec(c + d*x))^(5/2)), x, 6, (163*atanh((sqrt(a)*sqrt(sec(c + d*x))*sin(c + d*x))/(sqrt(2)*sqrt(a + a*sec(c + d*x)))))/(16*sqrt(2)*a^(5/2)*d) - sin(c + d*x)/(4*d*sqrt(sec(c + d*x))*(a + a*sec(c + d*x))^(5/2)) - (17*sin(c + d*x))/(16*a*d*sqrt(sec(c + d*x))*(a + a*sec(c + d*x))^(3/2)) + (95*sin(c + d*x))/(48*a^2*d*sqrt(sec(c + d*x))*sqrt(a + a*sec(c + d*x))) - (299*sqrt(sec(c + d*x))*sin(c + d*x))/(48*a^2*d*sqrt(a + a*sec(c + d*x)))]


    @test_int [sec(c + d*x)^(7/2)/sqrt(1 + sec(c + d*x)), x, 7, -((sqrt(2)*asinh(tan(c + d*x)/(1 + sec(c + d*x))))/d) + (7*asinh(tan(c + d*x)/sqrt(1 + sec(c + d*x))))/(4*d) - (sec(c + d*x)^(3/2)*sin(c + d*x))/(4*d*sqrt(1 + sec(c + d*x))) + (sec(c + d*x)^(5/2)*sin(c + d*x))/(2*d*sqrt(1 + sec(c + d*x)))]
    @test_int [sec(c + d*x)^(5/2)/sqrt(1 + sec(c + d*x)), x, 6, (sqrt(2)*asinh(tan(c + d*x)/(1 + sec(c + d*x))))/d - asinh(tan(c + d*x)/sqrt(1 + sec(c + d*x)))/d + (sec(c + d*x)^(3/2)*sin(c + d*x))/(d*sqrt(1 + sec(c + d*x)))]
    @test_int [sec(c + d*x)^(3/2)/sqrt(1 + sec(c + d*x)), x, 5, -((sqrt(2)*asinh(tan(c + d*x)/(1 + sec(c + d*x))))/d) + (2*asinh(tan(c + d*x)/sqrt(1 + sec(c + d*x))))/d]
    @test_int [sqrt(sec(c + d*x))/sqrt(1 + sec(c + d*x)), x, 2, (sqrt(2)*asinh(tan(c + d*x)/(1 + sec(c + d*x))))/d]
    @test_int [1/(sqrt(sec(c + d*x))*sqrt(1 + sec(c + d*x))), x, 3, -((sqrt(2)*asinh(tan(c + d*x)/(1 + sec(c + d*x))))/d) + (2*sqrt(sec(c + d*x))*sin(c + d*x))/(d*sqrt(1 + sec(c + d*x)))]
    @test_int [1/(sec(c + d*x)^(3/2)*sqrt(1 + sec(c + d*x))), x, 4, (sqrt(2)*asinh(tan(c + d*x)/(1 + sec(c + d*x))))/d + (2*sin(c + d*x))/(3*d*sqrt(sec(c + d*x))*sqrt(1 + sec(c + d*x))) - (2*sqrt(sec(c + d*x))*sin(c + d*x))/(3*d*sqrt(1 + sec(c + d*x)))]
    @test_int [1/(sec(c + d*x)^(5/2)*sqrt(1 + sec(c + d*x))), x, 5, -((sqrt(2)*asinh(tan(c + d*x)/(1 + sec(c + d*x))))/d) + (2*sin(c + d*x))/(5*d*sec(c + d*x)^(3/2)*sqrt(1 + sec(c + d*x))) - (2*sin(c + d*x))/(15*d*sqrt(sec(c + d*x))*sqrt(1 + sec(c + d*x))) + (26*sqrt(sec(c + d*x))*sin(c + d*x))/(15*d*sqrt(1 + sec(c + d*x)))]


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


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


    @test_int [sqrt(a + a*sec(c + d*x))*(e*sec(c + d*x))^(4/3), x, 4, (6*a*e*(e*sec(c + d*x))^(1/3)*tan(c + d*x))/(5*d*sqrt(a + a*sec(c + d*x))) + (4*3^(3/4)*sqrt(2 + sqrt(3))*a^2*e*Elliptic.F(asin(((1 - sqrt(3))*e^(1/3) - (e*sec(c + d*x))^(1/3))/((1 + sqrt(3))*e^(1/3) - (e*sec(c + d*x))^(1/3))), -7 - 4*sqrt(3))*(e^(1/3) - (e*sec(c + d*x))^(1/3))*sqrt((e^(2/3) + e^(1/3)*(e*sec(c + d*x))^(1/3) + (e*sec(c + d*x))^(2/3))/((1 + sqrt(3))*e^(1/3) - (e*sec(c + d*x))^(1/3))^2)*tan(c + d*x))/(5*d*(a - a*sec(c + d*x))*sqrt(a + a*sec(c + d*x))*sqrt((e^(1/3)*(e^(1/3) - (e*sec(c + d*x))^(1/3)))/((1 + sqrt(3))*e^(1/3) - (e*sec(c + d*x))^(1/3))^2))]
    @test_int [sqrt(a + a*sec(c + d*x))*(e*sec(c + d*x))^(1/3), x, 3, (2*3^(3/4)*sqrt(2 + sqrt(3))*a^2*Elliptic.F(asin(((1 - sqrt(3))*e^(1/3) - (e*sec(c + d*x))^(1/3))/((1 + sqrt(3))*e^(1/3) - (e*sec(c + d*x))^(1/3))), -7 - 4*sqrt(3))*(e^(1/3) - (e*sec(c + d*x))^(1/3))*sqrt((e^(2/3) + e^(1/3)*(e*sec(c + d*x))^(1/3) + (e*sec(c + d*x))^(2/3))/((1 + sqrt(3))*e^(1/3) - (e*sec(c + d*x))^(1/3))^2)*tan(c + d*x))/(d*(a - a*sec(c + d*x))*sqrt(a + a*sec(c + d*x))*sqrt((e^(1/3)*(e^(1/3) - (e*sec(c + d*x))^(1/3)))/((1 + sqrt(3))*e^(1/3) - (e*sec(c + d*x))^(1/3))^2))]
    @test_int [sqrt(a + a*sec(c + d*x))/(e*sec(c + d*x))^(2/3), x, 4, (3*a*tan(c + d*x))/(2*d*(e*sec(c + d*x))^(2/3)*sqrt(a + a*sec(c + d*x))) + (3^(3/4)*sqrt(2 + sqrt(3))*a^2*Elliptic.F(asin(((1 - sqrt(3))*e^(1/3) - (e*sec(c + d*x))^(1/3))/((1 + sqrt(3))*e^(1/3) - (e*sec(c + d*x))^(1/3))), -7 - 4*sqrt(3))*(e^(1/3) - (e*sec(c + d*x))^(1/3))*sqrt((e^(2/3) + e^(1/3)*(e*sec(c + d*x))^(1/3) + (e*sec(c + d*x))^(2/3))/((1 + sqrt(3))*e^(1/3) - (e*sec(c + d*x))^(1/3))^2)*tan(c + d*x))/(2*d*e*(a - a*sec(c + d*x))*sqrt(a + a*sec(c + d*x))*sqrt((e^(1/3)*(e^(1/3) - (e*sec(c + d*x))^(1/3)))/((1 + sqrt(3))*e^(1/3) - (e*sec(c + d*x))^(1/3))^2))]

    @test_int [sqrt(a + a*sec(c + d*x))*(e*sec(c + d*x))^(8/3), x, 7, (60*a*e^2*(e*sec(c + d*x))^(2/3)*tan(c + d*x))/(91*d*sqrt(a + a*sec(c + d*x))) + (6*a*e*(e*sec(c + d*x))^(5/3)*tan(c + d*x))/(13*d*sqrt(a + a*sec(c + d*x))) - (240*a*e^3*tan(c + d*x))/(91*d*sqrt(a + a*sec(c + d*x))*((1 + sqrt(3))*e^(1/3) - (e*sec(c + d*x))^(1/3))) + (120*3^(1/4)*sqrt(2 - sqrt(3))*a^2*e^(7/3)*Elliptic.E(asin(((1 - sqrt(3))*e^(1/3) - (e*sec(c + d*x))^(1/3))/((1 + sqrt(3))*e^(1/3) - (e*sec(c + d*x))^(1/3))), -7 - 4*sqrt(3))*(e^(1/3) - (e*sec(c + d*x))^(1/3))*sqrt((e^(2/3) + e^(1/3)*(e*sec(c + d*x))^(1/3) + (e*sec(c + d*x))^(2/3))/((1 + sqrt(3))*e^(1/3) - (e*sec(c + d*x))^(1/3))^2)*tan(c + d*x))/(91*d*(a - a*sec(c + d*x))*sqrt(a + a*sec(c + d*x))*sqrt((e^(1/3)*(e^(1/3) - (e*sec(c + d*x))^(1/3)))/((1 + sqrt(3))*e^(1/3) - (e*sec(c + d*x))^(1/3))^2)) - (80*sqrt(2)*3^(3/4)*a^2*e^(7/3)*Elliptic.F(asin(((1 - sqrt(3))*e^(1/3) - (e*sec(c + d*x))^(1/3))/((1 + sqrt(3))*e^(1/3) - (e*sec(c + d*x))^(1/3))), -7 - 4*sqrt(3))*(e^(1/3) - (e*sec(c + d*x))^(1/3))*sqrt((e^(2/3) + e^(1/3)*(e*sec(c + d*x))^(1/3) + (e*sec(c + d*x))^(2/3))/((1 + sqrt(3))*e^(1/3) - (e*sec(c + d*x))^(1/3))^2)*tan(c + d*x))/(91*d*(a - a*sec(c + d*x))*sqrt(a + a*sec(c + d*x))*sqrt((e^(1/3)*(e^(1/3) - (e*sec(c + d*x))^(1/3)))/((1 + sqrt(3))*e^(1/3) - (e*sec(c + d*x))^(1/3))^2))]
    @test_int [sqrt(a + a*sec(c + d*x))*(e*sec(c + d*x))^(5/3), x, 6, (6*a*e*(e*sec(c + d*x))^(2/3)*tan(c + d*x))/(7*d*sqrt(a + a*sec(c + d*x))) - (24*a*e^2*tan(c + d*x))/(7*d*sqrt(a + a*sec(c + d*x))*((1 + sqrt(3))*e^(1/3) - (e*sec(c + d*x))^(1/3))) + (12*3^(1/4)*sqrt(2 - sqrt(3))*a^2*e^(4/3)*Elliptic.E(asin(((1 - sqrt(3))*e^(1/3) - (e*sec(c + d*x))^(1/3))/((1 + sqrt(3))*e^(1/3) - (e*sec(c + d*x))^(1/3))), -7 - 4*sqrt(3))*(e^(1/3) - (e*sec(c + d*x))^(1/3))*sqrt((e^(2/3) + e^(1/3)*(e*sec(c + d*x))^(1/3) + (e*sec(c + d*x))^(2/3))/((1 + sqrt(3))*e^(1/3) - (e*sec(c + d*x))^(1/3))^2)*tan(c + d*x))/(7*d*(a - a*sec(c + d*x))*sqrt(a + a*sec(c + d*x))*sqrt((e^(1/3)*(e^(1/3) - (e*sec(c + d*x))^(1/3)))/((1 + sqrt(3))*e^(1/3) - (e*sec(c + d*x))^(1/3))^2)) - (8*sqrt(2)*3^(3/4)*a^2*e^(4/3)*Elliptic.F(asin(((1 - sqrt(3))*e^(1/3) - (e*sec(c + d*x))^(1/3))/((1 + sqrt(3))*e^(1/3) - (e*sec(c + d*x))^(1/3))), -7 - 4*sqrt(3))*(e^(1/3) - (e*sec(c + d*x))^(1/3))*sqrt((e^(2/3) + e^(1/3)*(e*sec(c + d*x))^(1/3) + (e*sec(c + d*x))^(2/3))/((1 + sqrt(3))*e^(1/3) - (e*sec(c + d*x))^(1/3))^2)*tan(c + d*x))/(7*d*(a - a*sec(c + d*x))*sqrt(a + a*sec(c + d*x))*sqrt((e^(1/3)*(e^(1/3) - (e*sec(c + d*x))^(1/3)))/((1 + sqrt(3))*e^(1/3) - (e*sec(c + d*x))^(1/3))^2))]
    @test_int [sqrt(a + a*sec(c + d*x))*(e*sec(c + d*x))^(2/3), x, 5, -((6*a*e*tan(c + d*x))/(d*sqrt(a + a*sec(c + d*x))*((1 + sqrt(3))*e^(1/3) - (e*sec(c + d*x))^(1/3)))) + (3*3^(1/4)*sqrt(2 - sqrt(3))*a^2*e^(1/3)*Elliptic.E(asin(((1 - sqrt(3))*e^(1/3) - (e*sec(c + d*x))^(1/3))/((1 + sqrt(3))*e^(1/3) - (e*sec(c + d*x))^(1/3))), -7 - 4*sqrt(3))*(e^(1/3) - (e*sec(c + d*x))^(1/3))*sqrt((e^(2/3) + e^(1/3)*(e*sec(c + d*x))^(1/3) + (e*sec(c + d*x))^(2/3))/((1 + sqrt(3))*e^(1/3) - (e*sec(c + d*x))^(1/3))^2)*tan(c + d*x))/(d*(a - a*sec(c + d*x))*sqrt(a + a*sec(c + d*x))*sqrt((e^(1/3)*(e^(1/3) - (e*sec(c + d*x))^(1/3)))/((1 + sqrt(3))*e^(1/3) - (e*sec(c + d*x))^(1/3))^2)) - (2*sqrt(2)*3^(3/4)*a^2*e^(1/3)*Elliptic.F(asin(((1 - sqrt(3))*e^(1/3) - (e*sec(c + d*x))^(1/3))/((1 + sqrt(3))*e^(1/3) - (e*sec(c + d*x))^(1/3))), -7 - 4*sqrt(3))*(e^(1/3) - (e*sec(c + d*x))^(1/3))*sqrt((e^(2/3) + e^(1/3)*(e*sec(c + d*x))^(1/3) + (e*sec(c + d*x))^(2/3))/((1 + sqrt(3))*e^(1/3) - (e*sec(c + d*x))^(1/3))^2)*tan(c + d*x))/(d*(a - a*sec(c + d*x))*sqrt(a + a*sec(c + d*x))*sqrt((e^(1/3)*(e^(1/3) - (e*sec(c + d*x))^(1/3)))/((1 + sqrt(3))*e^(1/3) - (e*sec(c + d*x))^(1/3))^2))]
    @test_int [sqrt(a + a*sec(c + d*x))/(e*sec(c + d*x))^(1/3), x, 6, (3*a*tan(c + d*x))/(d*(e*sec(c + d*x))^(1/3)*sqrt(a + a*sec(c + d*x))) + (3*a*tan(c + d*x))/(d*sqrt(a + a*sec(c + d*x))*((1 + sqrt(3))*e^(1/3) - (e*sec(c + d*x))^(1/3))) - (3*3^(1/4)*sqrt(2 - sqrt(3))*a^2*Elliptic.E(asin(((1 - sqrt(3))*e^(1/3) - (e*sec(c + d*x))^(1/3))/((1 + sqrt(3))*e^(1/3) - (e*sec(c + d*x))^(1/3))), -7 - 4*sqrt(3))*(e^(1/3) - (e*sec(c + d*x))^(1/3))*sqrt((e^(2/3) + e^(1/3)*(e*sec(c + d*x))^(1/3) + (e*sec(c + d*x))^(2/3))/((1 + sqrt(3))*e^(1/3) - (e*sec(c + d*x))^(1/3))^2)*tan(c + d*x))/(2*d*e^(2/3)*(a - a*sec(c + d*x))*sqrt(a + a*sec(c + d*x))*sqrt((e^(1/3)*(e^(1/3) - (e*sec(c + d*x))^(1/3)))/((1 + sqrt(3))*e^(1/3) - (e*sec(c + d*x))^(1/3))^2)) + (sqrt(2)*3^(3/4)*a^2*Elliptic.F(asin(((1 - sqrt(3))*e^(1/3) - (e*sec(c + d*x))^(1/3))/((1 + sqrt(3))*e^(1/3) - (e*sec(c + d*x))^(1/3))), -7 - 4*sqrt(3))*(e^(1/3) - (e*sec(c + d*x))^(1/3))*sqrt((e^(2/3) + e^(1/3)*(e*sec(c + d*x))^(1/3) + (e*sec(c + d*x))^(2/3))/((1 + sqrt(3))*e^(1/3) - (e*sec(c + d*x))^(1/3))^2)*tan(c + d*x))/(d*e^(2/3)*(a - a*sec(c + d*x))*sqrt(a + a*sec(c + d*x))*sqrt((e^(1/3)*(e^(1/3) - (e*sec(c + d*x))^(1/3)))/((1 + sqrt(3))*e^(1/3) - (e*sec(c + d*x))^(1/3))^2))]
    @test_int [sqrt(a + a*sec(c + d*x))/(e*sec(c + d*x))^(4/3), x, 7, (3*a*tan(c + d*x))/(4*d*(e*sec(c + d*x))^(4/3)*sqrt(a + a*sec(c + d*x))) + (15*a*tan(c + d*x))/(8*d*e*(e*sec(c + d*x))^(1/3)*sqrt(a + a*sec(c + d*x))) + (15*a*tan(c + d*x))/(8*d*e*sqrt(a + a*sec(c + d*x))*((1 + sqrt(3))*e^(1/3) - (e*sec(c + d*x))^(1/3))) - (15*3^(1/4)*sqrt(2 - sqrt(3))*a^2*Elliptic.E(asin(((1 - sqrt(3))*e^(1/3) - (e*sec(c + d*x))^(1/3))/((1 + sqrt(3))*e^(1/3) - (e*sec(c + d*x))^(1/3))), -7 - 4*sqrt(3))*(e^(1/3) - (e*sec(c + d*x))^(1/3))*sqrt((e^(2/3) + e^(1/3)*(e*sec(c + d*x))^(1/3) + (e*sec(c + d*x))^(2/3))/((1 + sqrt(3))*e^(1/3) - (e*sec(c + d*x))^(1/3))^2)*tan(c + d*x))/(16*d*e^(5/3)*(a - a*sec(c + d*x))*sqrt(a + a*sec(c + d*x))*sqrt((e^(1/3)*(e^(1/3) - (e*sec(c + d*x))^(1/3)))/((1 + sqrt(3))*e^(1/3) - (e*sec(c + d*x))^(1/3))^2)) + (5*3^(3/4)*a^2*Elliptic.F(asin(((1 - sqrt(3))*e^(1/3) - (e*sec(c + d*x))^(1/3))/((1 + sqrt(3))*e^(1/3) - (e*sec(c + d*x))^(1/3))), -7 - 4*sqrt(3))*(e^(1/3) - (e*sec(c + d*x))^(1/3))*sqrt((e^(2/3) + e^(1/3)*(e*sec(c + d*x))^(1/3) + (e*sec(c + d*x))^(2/3))/((1 + sqrt(3))*e^(1/3) - (e*sec(c + d*x))^(1/3))^2)*tan(c + d*x))/(4*sqrt(2)*d*e^(5/3)*(a - a*sec(c + d*x))*sqrt(a + a*sec(c + d*x))*sqrt((e^(1/3)*(e^(1/3) - (e*sec(c + d*x))^(1/3)))/((1 + sqrt(3))*e^(1/3) - (e*sec(c + d*x))^(1/3))^2))]


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


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


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


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


    @test_int [sec(c + d*x)^(4/3)*(a + a*sec(c + d*x))^(1/3), x, 3, (2^(5/6)*AppellF1(1/2, -(1/3), 1/6, 3/2, 1 - sec(c + d*x), (1/2)*(1 - sec(c + d*x)))*(a + a*sec(c + d*x))^(1/3)*tan(c + d*x))/(d*(1 + sec(c + d*x))^(5/6))]


    @test_int [sec(c + d*x)^(4/3)*(a + a*sec(c + d*x))^(2/3), x, 3, (2*2^(1/6)*AppellF1(1/2, -(1/3), -(1/6), 3/2, 1 - sec(c + d*x), (1/2)*(1 - sec(c + d*x)))*(a + a*sec(c + d*x))^(2/3)*tan(c + d*x))/(d*(1 + sec(c + d*x))^(7/6))]
    @test_int [sec(c + d*x)^(5/3)*(a + a*sec(c + d*x))^(2/3), x, -3, (9*sec(c + d*x)^(2/3)*(a*(1 + sec(c + d*x)))^(2/3)*sin(c + d*x))/(4*d) - (3*a*sec(c + d*x)^(5/3)*sin(c + d*x))/(2*d*(a*(1 + sec(c + d*x)))^(1/3)) - (9*(a*(1 + sec(c + d*x)))^(2/3)*tan(c + d*x))/(4*d*(1/(1 + cos(c + d*x)))^(1/3)*(1 + sec(c + d*x))^(7/3)) + (HypergeometricFunctions._₂F₁(1/4, 1/3, 5/4, tan((1/2)*(c + d*x))^4)*(cos(c + d*x)*sec((1/2)*(c + d*x))^4)^(1/3)*(a*(1 + sec(c + d*x)))^(2/3)*tan(c + d*x))/(8*d*(1/(1 + cos(c + d*x)))^(1/3)*(1 + sec(c + d*x))^(4/3)) - (5*HypergeometricFunctions._₂F₁(1/3, 3/4, 7/4, tan((1/2)*(c + d*x))^4)*(cos(c + d*x)*sec((1/2)*(c + d*x))^4)^(1/3)*(a*(1 + sec(c + d*x)))^(2/3)*tan(c + d*x)^3)/(8*d*(1/(1 + cos(c + d*x)))^(1/3)*(1 + sec(c + d*x))^(10/3))]


    @test_int [(a + a*sec(c + d*x))^(4/3)/sec(c + d*x)^(1/3), x, 3, (2*2^(5/6)*a*AppellF1(1/2, 4/3, -(5/6), 3/2, 1 - sec(c + d*x), (1/2)*(1 - sec(c + d*x)))*(a + a*sec(c + d*x))^(1/3)*tan(c + d*x))/(d*(1 + sec(c + d*x))^(5/6))]


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


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


    @test_int [sec(e + f*x)^n*(a + a*sec(e + f*x))^4, x, 8, If(13>=8, (a^4*(30 + 21*n + 4*n^2)*sec(e + f*x)^(1 + n)*sin(e + f*x))/(f*(1 + n)*(2 + n)*(3 + n)) + (sec(e + f*x)^(1 + n)*(a^2 + a^2*sec(e + f*x))^2*sin(e + f*x))/(f*(3 + n)) + (2*(4 + n)*sec(e + f*x)^(1 + n)*(a^4 + a^4*sec(e + f*x))*sin(e + f*x))/(f*(2 + n)*(3 + n)) - (a^4*(3 + 24*n + 8*n^2)*HypergeometricFunctions._₂F₁(1/2, (1 - n)/2, (3 - n)/2, cos(e + f*x)^2)*sec(e + f*x)^(-1 + n)*sin(e + f*x))/(f*(1 - n)*(1 + n)*(3 + n)*sqrt(sin(e + f*x)^2)) + (4*a^4*(3 + 2*n)*HypergeometricFunctions._₂F₁(1/2, -(n/2), (2 - n)/2, cos(e + f*x)^2)*sec(e + f*x)^n*sin(e + f*x))/(f*n*(2 + n)*sqrt(sin(e + f*x)^2)), (a^4*(30 + 21*n + 4*n^2)*sec(e + f*x)^(1 + n)*sin(e + f*x))/(f*(6 + 11*n + 6*n^2 + n^3)) + (sec(e + f*x)^(1 + n)*(a^2 + a^2*sec(e + f*x))^2*sin(e + f*x))/(f*(3 + n)) + (2*(4 + n)*sec(e + f*x)^(1 + n)*(a^4 + a^4*sec(e + f*x))*sin(e + f*x))/(f*(2 + n)*(3 + n)) - (a^4*(3 + 24*n + 8*n^2)*HypergeometricFunctions._₂F₁(1/2, (1 - n)/2, (3 - n)/2, cos(e + f*x)^2)*sec(e + f*x)^(-1 + n)*sin(e + f*x))/(f*(3 + n)*(1 - n^2)*sqrt(sin(e + f*x)^2)) + (4*a^4*(3 + 2*n)*HypergeometricFunctions._₂F₁(1/2, -(n/2), (2 - n)/2, cos(e + f*x)^2)*sec(e + f*x)^n*sin(e + f*x))/(f*n*(2 + n)*sqrt(sin(e + f*x)^2)))]
    @test_int [sec(e + f*x)^n*(a + a*sec(e + f*x))^3, x, 7, (a^3*(5 + 2*n)*sec(e + f*x)^(1 + n)*sin(e + f*x))/(f*(1 + n)*(2 + n)) + (sec(e + f*x)^(1 + n)*(a^3 + a^3*sec(e + f*x))*sin(e + f*x))/(f*(2 + n)) - (a^3*(1 + 4*n)*HypergeometricFunctions._₂F₁(1/2, (1 - n)/2, (3 - n)/2, cos(e + f*x)^2)*sec(e + f*x)^(-1 + n)*sin(e + f*x))/(f*(1 - n^2)*sqrt(sin(e + f*x)^2)) + (a^3*(7 + 4*n)*HypergeometricFunctions._₂F₁(1/2, -(n/2), (2 - n)/2, cos(e + f*x)^2)*sec(e + f*x)^n*sin(e + f*x))/(f*n*(2 + n)*sqrt(sin(e + f*x)^2))]
    @test_int [sec(e + f*x)^n*(a + a*sec(e + f*x))^2, x, 6, (a^2*sec(e + f*x)^(1 + n)*sin(e + f*x))/(f*(1 + n)) - (a^2*(1 + 2*n)*HypergeometricFunctions._₂F₁(1/2, (1 - n)/2, (3 - n)/2, cos(e + f*x)^2)*sec(e + f*x)^(-1 + n)*sin(e + f*x))/(f*(1 - n^2)*sqrt(sin(e + f*x)^2)) + (2*a^2*HypergeometricFunctions._₂F₁(1/2, -(n/2), (2 - n)/2, cos(e + f*x)^2)*sec(e + f*x)^n*sin(e + f*x))/(f*n*sqrt(sin(e + f*x)^2))]
    @test_int [sec(e + f*x)^n*(a + a*sec(e + f*x))^1, x, 5, -((a*HypergeometricFunctions._₂F₁(1/2, (1 - n)/2, (3 - n)/2, cos(e + f*x)^2)*sec(e + f*x)^(-1 + n)*sin(e + f*x))/(f*(1 - n)*sqrt(sin(e + f*x)^2))) + (a*HypergeometricFunctions._₂F₁(1/2, -(n/2), (2 - n)/2, cos(e + f*x)^2)*sec(e + f*x)^n*sin(e + f*x))/(f*n*sqrt(sin(e + f*x)^2))]
    @test_int [sec(e + f*x)^n/(a + a*sec(e + f*x))^1, x, 6, (sec(e + f*x)^n*sin(e + f*x))/(f*(a + a*sec(e + f*x))) + ((1 - n)*HypergeometricFunctions._₂F₁(1/2, (2 - n)/2, (4 - n)/2, cos(e + f*x)^2)*sec(e + f*x)^(-2 + n)*sin(e + f*x))/(a*f*(2 - n)*sqrt(sin(e + f*x)^2)) - (HypergeometricFunctions._₂F₁(1/2, (1 - n)/2, (3 - n)/2, cos(e + f*x)^2)*sec(e + f*x)^(-1 + n)*sin(e + f*x))/(a*f*sqrt(sin(e + f*x)^2))]
    @test_int [sec(e + f*x)^n/(a + a*sec(e + f*x))^2, x, 7, -((2*(2 - n)*sec(e + f*x)^(1 + n)*sin(e + f*x))/(3*a^2*f*(1 + sec(e + f*x)))) - (sec(e + f*x)^(1 + n)*sin(e + f*x))/(3*f*(a + a*sec(e + f*x))^2) - ((3 - 2*n)*HypergeometricFunctions._₂F₁(1/2, (1 - n)/2, (3 - n)/2, cos(e + f*x)^2)*sec(e + f*x)^(-1 + n)*sin(e + f*x))/(3*a^2*f*sqrt(sin(e + f*x)^2)) + (2*(2 - n)*HypergeometricFunctions._₂F₁(1/2, -(n/2), (2 - n)/2, cos(e + f*x)^2)*sec(e + f*x)^n*sin(e + f*x))/(3*a^2*f*sqrt(sin(e + f*x)^2))]


    @test_int [sec(e + f*x)^n*(1 + sec(e + f*x))^(5/2), x, 4, If(13>=8, (2*(7 + 4*n)*sec(e + f*x)^(1 + n)*sin(e + f*x))/(f*(1 + 2*n)*(3 + 2*n)*sqrt(1 + sec(e + f*x))) + (2*sec(e + f*x)^(1 + n)*sqrt(1 + sec(e + f*x))*sin(e + f*x))/(f*(3 + 2*n)) + (2*(3 + 24*n + 16*n^2)*HypergeometricFunctions._₂F₁(1/2, 1 - n, 3/2, 1 - sec(e + f*x))*tan(e + f*x))/(f*(1 + 2*n)*(3 + 2*n)*sqrt(1 + sec(e + f*x))), (2*(7 + 4*n)*sec(e + f*x)^(1 + n)*sin(e + f*x))/(f*(3 + 8*n + 4*n^2)*sqrt(1 + sec(e + f*x))) + (2*sec(e + f*x)^(1 + n)*sqrt(1 + sec(e + f*x))*sin(e + f*x))/(f*(3 + 2*n)) + (2*(3 + 24*n + 16*n^2)*HypergeometricFunctions._₂F₁(1/2, 1 - n, 3/2, 1 - sec(e + f*x))*tan(e + f*x))/(f*(3 + 8*n + 4*n^2)*sqrt(1 + sec(e + f*x))))]
    @test_int [sec(e + f*x)^n*(1 + sec(e + f*x))^(3/2), x, 4, (2*sec(e + f*x)^(1 + n)*sin(e + f*x))/(f*(1 + 2*n)*sqrt(1 + sec(e + f*x))) + (2*(1 + 4*n)*HypergeometricFunctions._₂F₁(1/2, 1 - n, 3/2, 1 - sec(e + f*x))*tan(e + f*x))/(f*(1 + 2*n)*sqrt(1 + sec(e + f*x)))]
    @test_int [sec(e + f*x)^n*(1 + sec(e + f*x))^(1/2), x, 2, (2*HypergeometricFunctions._₂F₁(1/2, 1 - n, 3/2, 1 - sec(e + f*x))*tan(e + f*x))/(f*sqrt(1 + sec(e + f*x)))]
    @test_int [sec(e + f*x)^n/(1 + sec(e + f*x))^(1/2), x, 3, (AppellF1(1/2, 1 - n, 1, 3/2, 1 - sec(e + f*x), (1/2)*(1 - sec(e + f*x)))*tan(e + f*x))/(f*sqrt(1 + sec(e + f*x)))]
    @test_int [sec(e + f*x)^n/(1 + sec(e + f*x))^(3/2), x, 3, (AppellF1(1/2, 1 - n, 2, 3/2, 1 - sec(e + f*x), (1/2)*(1 - sec(e + f*x)))*tan(e + f*x))/(2*f*sqrt(1 + sec(e + f*x)))]

    @test_int [(-sec(e + f*x))^n*(1 + sec(e + f*x))^(3/2), x, 4, (2*(-sec(e + f*x))^n*tan(e + f*x))/(f*(1 + 2*n)*sqrt(1 + sec(e + f*x))) - ((1 + 4*n)*HypergeometricFunctions._₂F₁(1/2, n, 1 + n, sec(e + f*x))*(-sec(e + f*x))^n*tan(e + f*x))/(f*n*(1 + 2*n)*sqrt(1 - sec(e + f*x))*sqrt(1 + sec(e + f*x)))]
    @test_int [(-sec(e + f*x))^n*(1 + sec(e + f*x))^(1/2), x, 2, -((HypergeometricFunctions._₂F₁(1/2, n, 1 + n, sec(e + f*x))*(-sec(e + f*x))^n*tan(e + f*x))/(f*n*sqrt(1 - sec(e + f*x))*sqrt(1 + sec(e + f*x))))]
    @test_int [(-sec(e + f*x))^n/(1 + sec(e + f*x))^(1/2), x, 2, -((AppellF1(n, 1/2, 1, 1 + n, sec(e + f*x), -sec(e + f*x))*(-sec(e + f*x))^n*tan(e + f*x))/(f*n*sqrt(1 - sec(e + f*x))*sqrt(1 + sec(e + f*x))))]
    @test_int [(-sec(e + f*x))^n/(1 + sec(e + f*x))^(3/2), x, 2, -((AppellF1(n, 1/2, 2, 1 + n, sec(e + f*x), -sec(e + f*x))*(-sec(e + f*x))^n*tan(e + f*x))/(f*n*sqrt(1 - sec(e + f*x))*sqrt(1 + sec(e + f*x))))]

    @test_int [(d*sec(e + f*x))^n*(1 + sec(e + f*x))^(3/2), x, 4, (2*(d*sec(e + f*x))^n*tan(e + f*x))/(f*(1 + 2*n)*sqrt(1 + sec(e + f*x))) - ((1 + 4*n)*HypergeometricFunctions._₂F₁(1/2, n, 1 + n, sec(e + f*x))*(d*sec(e + f*x))^n*tan(e + f*x))/(f*n*(1 + 2*n)*sqrt(1 - sec(e + f*x))*sqrt(1 + sec(e + f*x)))]
    @test_int [(d*sec(e + f*x))^n*(1 + sec(e + f*x))^(1/2), x, 2, -((HypergeometricFunctions._₂F₁(1/2, n, 1 + n, sec(e + f*x))*(d*sec(e + f*x))^n*tan(e + f*x))/(f*n*sqrt(1 - sec(e + f*x))*sqrt(1 + sec(e + f*x))))]
    @test_int [(d*sec(e + f*x))^n/(1 + sec(e + f*x))^(1/2), x, 2, -((AppellF1(n, 1/2, 1, 1 + n, sec(e + f*x), -sec(e + f*x))*(d*sec(e + f*x))^n*tan(e + f*x))/(f*n*sqrt(1 - sec(e + f*x))*sqrt(1 + sec(e + f*x))))]
    @test_int [(d*sec(e + f*x))^n/(1 + sec(e + f*x))^(3/2), x, 2, -((AppellF1(n, 1/2, 2, 1 + n, sec(e + f*x), -sec(e + f*x))*(d*sec(e + f*x))^n*tan(e + f*x))/(f*n*sqrt(1 - sec(e + f*x))*sqrt(1 + sec(e + f*x))))]


    @test_int [sec(e + f*x)^n*(a + a*sec(e + f*x))^(5/2), x, 4, If(13>=8, (2*a^3*(7 + 4*n)*sec(e + f*x)^(1 + n)*sin(e + f*x))/(f*(1 + 2*n)*(3 + 2*n)*sqrt(a + a*sec(e + f*x))) + (2*a^2*sec(e + f*x)^(1 + n)*sqrt(a + a*sec(e + f*x))*sin(e + f*x))/(f*(3 + 2*n)) + (2*a^3*(3 + 24*n + 16*n^2)*HypergeometricFunctions._₂F₁(1/2, 1 - n, 3/2, 1 - sec(e + f*x))*tan(e + f*x))/(f*(1 + 2*n)*(3 + 2*n)*sqrt(a + a*sec(e + f*x))), (2*a^3*(7 + 4*n)*sec(e + f*x)^(1 + n)*sin(e + f*x))/(f*(3 + 8*n + 4*n^2)*sqrt(a + a*sec(e + f*x))) + (2*a^2*sec(e + f*x)^(1 + n)*sqrt(a + a*sec(e + f*x))*sin(e + f*x))/(f*(3 + 2*n)) + (2*a^3*(3 + 24*n + 16*n^2)*HypergeometricFunctions._₂F₁(1/2, 1 - n, 3/2, 1 - sec(e + f*x))*tan(e + f*x))/(f*(3 + 8*n + 4*n^2)*sqrt(a + a*sec(e + f*x))))]
    @test_int [sec(e + f*x)^n*(a + a*sec(e + f*x))^(3/2), x, 4, (2*a^2*sec(e + f*x)^(1 + n)*sin(e + f*x))/(f*(1 + 2*n)*sqrt(a + a*sec(e + f*x))) + (2*a^2*(1 + 4*n)*HypergeometricFunctions._₂F₁(1/2, 1 - n, 3/2, 1 - sec(e + f*x))*tan(e + f*x))/(f*(1 + 2*n)*sqrt(a + a*sec(e + f*x)))]
    @test_int [sec(e + f*x)^n*(a + a*sec(e + f*x))^(1/2), x, 2, (2*a*HypergeometricFunctions._₂F₁(1/2, 1 - n, 3/2, 1 - sec(e + f*x))*tan(e + f*x))/(f*sqrt(a + a*sec(e + f*x)))]
    @test_int [sec(e + f*x)^n/(a + a*sec(e + f*x))^(1/2), x, 4, (AppellF1(1/2, 1 - n, 1, 3/2, 1 - sec(e + f*x), (1/2)*(1 - sec(e + f*x)))*tan(e + f*x))/(f*sqrt(a + a*sec(e + f*x)))]
    @test_int [sec(e + f*x)^n/(a + a*sec(e + f*x))^(3/2), x, 4, (AppellF1(1/2, 1 - n, 2, 3/2, 1 - sec(e + f*x), (1/2)*(1 - sec(e + f*x)))*tan(e + f*x))/(2*a*f*sqrt(a + a*sec(e + f*x)))]

    @test_int [(-sec(e + f*x))^n*(a + a*sec(e + f*x))^(3/2), x, 5, (2*a^2*(1 + 4*n)*HypergeometricFunctions._₂F₁(1/2, 1 - n, 3/2, 1 - sec(e + f*x))*(-sec(e + f*x))^n*sec(e + f*x)^(1 - n)*sin(e + f*x))/(f*(1 + 2*n)*sqrt(a + a*sec(e + f*x))) + (2*a^2*(-sec(e + f*x))^n*tan(e + f*x))/(f*(1 + 2*n)*sqrt(a + a*sec(e + f*x)))]
    @test_int [(-sec(e + f*x))^n*(a + a*sec(e + f*x))^(1/2), x, 3, (2*a*HypergeometricFunctions._₂F₁(1/2, 1 - n, 3/2, 1 - sec(e + f*x))*(-sec(e + f*x))^n*sec(e + f*x)^(1 - n)*sin(e + f*x))/(f*sqrt(a + a*sec(e + f*x)))]
    @test_int [(-sec(e + f*x))^n/(a + a*sec(e + f*x))^(1/2), x, 3, -((AppellF1(n, 1/2, 1, 1 + n, sec(e + f*x), -sec(e + f*x))*(-sec(e + f*x))^n*tan(e + f*x))/(f*n*sqrt(1 - sec(e + f*x))*sqrt(a + a*sec(e + f*x))))]
    @test_int [(-sec(e + f*x))^n/(a + a*sec(e + f*x))^(3/2), x, 3, -((AppellF1(n, 1/2, 2, 1 + n, sec(e + f*x), -sec(e + f*x))*(-sec(e + f*x))^n*tan(e + f*x))/(a*f*n*sqrt(1 - sec(e + f*x))*sqrt(a + a*sec(e + f*x))))]

    @test_int [(d*sec(e + f*x))^n*(a + a*sec(e + f*x))^(3/2), x, 5, (2*a^2*(1 + 4*n)*HypergeometricFunctions._₂F₁(1/2, 1 - n, 3/2, 1 - sec(e + f*x))*sec(e + f*x)^(1 - n)*(d*sec(e + f*x))^n*sin(e + f*x))/(f*(1 + 2*n)*sqrt(a + a*sec(e + f*x))) + (2*a^2*(d*sec(e + f*x))^n*tan(e + f*x))/(f*(1 + 2*n)*sqrt(a + a*sec(e + f*x)))]
    @test_int [(d*sec(e + f*x))^n*(a + a*sec(e + f*x))^(1/2), x, 3, (2*a*HypergeometricFunctions._₂F₁(1/2, 1 - n, 3/2, 1 - sec(e + f*x))*sec(e + f*x)^(1 - n)*(d*sec(e + f*x))^n*sin(e + f*x))/(f*sqrt(a + a*sec(e + f*x)))]
    @test_int [(d*sec(e + f*x))^n/(a + a*sec(e + f*x))^(1/2), x, 3, -((AppellF1(n, 1/2, 1, 1 + n, sec(e + f*x), -sec(e + f*x))*(d*sec(e + f*x))^n*tan(e + f*x))/(f*n*sqrt(1 - sec(e + f*x))*sqrt(a + a*sec(e + f*x))))]
    @test_int [(d*sec(e + f*x))^n/(a + a*sec(e + f*x))^(3/2), x, 3, -((AppellF1(n, 1/2, 2, 1 + n, sec(e + f*x), -sec(e + f*x))*(d*sec(e + f*x))^n*tan(e + f*x))/(a*f*n*sqrt(1 - sec(e + f*x))*sqrt(a + a*sec(e + f*x))))]


    @test_int [(-sec(e + f*x))^n*(a - a*sec(e + f*x))^(5/2), x, 4, If(13>=8, (2*a^3*(3 + 24*n + 16*n^2)*HypergeometricFunctions._₂F₁(1/2, 1 - n, 3/2, 1 + sec(e + f*x))*tan(e + f*x))/(f*(1 + 2*n)*(3 + 2*n)*sqrt(a - a*sec(e + f*x))) + (2*a^3*(7 + 4*n)*(-sec(e + f*x))^n*tan(e + f*x))/(f*(1 + 2*n)*(3 + 2*n)*sqrt(a - a*sec(e + f*x))) + (2*a^2*(-sec(e + f*x))^n*sqrt(a - a*sec(e + f*x))*tan(e + f*x))/(f*(3 + 2*n)), (2*a^3*(3 + 24*n + 16*n^2)*HypergeometricFunctions._₂F₁(1/2, 1 - n, 3/2, 1 + sec(e + f*x))*tan(e + f*x))/(f*(3 + 8*n + 4*n^2)*sqrt(a - a*sec(e + f*x))) + (2*a^3*(7 + 4*n)*(-sec(e + f*x))^n*tan(e + f*x))/(f*(3 + 8*n + 4*n^2)*sqrt(a - a*sec(e + f*x))) + (2*a^2*(-sec(e + f*x))^n*sqrt(a - a*sec(e + f*x))*tan(e + f*x))/(f*(3 + 2*n)))]
    @test_int [(-sec(e + f*x))^n*(a - a*sec(e + f*x))^(3/2), x, 4, (2*a^2*(1 + 4*n)*HypergeometricFunctions._₂F₁(1/2, 1 - n, 3/2, 1 + sec(e + f*x))*tan(e + f*x))/(f*(1 + 2*n)*sqrt(a - a*sec(e + f*x))) + (2*a^2*(-sec(e + f*x))^n*tan(e + f*x))/(f*(1 + 2*n)*sqrt(a - a*sec(e + f*x)))]
    @test_int [(-sec(e + f*x))^n*(a - a*sec(e + f*x))^(1/2), x, 2, (2*a*HypergeometricFunctions._₂F₁(1/2, 1 - n, 3/2, 1 + sec(e + f*x))*tan(e + f*x))/(f*sqrt(a - a*sec(e + f*x)))]
    @test_int [(-sec(e + f*x))^n/(a - a*sec(e + f*x))^(1/2), x, 4, (AppellF1(1/2, 1 - n, 1, 3/2, 1 + sec(e + f*x), (1/2)*(1 + sec(e + f*x)))*tan(e + f*x))/(f*sqrt(a - a*sec(e + f*x)))]
    @test_int [(-sec(e + f*x))^n/(a - a*sec(e + f*x))^(3/2), x, 4, (AppellF1(1/2, 1 - n, 2, 3/2, 1 + sec(e + f*x), (1/2)*(1 + sec(e + f*x)))*tan(e + f*x))/(2*a*f*sqrt(a - a*sec(e + f*x)))]

    @test_int [sec(e + f*x)^n*(a - a*sec(e + f*x))^(3/2), x, 5, (2*a^2*sec(e + f*x)^(1 + n)*sin(e + f*x))/(f*(1 + 2*n)*sqrt(a - a*sec(e + f*x))) + (2*a^2*(1 + 4*n)*HypergeometricFunctions._₂F₁(1/2, 1 - n, 3/2, 1 + sec(e + f*x))*sec(e + f*x)^(1 + n)*sin(e + f*x))/((-sec(e + f*x))^n*(f*(1 + 2*n)*sqrt(a - a*sec(e + f*x))))]
    @test_int [sec(e + f*x)^n*(a - a*sec(e + f*x))^(1/2), x, 3, (2*a*HypergeometricFunctions._₂F₁(1/2, 1 - n, 3/2, 1 + sec(e + f*x))*sec(e + f*x)^(1 + n)*sin(e + f*x))/((-sec(e + f*x))^n*(f*sqrt(a - a*sec(e + f*x))))]

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


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


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

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

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


    @test_int [sec(e + f*x)^4*(a + a*sec(e + f*x))^m, x, 6, If(13>=8, ((4 + m)*(a + a*sec(e + f*x))^m*tan(e + f*x))/(f*(1 + m)*(2 + m)*(3 + m)) + (sec(e + f*x)^2*(a + a*sec(e + f*x))^m*tan(e + f*x))/(f*(3 + m)) + (2^(1/2 + m)*m*(5 + 3*m + m^2)*HypergeometricFunctions._₂F₁(1/2, 1/2 - m, 3/2, (1/2)*(1 - sec(e + f*x)))*(1 + sec(e + f*x))^(-(1/2) - m)*(a + a*sec(e + f*x))^m*tan(e + f*x))/(f*(1 + m)*(2 + m)*(3 + m)) + (m*(a + a*sec(e + f*x))^(1 + m)*tan(e + f*x))/(a*f*(6 + 5*m + m^2)), ((4 + m)*(a + a*sec(e + f*x))^m*tan(e + f*x))/(f*(6 + 11*m + 6*m^2 + m^3)) + (sec(e + f*x)^2*(a + a*sec(e + f*x))^m*tan(e + f*x))/(f*(3 + m)) + (2^(1/2 + m)*m*(5 + 3*m + m^2)*HypergeometricFunctions._₂F₁(1/2, 1/2 - m, 3/2, (1/2)*(1 - sec(e + f*x)))*(1 + sec(e + f*x))^(-(1/2) - m)*(a + a*sec(e + f*x))^m*tan(e + f*x))/(f*(6 + 11*m + 6*m^2 + m^3)) + (m*(a + a*sec(e + f*x))^(1 + m)*tan(e + f*x))/(a*f*(6 + 5*m + m^2)))]
    @test_int [sec(e + f*x)^3*(a + a*sec(e + f*x))^m, x, 5, If(13>=8, -(((a + a*sec(e + f*x))^m*tan(e + f*x))/(f*(2 + 3*m + m^2))) + (2^(1/2 + m)*(1 + m + m^2)*HypergeometricFunctions._₂F₁(1/2, 1/2 - m, 3/2, (1/2)*(1 - sec(e + f*x)))*(1 + sec(e + f*x))^(-(1/2) - m)*(a + a*sec(e + f*x))^m*tan(e + f*x))/(f*(1 + m)*(2 + m)) + ((a + a*sec(e + f*x))^(1 + m)*tan(e + f*x))/(a*f*(2 + m)), -(((a + a*sec(e + f*x))^m*tan(e + f*x))/(f*(2 + 3*m + m^2))) + (2^(1/2 + m)*(1 + m + m^2)*HypergeometricFunctions._₂F₁(1/2, 1/2 - m, 3/2, (1/2)*(1 - sec(e + f*x)))*(1 + sec(e + f*x))^(-(1/2) - m)*(a + a*sec(e + f*x))^m*tan(e + f*x))/(f*(2 + 3*m + m^2)) + ((a + a*sec(e + f*x))^(1 + m)*tan(e + f*x))/(a*f*(2 + m)))]
    @test_int [sec(e + f*x)^2*(a + a*sec(e + f*x))^m, x, 4, ((a + a*sec(e + f*x))^m*tan(e + f*x))/(f*(1 + m)) + (2^(1/2 + m)*m*HypergeometricFunctions._₂F₁(1/2, 1/2 - m, 3/2, (1/2)*(1 - sec(e + f*x)))*(1 + sec(e + f*x))^(-(1/2) - m)*(a + a*sec(e + f*x))^m*tan(e + f*x))/(f*(1 + m))]
    @test_int [sec(e + f*x)^1*(a + a*sec(e + f*x))^m, x, 3, (2^(1/2 + m)*HypergeometricFunctions._₂F₁(1/2, 1/2 - m, 3/2, (1/2)*(1 - sec(e + f*x)))*(1 + sec(e + f*x))^(-(1/2) - m)*(a + a*sec(e + f*x))^m*tan(e + f*x))/f]
    @test_int [sec(e + f*x)^0*(a + a*sec(e + f*x))^m, x, 3, (sqrt(2)*AppellF1(1/2 + m, 1/2, 1, 3/2 + m, (1/2)*(1 + sec(e + f*x)), 1 + sec(e + f*x))*(a + a*sec(e + f*x))^m*tan(e + f*x))/(f*(1 + 2*m)*sqrt(1 - sec(e + f*x)))]
    @test_int [cos(e + f*x)^1*(a + a*sec(e + f*x))^m, x, 3, -((sqrt(2)*AppellF1(1/2 + m, 1/2, 2, 3/2 + m, (1/2)*(1 + sec(e + f*x)), 1 + sec(e + f*x))*(a + a*sec(e + f*x))^m*tan(e + f*x))/(f*(1 + 2*m)*sqrt(1 - sec(e + f*x))))]


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


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


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


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


    @test_int [cos(c + d*x)^(7/2)*(a + a*sec(c + d*x)), x, 7, (6*a*Elliptic.E((c + d*x)/2, 2))/(5*d) + (10*a*Elliptic.F((c + d*x)/2, 2))/(21*d) + (10*a*sqrt(cos(c + d*x))*sin(c + d*x))/(21*d) + (2*a*cos(c + d*x)^(3/2)*sin(c + d*x))/(5*d) + (2*a*cos(c + d*x)^(5/2)*sin(c + d*x))/(7*d)]
    @test_int [cos(c + d*x)^(5/2)*(a + a*sec(c + d*x)), x, 6, (6*a*Elliptic.E((c + d*x)/2, 2))/(5*d) + (2*a*Elliptic.F((c + d*x)/2, 2))/(3*d) + (2*a*sqrt(cos(c + d*x))*sin(c + d*x))/(3*d) + (2*a*cos(c + d*x)^(3/2)*sin(c + d*x))/(5*d)]
    @test_int [cos(c + d*x)^(3/2)*(a + a*sec(c + d*x)), x, 5, (2*a*Elliptic.E((c + d*x)/2, 2))/d + (2*a*Elliptic.F((c + d*x)/2, 2))/(3*d) + (2*a*sqrt(cos(c + d*x))*sin(c + d*x))/(3*d)]
    @test_int [cos(c + d*x)^(1/2)*(a + a*sec(c + d*x)), x, 4, (2*a*Elliptic.E((c + d*x)/2, 2))/d + (2*a*Elliptic.F((c + d*x)/2, 2))/d]
    @test_int [(a + a*sec(c + d*x))/cos(c + d*x)^(1/2), x, 5, (-2*a*Elliptic.E((c + d*x)/2, 2))/d + (2*a*Elliptic.F((c + d*x)/2, 2))/d + (2*a*sin(c + d*x))/(d*sqrt(cos(c + d*x)))]
    @test_int [(a + a*sec(c + d*x))/cos(c + d*x)^(3/2), x, 6, (-2*a*Elliptic.E((c + d*x)/2, 2))/d + (2*a*Elliptic.F((c + d*x)/2, 2))/(3*d) + (2*a*sin(c + d*x))/(3*d*cos(c + d*x)^(3/2)) + (2*a*sin(c + d*x))/(d*sqrt(cos(c + d*x)))]
    @test_int [(a + a*sec(c + d*x))/cos(c + d*x)^(5/2), x, 7, (-6*a*Elliptic.E((c + d*x)/2, 2))/(5*d) + (2*a*Elliptic.F((c + d*x)/2, 2))/(3*d) + (2*a*sin(c + d*x))/(5*d*cos(c + d*x)^(5/2)) + (2*a*sin(c + d*x))/(3*d*cos(c + d*x)^(3/2)) + (6*a*sin(c + d*x))/(5*d*sqrt(cos(c + d*x)))]
    @test_int [(a + a*sec(c + d*x))/cos(c + d*x)^(7/2), x, 8, (-6*a*Elliptic.E((c + d*x)/2, 2))/(5*d) + (10*a*Elliptic.F((c + d*x)/2, 2))/(21*d) + (2*a*sin(c + d*x))/(7*d*cos(c + d*x)^(7/2)) + (2*a*sin(c + d*x))/(5*d*cos(c + d*x)^(5/2)) + (10*a*sin(c + d*x))/(21*d*cos(c + d*x)^(3/2)) + (6*a*sin(c + d*x))/(5*d*sqrt(cos(c + d*x)))]


    @test_int [cos(c + d*x)^(9/2)*(a + a*sec(c + d*x))^2, x, 10, (32*a^2*Elliptic.E((c + d*x)/2, 2))/(15*d) + (20*a^2*Elliptic.F((c + d*x)/2, 2))/(21*d) + (20*a^2*sqrt(cos(c + d*x))*sin(c + d*x))/(21*d) + (32*a^2*cos(c + d*x)^(3/2)*sin(c + d*x))/(45*d) + (4*a^2*cos(c + d*x)^(5/2)*sin(c + d*x))/(7*d) + (2*a^2*cos(c + d*x)^(7/2)*sin(c + d*x))/(9*d)]
    @test_int [cos(c + d*x)^(7/2)*(a + a*sec(c + d*x))^2, x, 9, (12*a^2*Elliptic.E((c + d*x)/2, 2))/(5*d) + (8*a^2*Elliptic.F((c + d*x)/2, 2))/(7*d) + (8*a^2*sqrt(cos(c + d*x))*sin(c + d*x))/(7*d) + (4*a^2*cos(c + d*x)^(3/2)*sin(c + d*x))/(5*d) + (2*a^2*cos(c + d*x)^(5/2)*sin(c + d*x))/(7*d)]
    @test_int [cos(c + d*x)^(5/2)*(a + a*sec(c + d*x))^2, x, 8, (16*a^2*Elliptic.E((c + d*x)/2, 2))/(5*d) + (4*a^2*Elliptic.F((c + d*x)/2, 2))/(3*d) + (4*a^2*sqrt(cos(c + d*x))*sin(c + d*x))/(3*d) + (2*a^2*cos(c + d*x)^(3/2)*sin(c + d*x))/(5*d)]
    @test_int [cos(c + d*x)^(3/2)*(a + a*sec(c + d*x))^2, x, 7, (4*a^2*Elliptic.E((c + d*x)/2, 2))/d + (8*a^2*Elliptic.F((c + d*x)/2, 2))/(3*d) + (2*a^2*sqrt(cos(c + d*x))*sin(c + d*x))/(3*d)]
    @test_int [cos(c + d*x)^(1/2)*(a + a*sec(c + d*x))^2, x, 5, (4*a^2*Elliptic.F((c + d*x)/2, 2))/d + (2*a^2*sin(c + d*x))/(d*sqrt(cos(c + d*x)))]
    @test_int [(a + a*sec(c + d*x))^2/cos(c + d*x)^(1/2), x, 8, (-4*a^2*Elliptic.E((c + d*x)/2, 2))/d + (8*a^2*Elliptic.F((c + d*x)/2, 2))/(3*d) + (2*a^2*sin(c + d*x))/(3*d*cos(c + d*x)^(3/2)) + (4*a^2*sin(c + d*x))/(d*sqrt(cos(c + d*x)))]
    @test_int [(a + a*sec(c + d*x))^2/cos(c + d*x)^(3/2), x, 9, (-16*a^2*Elliptic.E((c + d*x)/2, 2))/(5*d) + (4*a^2*Elliptic.F((c + d*x)/2, 2))/(3*d) + (2*a^2*sin(c + d*x))/(5*d*cos(c + d*x)^(5/2)) + (4*a^2*sin(c + d*x))/(3*d*cos(c + d*x)^(3/2)) + (16*a^2*sin(c + d*x))/(5*d*sqrt(cos(c + d*x)))]
    @test_int [(a + a*sec(c + d*x))^2/cos(c + d*x)^(5/2), x, 10, (-12*a^2*Elliptic.E((c + d*x)/2, 2))/(5*d) + (8*a^2*Elliptic.F((c + d*x)/2, 2))/(7*d) + (2*a^2*sin(c + d*x))/(7*d*cos(c + d*x)^(7/2)) + (4*a^2*sin(c + d*x))/(5*d*cos(c + d*x)^(5/2)) + (8*a^2*sin(c + d*x))/(7*d*cos(c + d*x)^(3/2)) + (12*a^2*sin(c + d*x))/(5*d*sqrt(cos(c + d*x)))]


    @test_int [cos(c + d*x)^(9/2)*(a + a*sec(c + d*x))^3, x, 17, (68*a^3*Elliptic.E((1/2)*(c + d*x), 2))/(15*d) + (44*a^3*Elliptic.F((1/2)*(c + d*x), 2))/(21*d) + (44*a^3*sqrt(cos(c + d*x))*sin(c + d*x))/(21*d) + (68*a^3*cos(c + d*x)^(3/2)*sin(c + d*x))/(45*d) + (6*a^3*cos(c + d*x)^(5/2)*sin(c + d*x))/(7*d) + (2*a^3*cos(c + d*x)^(7/2)*sin(c + d*x))/(9*d)]
    @test_int [cos(c + d*x)^(7/2)*(a + a*sec(c + d*x))^3, x, 15, (28*a^3*Elliptic.E((1/2)*(c + d*x), 2))/(5*d) + (52*a^3*Elliptic.F((1/2)*(c + d*x), 2))/(21*d) + (52*a^3*sqrt(cos(c + d*x))*sin(c + d*x))/(21*d) + (6*a^3*cos(c + d*x)^(3/2)*sin(c + d*x))/(5*d) + (2*a^3*cos(c + d*x)^(5/2)*sin(c + d*x))/(7*d)]
    @test_int [cos(c + d*x)^(5/2)*(a + a*sec(c + d*x))^3, x, 13, (36*a^3*Elliptic.E((1/2)*(c + d*x), 2))/(5*d) + (4*a^3*Elliptic.F((1/2)*(c + d*x), 2))/d + (2*a^3*sqrt(cos(c + d*x))*sin(c + d*x))/d + (2*a^3*cos(c + d*x)^(3/2)*sin(c + d*x))/(5*d)]
    @test_int [cos(c + d*x)^(3/2)*(a + a*sec(c + d*x))^3, x, 13, (4*a^3*Elliptic.E((1/2)*(c + d*x), 2))/d + (20*a^3*Elliptic.F((1/2)*(c + d*x), 2))/(3*d) + (2*a^3*sin(c + d*x))/(d*sqrt(cos(c + d*x))) + (2*a^3*sqrt(cos(c + d*x))*sin(c + d*x))/(3*d)]
    @test_int [cos(c + d*x)^(1/2)*(a + a*sec(c + d*x))^3, x, 13, -((4*a^3*Elliptic.E((1/2)*(c + d*x), 2))/d) + (20*a^3*Elliptic.F((1/2)*(c + d*x), 2))/(3*d) + (2*a^3*sin(c + d*x))/(3*d*cos(c + d*x)^(3/2)) + (6*a^3*sin(c + d*x))/(d*sqrt(cos(c + d*x)))]
    @test_int [(a + a*sec(c + d*x))^3/cos(c + d*x)^(1/2), x, 15, -((36*a^3*Elliptic.E((1/2)*(c + d*x), 2))/(5*d)) + (4*a^3*Elliptic.F((1/2)*(c + d*x), 2))/d + (2*a^3*sin(c + d*x))/(5*d*cos(c + d*x)^(5/2)) + (2*a^3*sin(c + d*x))/(d*cos(c + d*x)^(3/2)) + (36*a^3*sin(c + d*x))/(5*d*sqrt(cos(c + d*x)))]
    @test_int [(a + a*sec(c + d*x))^3/cos(c + d*x)^(3/2), x, 17, -((28*a^3*Elliptic.E((1/2)*(c + d*x), 2))/(5*d)) + (52*a^3*Elliptic.F((1/2)*(c + d*x), 2))/(21*d) + (2*a^3*sin(c + d*x))/(7*d*cos(c + d*x)^(7/2)) + (6*a^3*sin(c + d*x))/(5*d*cos(c + d*x)^(5/2)) + (52*a^3*sin(c + d*x))/(21*d*cos(c + d*x)^(3/2)) + (28*a^3*sin(c + d*x))/(5*d*sqrt(cos(c + d*x)))]


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


    @test_int [cos(c + d*x)^(5/2)/(a + a*sec(c + d*x)), x, 9, (21*Elliptic.E((1/2)*(c + d*x), 2))/(5*a*d) - (5*Elliptic.F((1/2)*(c + d*x), 2))/(3*a*d) - (5*sqrt(cos(c + d*x))*sin(c + d*x))/(3*a*d) + (7*cos(c + d*x)^(3/2)*sin(c + d*x))/(5*a*d) - (cos(c + d*x)^(3/2)*sin(c + d*x))/(d*(a + a*sec(c + d*x)))]
    @test_int [cos(c + d*x)^(3/2)/(a + a*sec(c + d*x)), x, 8, -((3*Elliptic.E((1/2)*(c + d*x), 2))/(a*d)) + (5*Elliptic.F((1/2)*(c + d*x), 2))/(3*a*d) + (5*sqrt(cos(c + d*x))*sin(c + d*x))/(3*a*d) - (sqrt(cos(c + d*x))*sin(c + d*x))/(d*(a + a*sec(c + d*x)))]
    @test_int [cos(c + d*x)^(1/2)/(a + a*sec(c + d*x)), x, 7, (3*Elliptic.E((1/2)*(c + d*x), 2))/(a*d) - Elliptic.F((1/2)*(c + d*x), 2)/(a*d) - sin(c + d*x)/(d*sqrt(cos(c + d*x))*(a + a*sec(c + d*x)))]
    @test_int [1/(cos(c + d*x)^(1/2)*(a + a*sec(c + d*x))), x, 7, -(Elliptic.E((1/2)*(c + d*x), 2)/(a*d)) + Elliptic.F((1/2)*(c + d*x), 2)/(a*d) + sin(c + d*x)/(d*sqrt(cos(c + d*x))*(a + a*sec(c + d*x)))]
    @test_int [1/(cos(c + d*x)^(3/2)*(a + a*sec(c + d*x))), x, 7, Elliptic.E((1/2)*(c + d*x), 2)/(a*d) + Elliptic.F((1/2)*(c + d*x), 2)/(a*d) - sin(c + d*x)/(d*sqrt(cos(c + d*x))*(a + a*sec(c + d*x)))]
    @test_int [1/(cos(c + d*x)^(5/2)*(a + a*sec(c + d*x))), x, 8, -((3*Elliptic.E((1/2)*(c + d*x), 2))/(a*d)) - Elliptic.F((1/2)*(c + d*x), 2)/(a*d) + (3*sin(c + d*x))/(a*d*sqrt(cos(c + d*x))) - sin(c + d*x)/(d*cos(c + d*x)^(3/2)*(a + a*sec(c + d*x)))]
    @test_int [1/(cos(c + d*x)^(7/2)*(a + a*sec(c + d*x))), x, 9, (3*Elliptic.E((1/2)*(c + d*x), 2))/(a*d) + (5*Elliptic.F((1/2)*(c + d*x), 2))/(3*a*d) + (5*sin(c + d*x))/(3*a*d*cos(c + d*x)^(3/2)) - (3*sin(c + d*x))/(a*d*sqrt(cos(c + d*x))) - sin(c + d*x)/(d*cos(c + d*x)^(5/2)*(a + a*sec(c + d*x)))]


    @test_int [cos(c + d*x)^(5/2)/(a + a*sec(c + d*x))^2, x, 10, (56*Elliptic.E((1/2)*(c + d*x), 2))/(5*a^2*d) - (5*Elliptic.F((1/2)*(c + d*x), 2))/(a^2*d) - (5*sqrt(cos(c + d*x))*sin(c + d*x))/(a^2*d) + (56*cos(c + d*x)^(3/2)*sin(c + d*x))/(15*a^2*d) - (3*cos(c + d*x)^(3/2)*sin(c + d*x))/(a^2*d*(1 + sec(c + d*x))) - (cos(c + d*x)^(3/2)*sin(c + d*x))/(3*d*(a + a*sec(c + d*x))^2)]
    @test_int [cos(c + d*x)^(3/2)/(a + a*sec(c + d*x))^2, x, 9, -((7*Elliptic.E((1/2)*(c + d*x), 2))/(a^2*d)) + (10*Elliptic.F((1/2)*(c + d*x), 2))/(3*a^2*d) + (10*sqrt(cos(c + d*x))*sin(c + d*x))/(3*a^2*d) - (7*sqrt(cos(c + d*x))*sin(c + d*x))/(3*a^2*d*(1 + sec(c + d*x))) - (sqrt(cos(c + d*x))*sin(c + d*x))/(3*d*(a + a*sec(c + d*x))^2)]
    @test_int [cos(c + d*x)^(1/2)/(a + a*sec(c + d*x))^2, x, 8, (4*Elliptic.E((1/2)*(c + d*x), 2))/(a^2*d) - (5*Elliptic.F((1/2)*(c + d*x), 2))/(3*a^2*d) - (5*sin(c + d*x))/(3*a^2*d*sqrt(cos(c + d*x))*(1 + sec(c + d*x))) - sin(c + d*x)/(3*d*sqrt(cos(c + d*x))*(a + a*sec(c + d*x))^2)]
    @test_int [1/(cos(c + d*x)^(1/2)*(a + a*sec(c + d*x))^2), x, 8, -(Elliptic.E((1/2)*(c + d*x), 2)/(a^2*d)) + (2*Elliptic.F((1/2)*(c + d*x), 2))/(3*a^2*d) + sin(c + d*x)/(a^2*d*sqrt(cos(c + d*x))*(1 + sec(c + d*x))) - sin(c + d*x)/(3*d*cos(c + d*x)^(3/2)*(a + a*sec(c + d*x))^2)]
    @test_int [1/(cos(c + d*x)^(3/2)*(a + a*sec(c + d*x))^2), x, 5, Elliptic.F((1/2)*(c + d*x), 2)/(3*a^2*d) + sin(c + d*x)/(3*d*cos(c + d*x)^(3/2)*(a + a*sec(c + d*x))^2)]
    @test_int [1/(cos(c + d*x)^(5/2)*(a + a*sec(c + d*x))^2), x, 8, Elliptic.E((1/2)*(c + d*x), 2)/(a^2*d) + (2*Elliptic.F((1/2)*(c + d*x), 2))/(3*a^2*d) - sin(c + d*x)/(a^2*d*sqrt(cos(c + d*x))*(1 + sec(c + d*x))) - sin(c + d*x)/(3*d*cos(c + d*x)^(3/2)*(a + a*sec(c + d*x))^2)]
    @test_int [1/(cos(c + d*x)^(7/2)*(a + a*sec(c + d*x))^2), x, 9, -((4*Elliptic.E((1/2)*(c + d*x), 2))/(a^2*d)) - (5*Elliptic.F((1/2)*(c + d*x), 2))/(3*a^2*d) + (4*sin(c + d*x))/(a^2*d*sqrt(cos(c + d*x))) - (5*sin(c + d*x))/(3*a^2*d*cos(c + d*x)^(3/2)*(1 + sec(c + d*x))) - sin(c + d*x)/(3*d*cos(c + d*x)^(5/2)*(a + a*sec(c + d*x))^2)]
    @test_int [1/(cos(c + d*x)^(9/2)*(a + a*sec(c + d*x))^2), x, 10, (7*Elliptic.E((1/2)*(c + d*x), 2))/(a^2*d) + (10*Elliptic.F((1/2)*(c + d*x), 2))/(3*a^2*d) + (10*sin(c + d*x))/(3*a^2*d*cos(c + d*x)^(3/2)) - (7*sin(c + d*x))/(a^2*d*sqrt(cos(c + d*x))) - (7*sin(c + d*x))/(3*a^2*d*cos(c + d*x)^(5/2)*(1 + sec(c + d*x))) - sin(c + d*x)/(3*d*cos(c + d*x)^(7/2)*(a + a*sec(c + d*x))^2)]


    @test_int [cos(c + d*x)^(5/2)/(a + a*sec(c + d*x))^3, x, 11, (231*Elliptic.E((1/2)*(c + d*x), 2))/(10*a^3*d) - (21*Elliptic.F((1/2)*(c + d*x), 2))/(2*a^3*d) - (21*sqrt(cos(c + d*x))*sin(c + d*x))/(2*a^3*d) + (77*cos(c + d*x)^(3/2)*sin(c + d*x))/(10*a^3*d) - (cos(c + d*x)^(3/2)*sin(c + d*x))/(5*d*(a + a*sec(c + d*x))^3) - (4*cos(c + d*x)^(3/2)*sin(c + d*x))/(5*a*d*(a + a*sec(c + d*x))^2) - (63*cos(c + d*x)^(3/2)*sin(c + d*x))/(10*d*(a^3 + a^3*sec(c + d*x)))]
    @test_int [cos(c + d*x)^(3/2)/(a + a*sec(c + d*x))^3, x, 10, -((119*Elliptic.E((1/2)*(c + d*x), 2))/(10*a^3*d)) + (11*Elliptic.F((1/2)*(c + d*x), 2))/(2*a^3*d) + (11*sqrt(cos(c + d*x))*sin(c + d*x))/(2*a^3*d) - (sqrt(cos(c + d*x))*sin(c + d*x))/(5*d*(a + a*sec(c + d*x))^3) - (2*sqrt(cos(c + d*x))*sin(c + d*x))/(3*a*d*(a + a*sec(c + d*x))^2) - (119*sqrt(cos(c + d*x))*sin(c + d*x))/(30*d*(a^3 + a^3*sec(c + d*x)))]
    @test_int [cos(c + d*x)^(1/2)/(a + a*sec(c + d*x))^3, x, 9, (49*Elliptic.E((1/2)*(c + d*x), 2))/(10*a^3*d) - (13*Elliptic.F((1/2)*(c + d*x), 2))/(6*a^3*d) - sin(c + d*x)/(5*d*sqrt(cos(c + d*x))*(a + a*sec(c + d*x))^3) - (8*sin(c + d*x))/(15*a*d*sqrt(cos(c + d*x))*(a + a*sec(c + d*x))^2) - (13*sin(c + d*x))/(6*d*sqrt(cos(c + d*x))*(a^3 + a^3*sec(c + d*x)))]
    @test_int [1/(cos(c + d*x)^(1/2)*(a + a*sec(c + d*x))^3), x, 9, -((9*Elliptic.E((1/2)*(c + d*x), 2))/(10*a^3*d)) + Elliptic.F((1/2)*(c + d*x), 2)/(2*a^3*d) - sin(c + d*x)/(5*d*cos(c + d*x)^(3/2)*(a + a*sec(c + d*x))^3) + (2*sin(c + d*x))/(5*a*d*sqrt(cos(c + d*x))*(a + a*sec(c + d*x))^2) + sin(c + d*x)/(2*d*sqrt(cos(c + d*x))*(a^3 + a^3*sec(c + d*x)))]
    @test_int [1/(cos(c + d*x)^(3/2)*(a + a*sec(c + d*x))^3), x, 9, -(Elliptic.E((1/2)*(c + d*x), 2)/(10*a^3*d)) + Elliptic.F((1/2)*(c + d*x), 2)/(6*a^3*d) + sin(c + d*x)/(5*d*cos(c + d*x)^(3/2)*(a + a*sec(c + d*x))^3) - sin(c + d*x)/(15*a*d*sqrt(cos(c + d*x))*(a + a*sec(c + d*x))^2) + sin(c + d*x)/(6*d*sqrt(cos(c + d*x))*(a^3 + a^3*sec(c + d*x)))]
    @test_int [1/(cos(c + d*x)^(5/2)*(a + a*sec(c + d*x))^3), x, 9, Elliptic.E((1/2)*(c + d*x), 2)/(10*a^3*d) + Elliptic.F((1/2)*(c + d*x), 2)/(6*a^3*d) - sin(c + d*x)/(5*d*cos(c + d*x)^(3/2)*(a + a*sec(c + d*x))^3) - (4*sin(c + d*x))/(15*a*d*sqrt(cos(c + d*x))*(a + a*sec(c + d*x))^2) + sin(c + d*x)/(6*d*sqrt(cos(c + d*x))*(a^3 + a^3*sec(c + d*x)))]
    @test_int [1/(cos(c + d*x)^(7/2)*(a + a*sec(c + d*x))^3), x, 9, (9*Elliptic.E((1/2)*(c + d*x), 2))/(10*a^3*d) + Elliptic.F((1/2)*(c + d*x), 2)/(2*a^3*d) - sin(c + d*x)/(5*d*cos(c + d*x)^(5/2)*(a + a*sec(c + d*x))^3) - (2*sin(c + d*x))/(5*a*d*cos(c + d*x)^(3/2)*(a + a*sec(c + d*x))^2) - (9*sin(c + d*x))/(10*d*sqrt(cos(c + d*x))*(a^3 + a^3*sec(c + d*x)))]
    @test_int [1/(cos(c + d*x)^(9/2)*(a + a*sec(c + d*x))^3), x, 10, -((49*Elliptic.E((1/2)*(c + d*x), 2))/(10*a^3*d)) - (13*Elliptic.F((1/2)*(c + d*x), 2))/(6*a^3*d) + (49*sin(c + d*x))/(10*a^3*d*sqrt(cos(c + d*x))) - sin(c + d*x)/(5*d*cos(c + d*x)^(7/2)*(a + a*sec(c + d*x))^3) - (8*sin(c + d*x))/(15*a*d*cos(c + d*x)^(5/2)*(a + a*sec(c + d*x))^2) - (13*sin(c + d*x))/(6*d*cos(c + d*x)^(3/2)*(a^3 + a^3*sec(c + d*x)))]
    @test_int [1/(cos(c + d*x)^(11/2)*(a + a*sec(c + d*x))^3), x, 11, (119*Elliptic.E((1/2)*(c + d*x), 2))/(10*a^3*d) + (11*Elliptic.F((1/2)*(c + d*x), 2))/(2*a^3*d) + (11*sin(c + d*x))/(2*a^3*d*cos(c + d*x)^(3/2)) - (119*sin(c + d*x))/(10*a^3*d*sqrt(cos(c + d*x))) - sin(c + d*x)/(5*d*cos(c + d*x)^(9/2)*(a + a*sec(c + d*x))^3) - (2*sin(c + d*x))/(3*a*d*cos(c + d*x)^(7/2)*(a + a*sec(c + d*x))^2) - (119*sin(c + d*x))/(30*d*cos(c + d*x)^(5/2)*(a^3 + a^3*sec(c + d*x)))]


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


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


    @test_int [cos(c + d*x)^(7/2)*sqrt(a + a*sec(c + d*x)), x, 5, (32*a*sin(c + d*x))/(35*d*sqrt(cos(c + d*x))*sqrt(a + a*sec(c + d*x))) + (16*a*sqrt(cos(c + d*x))*sin(c + d*x))/(35*d*sqrt(a + a*sec(c + d*x))) + (12*a*cos(c + d*x)^(3/2)*sin(c + d*x))/(35*d*sqrt(a + a*sec(c + d*x))) + (2*a*cos(c + d*x)^(5/2)*sin(c + d*x))/(7*d*sqrt(a + a*sec(c + d*x)))]
    @test_int [cos(c + d*x)^(5/2)*sqrt(a + a*sec(c + d*x)), x, 4, (16*a*sin(c + d*x))/(15*d*sqrt(cos(c + d*x))*sqrt(a + a*sec(c + d*x))) + (8*a*sqrt(cos(c + d*x))*sin(c + d*x))/(15*d*sqrt(a + a*sec(c + d*x))) + (2*a*cos(c + d*x)^(3/2)*sin(c + d*x))/(5*d*sqrt(a + a*sec(c + d*x)))]
    @test_int [cos(c + d*x)^(3/2)*sqrt(a + a*sec(c + d*x)), x, 3, (4*a*sin(c + d*x))/(3*d*sqrt(cos(c + d*x))*sqrt(a + a*sec(c + d*x))) + (2*a*sqrt(cos(c + d*x))*sin(c + d*x))/(3*d*sqrt(a + a*sec(c + d*x)))]
    @test_int [sqrt(cos(c + d*x))*sqrt(a + a*sec(c + d*x)), x, 2, (2*a*sin(c + d*x))/(d*sqrt(cos(c + d*x))*sqrt(a + a*sec(c + d*x)))]
    @test_int [sqrt(a + a*sec(c + d*x))/sqrt(cos(c + d*x)), x, 3, (2*sqrt(a)*asinh((sqrt(a)*tan(c + d*x))/sqrt(a + a*sec(c + d*x)))*sqrt(cos(c + d*x))*sqrt(sec(c + d*x)))/d]
    @test_int [sqrt(a + a*sec(c + d*x))/cos(c + d*x)^(3/2), x, 4, (sqrt(a)*asinh((sqrt(a)*tan(c + d*x))/sqrt(a + a*sec(c + d*x)))*sqrt(cos(c + d*x))*sqrt(sec(c + d*x)))/d + (a*sin(c + d*x))/(d*cos(c + d*x)^(3/2)*sqrt(a + a*sec(c + d*x)))]
    @test_int [sqrt(a + a*sec(c + d*x))/cos(c + d*x)^(5/2), x, 5, (3*sqrt(a)*asinh((sqrt(a)*tan(c + d*x))/sqrt(a + a*sec(c + d*x)))*sqrt(cos(c + d*x))*sqrt(sec(c + d*x)))/(4*d) + (a*sin(c + d*x))/(2*d*cos(c + d*x)^(5/2)*sqrt(a + a*sec(c + d*x))) + (3*a*sin(c + d*x))/(4*d*cos(c + d*x)^(3/2)*sqrt(a + a*sec(c + d*x)))]


    @test_int [cos(c + d*x)^(7/2)*(a + a*sec(c + d*x))^(3/2), x, 6, (208*a^2*sin(c + d*x))/(105*d*sqrt(cos(c + d*x))*sqrt(a + a*sec(c + d*x))) + (104*a^2*sqrt(cos(c + d*x))*sin(c + d*x))/(105*d*sqrt(a + a*sec(c + d*x))) + (26*a^2*cos(c + d*x)^(3/2)*sin(c + d*x))/(35*d*sqrt(a + a*sec(c + d*x))) + (2*a^2*cos(c + d*x)^(5/2)*sin(c + d*x))/(7*d*sqrt(a + a*sec(c + d*x)))]
    @test_int [cos(c + d*x)^(5/2)*(a + a*sec(c + d*x))^(3/2), x, 4, (8*a^2*sin(c + d*x))/(5*d*sqrt(cos(c + d*x))*sqrt(a + a*sec(c + d*x))) + (2*a*sqrt(cos(c + d*x))*sqrt(a + a*sec(c + d*x))*sin(c + d*x))/(5*d) + (2*cos(c + d*x)^(3/2)*(a + a*sec(c + d*x))^(3/2)*sin(c + d*x))/(5*d)]
    @test_int [cos(c + d*x)^(3/2)*(a + a*sec(c + d*x))^(3/2), x, 3, (8*a^2*sin(c + d*x))/(3*d*sqrt(cos(c + d*x))*sqrt(a + a*sec(c + d*x))) + (2*a*sqrt(cos(c + d*x))*sqrt(a + a*sec(c + d*x))*sin(c + d*x))/(3*d)]
    @test_int [sqrt(cos(c + d*x))*(a + a*sec(c + d*x))^(3/2), x, 5, (2*a^(3/2)*asinh((sqrt(a)*tan(c + d*x))/sqrt(a + a*sec(c + d*x)))*sqrt(cos(c + d*x))*sqrt(sec(c + d*x)))/d + (2*a^2*sin(c + d*x))/(d*sqrt(cos(c + d*x))*sqrt(a + a*sec(c + d*x)))]
    @test_int [(a + a*sec(c + d*x))^(3/2)/sqrt(cos(c + d*x)), x, 5, (3*a^(3/2)*asinh((sqrt(a)*tan(c + d*x))/sqrt(a + a*sec(c + d*x)))*sqrt(cos(c + d*x))*sqrt(sec(c + d*x)))/d + (a^2*sin(c + d*x))/(d*cos(c + d*x)^(3/2)*sqrt(a + a*sec(c + d*x)))]
    @test_int [(a + a*sec(c + d*x))^(3/2)/cos(c + d*x)^(3/2), x, 6, (7*a^(3/2)*asinh((sqrt(a)*tan(c + d*x))/sqrt(a + a*sec(c + d*x)))*sqrt(cos(c + d*x))*sqrt(sec(c + d*x)))/(4*d) + (a^2*sin(c + d*x))/(2*d*cos(c + d*x)^(5/2)*sqrt(a + a*sec(c + d*x))) + (7*a^2*sin(c + d*x))/(4*d*cos(c + d*x)^(3/2)*sqrt(a + a*sec(c + d*x)))]
    @test_int [(a + a*sec(c + d*x))^(3/2)/cos(c + d*x)^(5/2), x, 7, (11*a^(3/2)*asinh((sqrt(a)*tan(c + d*x))/sqrt(a + a*sec(c + d*x)))*sqrt(cos(c + d*x))*sqrt(sec(c + d*x)))/(8*d) + (a^2*sin(c + d*x))/(3*d*cos(c + d*x)^(7/2)*sqrt(a + a*sec(c + d*x))) + (11*a^2*sin(c + d*x))/(12*d*cos(c + d*x)^(5/2)*sqrt(a + a*sec(c + d*x))) + (11*a^2*sin(c + d*x))/(8*d*cos(c + d*x)^(3/2)*sqrt(a + a*sec(c + d*x)))]


    @test_int [cos(c + d*x)^(9/2)*(a + a*sec(c + d*x))^(5/2), x, 6, (1168*a^3*sin(c + d*x))/(315*d*sqrt(cos(c + d*x))*sqrt(a + a*sec(c + d*x))) + (584*a^3*sqrt(cos(c + d*x))*sin(c + d*x))/(315*d*sqrt(a + a*sec(c + d*x))) + (146*a^3*cos(c + d*x)^(3/2)*sin(c + d*x))/(105*d*sqrt(a + a*sec(c + d*x))) + (38*a^3*cos(c + d*x)^(5/2)*sin(c + d*x))/(63*d*sqrt(a + a*sec(c + d*x))) + (2*a^2*cos(c + d*x)^(7/2)*sqrt(a + a*sec(c + d*x))*sin(c + d*x))/(9*d)]
    @test_int [cos(c + d*x)^(7/2)*(a + a*sec(c + d*x))^(5/2), x, 5, (64*a^3*sin(c + d*x))/(21*d*sqrt(cos(c + d*x))*sqrt(a + a*sec(c + d*x))) + (16*a^2*sqrt(cos(c + d*x))*sqrt(a + a*sec(c + d*x))*sin(c + d*x))/(21*d) + (2*a*cos(c + d*x)^(3/2)*(a + a*sec(c + d*x))^(3/2)*sin(c + d*x))/(7*d) + (2*cos(c + d*x)^(5/2)*(a + a*sec(c + d*x))^(5/2)*sin(c + d*x))/(7*d)]
    @test_int [cos(c + d*x)^(5/2)*(a + a*sec(c + d*x))^(5/2), x, 4, (64*a^3*sin(c + d*x))/(15*d*sqrt(cos(c + d*x))*sqrt(a + a*sec(c + d*x))) + (16*a^2*sqrt(cos(c + d*x))*sqrt(a + a*sec(c + d*x))*sin(c + d*x))/(15*d) + (2*a*cos(c + d*x)^(3/2)*(a + a*sec(c + d*x))^(3/2)*sin(c + d*x))/(5*d)]
    @test_int [cos(c + d*x)^(3/2)*(a + a*sec(c + d*x))^(5/2), x, 5, (2*a^(5/2)*asinh((sqrt(a)*tan(c + d*x))/sqrt(a + a*sec(c + d*x)))*sqrt(cos(c + d*x))*sqrt(sec(c + d*x)))/d + (14*a^3*sin(c + d*x))/(3*d*sqrt(cos(c + d*x))*sqrt(a + a*sec(c + d*x))) + (2*a^2*sqrt(cos(c + d*x))*sqrt(a + a*sec(c + d*x))*sin(c + d*x))/(3*d)]
    @test_int [sqrt(cos(c + d*x))*(a + a*sec(c + d*x))^(5/2), x, 5, (5*a^(5/2)*asinh((sqrt(a)*tan(c + d*x))/sqrt(a + a*sec(c + d*x)))*sqrt(cos(c + d*x))*sqrt(sec(c + d*x)))/d + (a^3*sin(c + d*x))/(d*sqrt(cos(c + d*x))*sqrt(a + a*sec(c + d*x))) + (a^2*sqrt(a + a*sec(c + d*x))*sin(c + d*x))/(d*sqrt(cos(c + d*x)))]
    @test_int [(a + a*sec(c + d*x))^(5/2)/sqrt(cos(c + d*x)), x, 5, (19*a^(5/2)*asinh((sqrt(a)*tan(c + d*x))/sqrt(a + a*sec(c + d*x)))*sqrt(cos(c + d*x))*sqrt(sec(c + d*x)))/(4*d) + (9*a^3*sin(c + d*x))/(4*d*cos(c + d*x)^(3/2)*sqrt(a + a*sec(c + d*x))) + (a^2*sqrt(a + a*sec(c + d*x))*sin(c + d*x))/(2*d*cos(c + d*x)^(3/2))]
    @test_int [(a + a*sec(c + d*x))^(5/2)/cos(c + d*x)^(3/2), x, 6, (25*a^(5/2)*asinh((sqrt(a)*tan(c + d*x))/sqrt(a + a*sec(c + d*x)))*sqrt(cos(c + d*x))*sqrt(sec(c + d*x)))/(8*d) + (13*a^3*sin(c + d*x))/(12*d*cos(c + d*x)^(5/2)*sqrt(a + a*sec(c + d*x))) + (25*a^3*sin(c + d*x))/(8*d*cos(c + d*x)^(3/2)*sqrt(a + a*sec(c + d*x))) + (a^2*sqrt(a + a*sec(c + d*x))*sin(c + d*x))/(3*d*cos(c + d*x)^(5/2))]
    @test_int [(a + a*sec(c + d*x))^(5/2)/cos(c + d*x)^(5/2), x, 7, (163*a^(5/2)*asinh((sqrt(a)*tan(c + d*x))/sqrt(a + a*sec(c + d*x)))*sqrt(cos(c + d*x))*sqrt(sec(c + d*x)))/(64*d) + (17*a^3*sin(c + d*x))/(24*d*cos(c + d*x)^(7/2)*sqrt(a + a*sec(c + d*x))) + (163*a^3*sin(c + d*x))/(96*d*cos(c + d*x)^(5/2)*sqrt(a + a*sec(c + d*x))) + (163*a^3*sin(c + d*x))/(64*d*cos(c + d*x)^(3/2)*sqrt(a + a*sec(c + d*x))) + (a^2*sqrt(a + a*sec(c + d*x))*sin(c + d*x))/(4*d*cos(c + d*x)^(7/2))]


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


    @test_int [cos(c + d*x)^(5/2)/sqrt(a + a*sec(c + d*x)), x, 6, -((sqrt(2)*atanh((sqrt(a)*sqrt(sec(c + d*x))*sin(c + d*x))/(sqrt(2)*sqrt(a + a*sec(c + d*x))))*sqrt(cos(c + d*x))*sqrt(sec(c + d*x)))/(sqrt(a)*d)) + (26*sin(c + d*x))/(15*d*sqrt(cos(c + d*x))*sqrt(a + a*sec(c + d*x))) - (2*sqrt(cos(c + d*x))*sin(c + d*x))/(15*d*sqrt(a + a*sec(c + d*x))) + (2*cos(c + d*x)^(3/2)*sin(c + d*x))/(5*d*sqrt(a + a*sec(c + d*x)))]
    @test_int [cos(c + d*x)^(3/2)/sqrt(a + a*sec(c + d*x)), x, 5, (sqrt(2)*atanh((sqrt(a)*sqrt(sec(c + d*x))*sin(c + d*x))/(sqrt(2)*sqrt(a + a*sec(c + d*x))))*sqrt(cos(c + d*x))*sqrt(sec(c + d*x)))/(sqrt(a)*d) - (2*sin(c + d*x))/(3*d*sqrt(cos(c + d*x))*sqrt(a + a*sec(c + d*x))) + (2*sqrt(cos(c + d*x))*sin(c + d*x))/(3*d*sqrt(a + a*sec(c + d*x)))]
    @test_int [sqrt(cos(c + d*x))/sqrt(a + a*sec(c + d*x)), x, 4, -((sqrt(2)*atanh((sqrt(a)*sqrt(sec(c + d*x))*sin(c + d*x))/(sqrt(2)*sqrt(a + a*sec(c + d*x))))*sqrt(cos(c + d*x))*sqrt(sec(c + d*x)))/(sqrt(a)*d)) + (2*sin(c + d*x))/(d*sqrt(cos(c + d*x))*sqrt(a + a*sec(c + d*x)))]
    @test_int [1/(sqrt(cos(c + d*x))*sqrt(a + a*sec(c + d*x))), x, 3, (sqrt(2)*atanh((sqrt(a)*sin(c + d*x))/(sqrt(2)*sqrt(cos(c + d*x))*sqrt(a + a*sec(c + d*x)))))/(sqrt(a)*d), (sqrt(2)*atanh((sqrt(a)*sqrt(sec(c + d*x))*sin(c + d*x))/(sqrt(2)*sqrt(a + a*sec(c + d*x))))*sqrt(cos(c + d*x))*sqrt(sec(c + d*x)))/(sqrt(a)*d)]
    @test_int [1/(cos(c + d*x)^(3/2)*sqrt(a + a*sec(c + d*x))), x, 6, (2*asinh((sqrt(a)*tan(c + d*x))/sqrt(a + a*sec(c + d*x)))*sqrt(cos(c + d*x))*sqrt(sec(c + d*x)))/(sqrt(a)*d) - (sqrt(2)*atanh((sqrt(a)*sqrt(sec(c + d*x))*sin(c + d*x))/(sqrt(2)*sqrt(a + a*sec(c + d*x))))*sqrt(cos(c + d*x))*sqrt(sec(c + d*x)))/(sqrt(a)*d)]
    @test_int [1/(cos(c + d*x)^(5/2)*sqrt(a + a*sec(c + d*x))), x, 7, -((asinh((sqrt(a)*tan(c + d*x))/sqrt(a + a*sec(c + d*x)))*sqrt(cos(c + d*x))*sqrt(sec(c + d*x)))/(sqrt(a)*d)) + (sqrt(2)*atanh((sqrt(a)*sqrt(sec(c + d*x))*sin(c + d*x))/(sqrt(2)*sqrt(a + a*sec(c + d*x))))*sqrt(cos(c + d*x))*sqrt(sec(c + d*x)))/(sqrt(a)*d) + sin(c + d*x)/(d*cos(c + d*x)^(3/2)*sqrt(a + a*sec(c + d*x)))]
    @test_int [1/(cos(c + d*x)^(7/2)*sqrt(a + a*sec(c + d*x))), x, 8, (7*asinh((sqrt(a)*tan(c + d*x))/sqrt(a + a*sec(c + d*x)))*sqrt(cos(c + d*x))*sqrt(sec(c + d*x)))/(4*sqrt(a)*d) - (sqrt(2)*atanh((sqrt(a)*sqrt(sec(c + d*x))*sin(c + d*x))/(sqrt(2)*sqrt(a + a*sec(c + d*x))))*sqrt(cos(c + d*x))*sqrt(sec(c + d*x)))/(sqrt(a)*d) + sin(c + d*x)/(2*d*cos(c + d*x)^(5/2)*sqrt(a + a*sec(c + d*x))) - sin(c + d*x)/(4*d*cos(c + d*x)^(3/2)*sqrt(a + a*sec(c + d*x)))]


    @test_int [cos(c + d*x)^(5/2)/(a + a*sec(c + d*x))^(3/2), x, 7, -((15*atanh((sqrt(a)*sqrt(sec(c + d*x))*sin(c + d*x))/(sqrt(2)*sqrt(a + a*sec(c + d*x))))*sqrt(cos(c + d*x))*sqrt(sec(c + d*x)))/(2*sqrt(2)*a^(3/2)*d)) - (cos(c + d*x)^(3/2)*sin(c + d*x))/(2*d*(a + a*sec(c + d*x))^(3/2)) + (49*sin(c + d*x))/(10*a*d*sqrt(cos(c + d*x))*sqrt(a + a*sec(c + d*x))) - (13*sqrt(cos(c + d*x))*sin(c + d*x))/(10*a*d*sqrt(a + a*sec(c + d*x))) + (9*cos(c + d*x)^(3/2)*sin(c + d*x))/(10*a*d*sqrt(a + a*sec(c + d*x)))]
    @test_int [cos(c + d*x)^(3/2)/(a + a*sec(c + d*x))^(3/2), x, 6, (11*atanh((sqrt(a)*sqrt(sec(c + d*x))*sin(c + d*x))/(sqrt(2)*sqrt(a + a*sec(c + d*x))))*sqrt(cos(c + d*x))*sqrt(sec(c + d*x)))/(2*sqrt(2)*a^(3/2)*d) - (sqrt(cos(c + d*x))*sin(c + d*x))/(2*d*(a + a*sec(c + d*x))^(3/2)) - (19*sin(c + d*x))/(6*a*d*sqrt(cos(c + d*x))*sqrt(a + a*sec(c + d*x))) + (7*sqrt(cos(c + d*x))*sin(c + d*x))/(6*a*d*sqrt(a + a*sec(c + d*x)))]
    @test_int [sqrt(cos(c + d*x))/(a + a*sec(c + d*x))^(3/2), x, 5, -((7*atanh((sqrt(a)*sqrt(sec(c + d*x))*sin(c + d*x))/(sqrt(2)*sqrt(a + a*sec(c + d*x))))*sqrt(cos(c + d*x))*sqrt(sec(c + d*x)))/(2*sqrt(2)*a^(3/2)*d)) - sin(c + d*x)/(2*d*sqrt(cos(c + d*x))*(a + a*sec(c + d*x))^(3/2)) + (5*sin(c + d*x))/(2*a*d*sqrt(cos(c + d*x))*sqrt(a + a*sec(c + d*x)))]
    @test_int [1/(sqrt(cos(c + d*x))*(a + a*sec(c + d*x))^(3/2)), x, 4, (3*atanh((sqrt(a)*sqrt(sec(c + d*x))*sin(c + d*x))/(sqrt(2)*sqrt(a + a*sec(c + d*x))))*sqrt(cos(c + d*x))*sqrt(sec(c + d*x)))/(2*sqrt(2)*a^(3/2)*d) - sin(c + d*x)/(2*d*cos(c + d*x)^(3/2)*(a + a*sec(c + d*x))^(3/2))]
    @test_int [1/(cos(c + d*x)^(3/2)*(a + a*sec(c + d*x))^(3/2)), x, 4, (atanh((sqrt(a)*sqrt(sec(c + d*x))*sin(c + d*x))/(sqrt(2)*sqrt(a + a*sec(c + d*x))))*sqrt(cos(c + d*x))*sqrt(sec(c + d*x)))/(2*sqrt(2)*a^(3/2)*d) + sin(c + d*x)/(2*d*cos(c + d*x)^(3/2)*(a + a*sec(c + d*x))^(3/2))]
    @test_int [1/(cos(c + d*x)^(5/2)*(a + a*sec(c + d*x))^(3/2)), x, 7, (2*asinh((sqrt(a)*tan(c + d*x))/sqrt(a + a*sec(c + d*x)))*sqrt(cos(c + d*x))*sqrt(sec(c + d*x)))/(a^(3/2)*d) - (5*atanh((sqrt(a)*sqrt(sec(c + d*x))*sin(c + d*x))/(sqrt(2)*sqrt(a + a*sec(c + d*x))))*sqrt(cos(c + d*x))*sqrt(sec(c + d*x)))/(2*sqrt(2)*a^(3/2)*d) - sin(c + d*x)/(2*d*cos(c + d*x)^(3/2)*(a + a*sec(c + d*x))^(3/2))]
    @test_int [1/(cos(c + d*x)^(7/2)*(a + a*sec(c + d*x))^(3/2)), x, 8, -((3*asinh((sqrt(a)*tan(c + d*x))/sqrt(a + a*sec(c + d*x)))*sqrt(cos(c + d*x))*sqrt(sec(c + d*x)))/(a^(3/2)*d)) + (9*atanh((sqrt(a)*sqrt(sec(c + d*x))*sin(c + d*x))/(sqrt(2)*sqrt(a + a*sec(c + d*x))))*sqrt(cos(c + d*x))*sqrt(sec(c + d*x)))/(2*sqrt(2)*a^(3/2)*d) - sin(c + d*x)/(2*d*cos(c + d*x)^(5/2)*(a + a*sec(c + d*x))^(3/2)) + (3*sin(c + d*x))/(2*a*d*cos(c + d*x)^(3/2)*sqrt(a + a*sec(c + d*x)))]


    @test_int [cos(c + d*x)^(3/2)/(a + a*sec(c + d*x))^(5/2), x, 7, (163*atanh((sqrt(a)*sqrt(sec(c + d*x))*sin(c + d*x))/(sqrt(2)*sqrt(a + a*sec(c + d*x))))*sqrt(cos(c + d*x))*sqrt(sec(c + d*x)))/(16*sqrt(2)*a^(5/2)*d) - (sqrt(cos(c + d*x))*sin(c + d*x))/(4*d*(a + a*sec(c + d*x))^(5/2)) - (17*sqrt(cos(c + d*x))*sin(c + d*x))/(16*a*d*(a + a*sec(c + d*x))^(3/2)) - (299*sin(c + d*x))/(48*a^2*d*sqrt(cos(c + d*x))*sqrt(a + a*sec(c + d*x))) + (95*sqrt(cos(c + d*x))*sin(c + d*x))/(48*a^2*d*sqrt(a + a*sec(c + d*x)))]
    @test_int [sqrt(cos(c + d*x))/(a + a*sec(c + d*x))^(5/2), x, 6, -((75*atanh((sqrt(a)*sqrt(sec(c + d*x))*sin(c + d*x))/(sqrt(2)*sqrt(a + a*sec(c + d*x))))*sqrt(cos(c + d*x))*sqrt(sec(c + d*x)))/(16*sqrt(2)*a^(5/2)*d)) - sin(c + d*x)/(4*d*sqrt(cos(c + d*x))*(a + a*sec(c + d*x))^(5/2)) - (13*sin(c + d*x))/(16*a*d*sqrt(cos(c + d*x))*(a + a*sec(c + d*x))^(3/2)) + (49*sin(c + d*x))/(16*a^2*d*sqrt(cos(c + d*x))*sqrt(a + a*sec(c + d*x)))]
    @test_int [1/(sqrt(cos(c + d*x))*(a + a*sec(c + d*x))^(5/2)), x, 5, (19*atanh((sqrt(a)*sqrt(sec(c + d*x))*sin(c + d*x))/(sqrt(2)*sqrt(a + a*sec(c + d*x))))*sqrt(cos(c + d*x))*sqrt(sec(c + d*x)))/(16*sqrt(2)*a^(5/2)*d) - sin(c + d*x)/(4*d*cos(c + d*x)^(3/2)*(a + a*sec(c + d*x))^(5/2)) - (9*sin(c + d*x))/(16*a*d*cos(c + d*x)^(3/2)*(a + a*sec(c + d*x))^(3/2))]
    @test_int [1/(cos(c + d*x)^(3/2)*(a + a*sec(c + d*x))^(5/2)), x, 5, (5*atanh((sqrt(a)*sqrt(sec(c + d*x))*sin(c + d*x))/(sqrt(2)*sqrt(a + a*sec(c + d*x))))*sqrt(cos(c + d*x))*sqrt(sec(c + d*x)))/(16*sqrt(2)*a^(5/2)*d) - sin(c + d*x)/(4*d*cos(c + d*x)^(5/2)*(a + a*sec(c + d*x))^(5/2)) + (5*sin(c + d*x))/(16*a*d*cos(c + d*x)^(3/2)*(a + a*sec(c + d*x))^(3/2))]
    @test_int [1/(cos(c + d*x)^(5/2)*(a + a*sec(c + d*x))^(5/2)), x, 5, (3*atanh((sqrt(a)*sqrt(sec(c + d*x))*sin(c + d*x))/(sqrt(2)*sqrt(a + a*sec(c + d*x))))*sqrt(cos(c + d*x))*sqrt(sec(c + d*x)))/(16*sqrt(2)*a^(5/2)*d) + sin(c + d*x)/(4*d*cos(c + d*x)^(5/2)*(a + a*sec(c + d*x))^(5/2)) + (3*sin(c + d*x))/(16*a*d*cos(c + d*x)^(3/2)*(a + a*sec(c + d*x))^(3/2))]
    @test_int [1/(cos(c + d*x)^(7/2)*(a + a*sec(c + d*x))^(5/2)), x, 8, (2*asinh((sqrt(a)*tan(c + d*x))/sqrt(a + a*sec(c + d*x)))*sqrt(cos(c + d*x))*sqrt(sec(c + d*x)))/(a^(5/2)*d) - (43*atanh((sqrt(a)*sqrt(sec(c + d*x))*sin(c + d*x))/(sqrt(2)*sqrt(a + a*sec(c + d*x))))*sqrt(cos(c + d*x))*sqrt(sec(c + d*x)))/(16*sqrt(2)*a^(5/2)*d) - sin(c + d*x)/(4*d*cos(c + d*x)^(5/2)*(a + a*sec(c + d*x))^(5/2)) - (11*sin(c + d*x))/(16*a*d*cos(c + d*x)^(3/2)*(a + a*sec(c + d*x))^(3/2))]
    @test_int [1/(cos(c + d*x)^(9/2)*(a + a*sec(c + d*x))^(5/2)), x, 9, -((5*asinh((sqrt(a)*tan(c + d*x))/sqrt(a + a*sec(c + d*x)))*sqrt(cos(c + d*x))*sqrt(sec(c + d*x)))/(a^(5/2)*d)) + (115*atanh((sqrt(a)*sqrt(sec(c + d*x))*sin(c + d*x))/(sqrt(2)*sqrt(a + a*sec(c + d*x))))*sqrt(cos(c + d*x))*sqrt(sec(c + d*x)))/(16*sqrt(2)*a^(5/2)*d) - sin(c + d*x)/(4*d*cos(c + d*x)^(7/2)*(a + a*sec(c + d*x))^(5/2)) - (15*sin(c + d*x))/(16*a*d*cos(c + d*x)^(5/2)*(a + a*sec(c + d*x))^(3/2)) + (35*sin(c + d*x))/(16*a^2*d*cos(c + d*x)^(3/2)*sqrt(a + a*sec(c + d*x)))]


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


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


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


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


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


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


    @test_int [sec(c + d*x)^4*(a + b*sec(c + d*x))^2, x, 7, (3*a*b*atanh(sin(c + d*x)))/(4*d) + ((5*a^2 + 4*b^2)*tan(c + d*x))/(5*d) + (3*a*b*sec(c + d*x)*tan(c + d*x))/(4*d) + (a*b*sec(c + d*x)^3*tan(c + d*x))/(2*d) + (b^2*sec(c + d*x)^4*tan(c + d*x))/(5*d) + ((5*a^2 + 4*b^2)*tan(c + d*x)^3)/(15*d)]
    @test_int [sec(c + d*x)^3*(a + b*sec(c + d*x))^2, x, 6, ((4*a^2 + 3*b^2)*atanh(sin(c + d*x)))/(8*d) + (2*a*b*tan(c + d*x))/d + ((4*a^2 + 3*b^2)*sec(c + d*x)*tan(c + d*x))/(8*d) + (b^2*sec(c + d*x)^3*tan(c + d*x))/(4*d) + (2*a*b*tan(c + d*x)^3)/(3*d)]
    @test_int [sec(c + d*x)^2*(a + b*sec(c + d*x))^2, x, 6, (a*b*atanh(sin(c + d*x)))/d + ((3*a^2 + 2*b^2)*tan(c + d*x))/(3*d) + (a*b*sec(c + d*x)*tan(c + d*x))/d + (b^2*sec(c + d*x)^2*tan(c + d*x))/(3*d)]
    @test_int [sec(c + d*x)^1*(a + b*sec(c + d*x))^2, x, 5, ((2*a^2 + b^2)*atanh(sin(c + d*x)))/(2*d) + (2*a*b*tan(c + d*x))/d + (b^2*sec(c + d*x)*tan(c + d*x))/(2*d)]
    @test_int [sec(c + d*x)^0*(a + b*sec(c + d*x))^2, x, 4, a^2*x + (2*a*b*atanh(sin(c + d*x)))/d + (b^2*tan(c + d*x))/d]
    @test_int [cos(c + d*x)^1*(a + b*sec(c + d*x))^2, x, 4, 2*a*b*x + (b^2*atanh(sin(c + d*x)))/d + (a^2*sin(c + d*x))/d]
    @test_int [cos(c + d*x)^2*(a + b*sec(c + d*x))^2, x, 4, (1/2)*(a^2 + 2*b^2)*x + (2*a*b*sin(c + d*x))/d + (a^2*cos(c + d*x)*sin(c + d*x))/(2*d)]
    @test_int [cos(c + d*x)^3*(a + b*sec(c + d*x))^2, x, 6, a*b*x + ((a^2 + b^2)*sin(c + d*x))/d + (a*b*cos(c + d*x)*sin(c + d*x))/d - (a^2*sin(c + d*x)^3)/(3*d)]
    @test_int [cos(c + d*x)^4*(a + b*sec(c + d*x))^2, x, 6, (1/8)*(3*a^2 + 4*b^2)*x + (2*a*b*sin(c + d*x))/d + ((3*a^2 + 4*b^2)*cos(c + d*x)*sin(c + d*x))/(8*d) + (a^2*cos(c + d*x)^3*sin(c + d*x))/(4*d) - (2*a*b*sin(c + d*x)^3)/(3*d)]
    @test_int [cos(c + d*x)^5*(a + b*sec(c + d*x))^2, x, 8, (3*a*b*x)/4 + ((a^2 + b^2)*sin(c + d*x))/d + (3*a*b*cos(c + d*x)*sin(c + d*x))/(4*d) + (a*b*cos(c + d*x)^3*sin(c + d*x))/(2*d) - ((2*a^2 + b^2)*sin(c + d*x)^3)/(3*d) + (a^2*sin(c + d*x)^5)/(5*d)]


    @test_int [sec(c + d*x)^3*(a + b*sec(c + d*x))^3, x, 8, (a*(4*a^2 + 9*b^2)*atanh(sin(c + d*x)))/(8*d) - ((3*a^4 - 52*a^2*b^2 - 16*b^4)*tan(c + d*x))/(30*b*d) - (a*(6*a^2 - 71*b^2)*sec(c + d*x)*tan(c + d*x))/(120*d) - ((3*a^2 - 16*b^2)*(a + b*sec(c + d*x))^2*tan(c + d*x))/(60*b*d) - (a*(a + b*sec(c + d*x))^3*tan(c + d*x))/(20*b*d) + ((a + b*sec(c + d*x))^4*tan(c + d*x))/(5*b*d)]
    @test_int [sec(c + d*x)^2*(a + b*sec(c + d*x))^3, x, 7, (3*b*(4*a^2 + b^2)*atanh(sin(c + d*x)))/(8*d) + (a*(a^2 + 4*b^2)*tan(c + d*x))/(2*d) + (b*(2*a^2 + 3*b^2)*sec(c + d*x)*tan(c + d*x))/(8*d) + (a*(a + b*sec(c + d*x))^2*tan(c + d*x))/(4*d) + ((a + b*sec(c + d*x))^3*tan(c + d*x))/(4*d)]
    @test_int [sec(c + d*x)^1*(a + b*sec(c + d*x))^3, x, 6, (a*(2*a^2 + 3*b^2)*atanh(sin(c + d*x)))/(2*d) + (2*b*(4*a^2 + b^2)*tan(c + d*x))/(3*d) + (5*a*b^2*sec(c + d*x)*tan(c + d*x))/(6*d) + (b*(a + b*sec(c + d*x))^2*tan(c + d*x))/(3*d)]
    @test_int [sec(c + d*x)^0*(a + b*sec(c + d*x))^3, x, 5, a^3*x + (b*(6*a^2 + b^2)*atanh(sin(c + d*x)))/(2*d) + (5*a*b^2*tan(c + d*x))/(2*d) + (b^2*(a + b*sec(c + d*x))*tan(c + d*x))/(2*d)]
    @test_int [cos(c + d*x)^1*(a + b*sec(c + d*x))^3, x, 5, 3*a^2*b*x + (3*a*b^2*atanh(sin(c + d*x)))/d + (a*(a^2 - b^2)*sin(c + d*x))/d + (b^2*(a + b*sec(c + d*x))*sin(c + d*x))/d]
    @test_int [cos(c + d*x)^2*(a + b*sec(c + d*x))^3, x, 5, (1/2)*a*(a^2 + 6*b^2)*x + (b^3*atanh(sin(c + d*x)))/d + (5*a^2*b*sin(c + d*x))/(2*d) + (a^2*cos(c + d*x)*(a + b*sec(c + d*x))*sin(c + d*x))/(2*d)]
    @test_int [cos(c + d*x)^3*(a + b*sec(c + d*x))^3, x, 5, (1/2)*b*(3*a^2 + 2*b^2)*x + (a*(2*a^2 + 9*b^2)*sin(c + d*x))/(3*d) + (7*a^2*b*cos(c + d*x)*sin(c + d*x))/(6*d) + (a^2*cos(c + d*x)^2*(a + b*sec(c + d*x))*sin(c + d*x))/(3*d)]
    @test_int [cos(c + d*x)^4*(a + b*sec(c + d*x))^3, x, 7, (3/8)*a*(a^2 + 4*b^2)*x + (b*(11*a^2 + 4*b^2)*sin(c + d*x))/(4*d) + (3*a*(a^2 + 4*b^2)*cos(c + d*x)*sin(c + d*x))/(8*d) + (a^2*cos(c + d*x)^3*(a + b*sec(c + d*x))*sin(c + d*x))/(4*d) - (3*a^2*b*sin(c + d*x)^3)/(4*d)]
    @test_int [cos(c + d*x)^5*(a + b*sec(c + d*x))^3, x, 7, (1/8)*b*(9*a^2 + 4*b^2)*x + (a*(4*a^2 + 15*b^2)*sin(c + d*x))/(5*d) + (b*(9*a^2 + 4*b^2)*cos(c + d*x)*sin(c + d*x))/(8*d) + (11*a^2*b*cos(c + d*x)^3*sin(c + d*x))/(20*d) + (a^2*cos(c + d*x)^4*(a + b*sec(c + d*x))*sin(c + d*x))/(5*d) - (a*(4*a^2 + 15*b^2)*sin(c + d*x)^3)/(15*d)]
    @test_int [cos(c + d*x)^6*(a + b*sec(c + d*x))^3, x, 9, (1/16)*a*(5*a^2 + 18*b^2)*x + (b*(17*a^2 + 6*b^2)*sin(c + d*x))/(6*d) + (a*(5*a^2 + 18*b^2)*cos(c + d*x)*sin(c + d*x))/(16*d) + (a*(5*a^2 + 18*b^2)*cos(c + d*x)^3*sin(c + d*x))/(24*d) + (a^2*cos(c + d*x)^5*(a + b*sec(c + d*x))*sin(c + d*x))/(6*d) - (b*(5*a^2 + b^2)*sin(c + d*x)^3)/(3*d) + (13*a^2*b*sin(c + d*x)^5)/(30*d)]


    @test_int [sec(c + d*x)^3*(a + b*sec(c + d*x))^4, x, 9, ((8*a^4 + 36*a^2*b^2 + 5*b^4)*atanh(sin(c + d*x)))/(16*d) - (a*(4*a^4 - 121*a^2*b^2 - 128*b^4)*tan(c + d*x))/(60*b*d) - ((8*a^4 - 178*a^2*b^2 - 75*b^4)*sec(c + d*x)*tan(c + d*x))/(240*d) - (a*(4*a^2 - 53*b^2)*(a + b*sec(c + d*x))^2*tan(c + d*x))/(120*b*d) - ((4*a^2 - 25*b^2)*(a + b*sec(c + d*x))^3*tan(c + d*x))/(120*b*d) - (a*(a + b*sec(c + d*x))^4*tan(c + d*x))/(30*b*d) + ((a + b*sec(c + d*x))^5*tan(c + d*x))/(6*b*d)]
    @test_int [sec(c + d*x)^2*(a + b*sec(c + d*x))^4, x, 8, (a*b*(4*a^2 + 3*b^2)*atanh(sin(c + d*x)))/(2*d) + (2*(3*a^4 + 28*a^2*b^2 + 4*b^4)*tan(c + d*x))/(15*d) + (a*b*(6*a^2 + 29*b^2)*sec(c + d*x)*tan(c + d*x))/(30*d) + ((3*a^2 + 4*b^2)*(a + b*sec(c + d*x))^2*tan(c + d*x))/(15*d) + (a*(a + b*sec(c + d*x))^3*tan(c + d*x))/(5*d) + ((a + b*sec(c + d*x))^4*tan(c + d*x))/(5*d)]
    @test_int [sec(c + d*x)^1*(a + b*sec(c + d*x))^4, x, 7, ((8*a^4 + 24*a^2*b^2 + 3*b^4)*atanh(sin(c + d*x)))/(8*d) + (a*b*(19*a^2 + 16*b^2)*tan(c + d*x))/(6*d) + (b^2*(26*a^2 + 9*b^2)*sec(c + d*x)*tan(c + d*x))/(24*d) + (7*a*b*(a + b*sec(c + d*x))^2*tan(c + d*x))/(12*d) + (b*(a + b*sec(c + d*x))^3*tan(c + d*x))/(4*d)]
    @test_int [sec(c + d*x)^0*(a + b*sec(c + d*x))^4, x, 6, a^4*x + (2*a*b*(2*a^2 + b^2)*atanh(sin(c + d*x)))/d + (b^2*(17*a^2 + 2*b^2)*tan(c + d*x))/(3*d) + (4*a*b^3*sec(c + d*x)*tan(c + d*x))/(3*d) + (b^2*(a + b*sec(c + d*x))^2*tan(c + d*x))/(3*d)]
    @test_int [cos(c + d*x)^1*(a + b*sec(c + d*x))^4, x, 6, 4*a^3*b*x + (b^2*(12*a^2 + b^2)*atanh(sin(c + d*x)))/(2*d) + (a^2*(2*a^2 - b^2)*sin(c + d*x))/(2*d) + (b^2*(a + b*sec(c + d*x))^2*sin(c + d*x))/(2*d) + (3*a*b^3*tan(c + d*x))/d]
    @test_int [cos(c + d*x)^2*(a + b*sec(c + d*x))^4, x, 6, (1/2)*a^2*(a^2 + 12*b^2)*x + (4*a*b^3*atanh(sin(c + d*x)))/d + (3*a^3*b*sin(c + d*x))/d + (a^2*cos(c + d*x)*(a + b*sec(c + d*x))^2*sin(c + d*x))/(2*d) - (b^2*(a^2 - 2*b^2)*tan(c + d*x))/(2*d)]
    @test_int [cos(c + d*x)^3*(a + b*sec(c + d*x))^4, x, 6, 2*a*b*(a^2 + 2*b^2)*x + (b^4*atanh(sin(c + d*x)))/d + (a^2*(2*a^2 + 17*b^2)*sin(c + d*x))/(3*d) + (4*a^3*b*cos(c + d*x)*sin(c + d*x))/(3*d) + (a^2*cos(c + d*x)^2*(a + b*sec(c + d*x))^2*sin(c + d*x))/(3*d)]
    @test_int [cos(c + d*x)^4*(a + b*sec(c + d*x))^4, x, 6, (1/8)*(3*a^4 + 24*a^2*b^2 + 8*b^4)*x + (4*a*b*(2*a^2 + 3*b^2)*sin(c + d*x))/(3*d) + (a^2*(3*a^2 + 22*b^2)*cos(c + d*x)*sin(c + d*x))/(8*d) + (5*a^3*b*cos(c + d*x)^2*sin(c + d*x))/(6*d) + (a^2*cos(c + d*x)^3*(a + b*sec(c + d*x))^2*sin(c + d*x))/(4*d)]
    @test_int [cos(c + d*x)^5*(a + b*sec(c + d*x))^4, x, 8, (1/2)*a*b*(3*a^2 + 4*b^2)*x + ((4*a^4 + 29*a^2*b^2 + 5*b^4)*sin(c + d*x))/(5*d) + (a*b*(3*a^2 + 4*b^2)*cos(c + d*x)*sin(c + d*x))/(2*d) + (3*a^3*b*cos(c + d*x)^3*sin(c + d*x))/(5*d) + (a^2*cos(c + d*x)^4*(a + b*sec(c + d*x))^2*sin(c + d*x))/(5*d) - (a^2*(4*a^2 + 27*b^2)*sin(c + d*x)^3)/(15*d)]
    @test_int [cos(c + d*x)^6*(a + b*sec(c + d*x))^4, x, 8, (1/16)*(5*a^4 + 36*a^2*b^2 + 8*b^4)*x + (4*a*b*(4*a^2 + 5*b^2)*sin(c + d*x))/(5*d) + ((5*a^4 + 36*a^2*b^2 + 8*b^4)*cos(c + d*x)*sin(c + d*x))/(16*d) + (a^2*(5*a^2 + 32*b^2)*cos(c + d*x)^3*sin(c + d*x))/(24*d) + (7*a^3*b*cos(c + d*x)^4*sin(c + d*x))/(15*d) + (a^2*cos(c + d*x)^5*(a + b*sec(c + d*x))^2*sin(c + d*x))/(6*d) - (4*a*b*(4*a^2 + 5*b^2)*sin(c + d*x)^3)/(15*d)]


    @test_int [(a + b*sec(c + d*x))^5, x, 7, a^5*x + (b*(40*a^4 + 40*a^2*b^2 + 3*b^4)*atanh(sin(c + d*x)))/(8*d) + (a*b^2*(53*a^2 + 20*b^2)*tan(c + d*x))/(6*d) + (b^3*(58*a^2 + 9*b^2)*sec(c + d*x)*tan(c + d*x))/(24*d) + (11*a*b^2*(a + b*sec(c + d*x))^2*tan(c + d*x))/(12*d) + (b^2*(a + b*sec(c + d*x))^3*tan(c + d*x))/(4*d)]


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


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


    @test_int [sec(c + d*x)^5/(a + b*sec(c + d*x))^2, x, 8, ((6*a^2 + b^2)*atanh(sin(c + d*x)))/(2*b^4*d) - (2*a^3*(3*a^2 - 4*b^2)*atanh((sqrt(a - b)*tan((1/2)*(c + d*x)))/sqrt(a + b)))/((a - b)^(3/2)*b^4*(a + b)^(3/2)*d) - (a*(3*a^2 - 2*b^2)*tan(c + d*x))/(b^3*(a^2 - b^2)*d) + ((3*a^2 - b^2)*sec(c + d*x)*tan(c + d*x))/(2*b^2*(a^2 - b^2)*d) - (a^2*sec(c + d*x)^2*tan(c + d*x))/(b*(a^2 - b^2)*d*(a + b*sec(c + d*x)))]
    @test_int [sec(c + d*x)^4/(a + b*sec(c + d*x))^2, x, 7, -((2*a*atanh(sin(c + d*x)))/(b^3*d)) + (2*a^2*(2*a^2 - 3*b^2)*atanh((sqrt(a - b)*tan((1/2)*(c + d*x)))/sqrt(a + b)))/((a - b)^(3/2)*b^3*(a + b)^(3/2)*d) + ((2*a^2 - b^2)*tan(c + d*x))/(b^2*(a^2 - b^2)*d) - (a^2*sec(c + d*x)*tan(c + d*x))/(b*(a^2 - b^2)*d*(a + b*sec(c + d*x)))]
    @test_int [sec(c + d*x)^3/(a + b*sec(c + d*x))^2, x, 6, atanh(sin(c + d*x))/(b^2*d) - (2*a*(a^2 - 2*b^2)*atanh((sqrt(a - b)*tan((1/2)*(c + d*x)))/sqrt(a + b)))/((a - b)^(3/2)*b^2*(a + b)^(3/2)*d) - (a^2*tan(c + d*x))/(b*(a^2 - b^2)*d*(a + b*sec(c + d*x)))]
    @test_int [sec(c + d*x)^2/(a + b*sec(c + d*x))^2, x, 5, -((2*b*atanh((sqrt(a - b)*tan((1/2)*(c + d*x)))/sqrt(a + b)))/((a - b)^(3/2)*(a + b)^(3/2)*d)) + (a*tan(c + d*x))/((a^2 - b^2)*d*(a + b*sec(c + d*x)))]
    @test_int [sec(c + d*x)^1/(a + b*sec(c + d*x))^2, x, 5, (2*a*atanh((sqrt(a - b)*tan((1/2)*(c + d*x)))/sqrt(a + b)))/((a - b)^(3/2)*(a + b)^(3/2)*d) - (b*tan(c + d*x))/((a^2 - b^2)*d*(a + b*sec(c + d*x)))]
    @test_int [sec(c + d*x)^0/(a + b*sec(c + d*x))^2, x, 5, x/a^2 - (2*b*(2*a^2 - b^2)*atanh((sqrt(a - b)*tan((1/2)*(c + d*x)))/sqrt(a + b)))/(a^2*(a - b)^(3/2)*(a + b)^(3/2)*d) + (b^2*tan(c + d*x))/(a*(a^2 - b^2)*d*(a + b*sec(c + d*x)))]
    @test_int [cos(c + d*x)^1/(a + b*sec(c + d*x))^2, x, 6, -((2*b*x)/a^3) + (2*b^2*(3*a^2 - 2*b^2)*atanh((sqrt(a - b)*tan((1/2)*(c + d*x)))/sqrt(a + b)))/(a^3*(a - b)^(3/2)*(a + b)^(3/2)*d) + ((a^2 - 2*b^2)*sin(c + d*x))/(a^2*(a^2 - b^2)*d) + (b^2*sin(c + d*x))/(a*(a^2 - b^2)*d*(a + b*sec(c + d*x)))]
    @test_int [cos(c + d*x)^2/(a + b*sec(c + d*x))^2, x, 7, ((a^2 + 6*b^2)*x)/(2*a^4) - (2*b^3*(4*a^2 - 3*b^2)*atanh((sqrt(a - b)*tan((1/2)*(c + d*x)))/sqrt(a + b)))/(a^4*(a - b)^(3/2)*(a + b)^(3/2)*d) - (b*(2*a^2 - 3*b^2)*sin(c + d*x))/(a^3*(a^2 - b^2)*d) + ((a^2 - 3*b^2)*cos(c + d*x)*sin(c + d*x))/(2*a^2*(a^2 - b^2)*d) + (b^2*cos(c + d*x)*sin(c + d*x))/(a*(a^2 - b^2)*d*(a + b*sec(c + d*x)))]
    @test_int [cos(c + d*x)^3/(a + b*sec(c + d*x))^2, x, 8, -((b*(a^2 + 4*b^2)*x)/a^5) + (2*b^4*(5*a^2 - 4*b^2)*atanh((sqrt(a - b)*tan((1/2)*(c + d*x)))/sqrt(a + b)))/(a^5*(a - b)^(3/2)*(a + b)^(3/2)*d) + ((2*a^4 + 7*a^2*b^2 - 12*b^4)*sin(c + d*x))/(3*a^4*(a^2 - b^2)*d) - (b*(a^2 - 2*b^2)*cos(c + d*x)*sin(c + d*x))/(a^3*(a^2 - b^2)*d) + ((a^2 - 4*b^2)*cos(c + d*x)^2*sin(c + d*x))/(3*a^2*(a^2 - b^2)*d) + (b^2*cos(c + d*x)^2*sin(c + d*x))/(a*(a^2 - b^2)*d*(a + b*sec(c + d*x)))]


    @test_int [sec(c + d*x)^5/(a + b*sec(c + d*x))^3, x, 8, -((3*a*atanh(sin(c + d*x)))/(b^4*d)) + (3*a^2*(2*a^4 - 5*a^2*b^2 + 4*b^4)*atanh((sqrt(a - b)*tan((1/2)*(c + d*x)))/sqrt(a + b)))/((a - b)^(5/2)*b^4*(a + b)^(5/2)*d) + ((3*a^2 - 2*b^2)*tan(c + d*x))/(2*b^3*(a^2 - b^2)*d) - (a^2*sec(c + d*x)^2*tan(c + d*x))/(2*b*(a^2 - b^2)*d*(a + b*sec(c + d*x))^2) + (3*a^3*(a^2 - 2*b^2)*tan(c + d*x))/(2*b^3*(a^2 - b^2)^2*d*(a + b*sec(c + d*x)))]
    @test_int [sec(c + d*x)^4/(a + b*sec(c + d*x))^3, x, 7, atanh(sin(c + d*x))/(b^3*d) - (a*(2*a^4 - 5*a^2*b^2 + 6*b^4)*atanh((sqrt(a - b)*tan((1/2)*(c + d*x)))/sqrt(a + b)))/((a - b)^(5/2)*b^3*(a + b)^(5/2)*d) - (a^2*sec(c + d*x)*tan(c + d*x))/(2*b*(a^2 - b^2)*d*(a + b*sec(c + d*x))^2) - (a^2*(2*a^2 - 5*b^2)*tan(c + d*x))/(2*b^2*(a^2 - b^2)^2*d*(a + b*sec(c + d*x)))]
    @test_int [sec(c + d*x)^3/(a + b*sec(c + d*x))^3, x, 6, ((a^2 + 2*b^2)*atanh((sqrt(a - b)*tan((1/2)*(c + d*x)))/sqrt(a + b)))/((a - b)^(5/2)*(a + b)^(5/2)*d) - (a^2*tan(c + d*x))/(2*b*(a^2 - b^2)*d*(a + b*sec(c + d*x))^2) + (a*(a^2 - 4*b^2)*tan(c + d*x))/(2*b*(a^2 - b^2)^2*d*(a + b*sec(c + d*x)))]
    @test_int [sec(c + d*x)^2/(a + b*sec(c + d*x))^3, x, 6, -((3*a*b*atanh((sqrt(a - b)*tan((1/2)*(c + d*x)))/sqrt(a + b)))/((a - b)^(5/2)*(a + b)^(5/2)*d)) + (a*tan(c + d*x))/(2*(a^2 - b^2)*d*(a + b*sec(c + d*x))^2) + ((a^2 + 2*b^2)*tan(c + d*x))/(2*(a^2 - b^2)^2*d*(a + b*sec(c + d*x)))]
    @test_int [sec(c + d*x)^1/(a + b*sec(c + d*x))^3, x, 6, ((2*a^2 + b^2)*atanh((sqrt(a - b)*tan((1/2)*(c + d*x)))/sqrt(a + b)))/((a - b)^(5/2)*(a + b)^(5/2)*d) - (b*tan(c + d*x))/(2*(a^2 - b^2)*d*(a + b*sec(c + d*x))^2) - (3*a*b*tan(c + d*x))/(2*(a^2 - b^2)^2*d*(a + b*sec(c + d*x)))]
    @test_int [sec(c + d*x)^0/(a + b*sec(c + d*x))^3, x, 6, x/a^3 - (b*(6*a^4 - 5*a^2*b^2 + 2*b^4)*atanh((sqrt(a - b)*tan((1/2)*(c + d*x)))/sqrt(a + b)))/(a^3*(a - b)^(5/2)*(a + b)^(5/2)*d) + (b^2*tan(c + d*x))/(2*a*(a^2 - b^2)*d*(a + b*sec(c + d*x))^2) + (b^2*(5*a^2 - 2*b^2)*tan(c + d*x))/(2*a^2*(a^2 - b^2)^2*d*(a + b*sec(c + d*x)))]
    @test_int [cos(c + d*x)^1/(a + b*sec(c + d*x))^3, x, 7, -((3*b*x)/a^4) + (3*b^2*(4*a^4 - 5*a^2*b^2 + 2*b^4)*atanh((sqrt(a - b)*tan((1/2)*(c + d*x)))/sqrt(a + b)))/(a^4*(a - b)^(5/2)*(a + b)^(5/2)*d) + ((2*a^4 - 11*a^2*b^2 + 6*b^4)*sin(c + d*x))/(2*a^3*(a^2 - b^2)^2*d) + (b^2*sin(c + d*x))/(2*a*(a^2 - b^2)*d*(a + b*sec(c + d*x))^2) + (3*b^2*(2*a^2 - b^2)*sin(c + d*x))/(2*a^2*(a^2 - b^2)^2*d*(a + b*sec(c + d*x)))]
    @test_int [cos(c + d*x)^2/(a + b*sec(c + d*x))^3, x, 8, ((a^2 + 12*b^2)*x)/(2*a^5) - (b^3*(20*a^4 - 29*a^2*b^2 + 12*b^4)*atanh((sqrt(a - b)*tan((1/2)*(c + d*x)))/sqrt(a + b)))/(a^5*(a - b)^(5/2)*(a + b)^(5/2)*d) - (3*b*(2*a^4 - 7*a^2*b^2 + 4*b^4)*sin(c + d*x))/(2*a^4*(a^2 - b^2)^2*d) + ((a^4 - 10*a^2*b^2 + 6*b^4)*cos(c + d*x)*sin(c + d*x))/(2*a^3*(a^2 - b^2)^2*d) + (b^2*cos(c + d*x)*sin(c + d*x))/(2*a*(a^2 - b^2)*d*(a + b*sec(c + d*x))^2) + (b^2*(7*a^2 - 4*b^2)*cos(c + d*x)*sin(c + d*x))/(2*a^2*(a^2 - b^2)^2*d*(a + b*sec(c + d*x)))]


    @test_int [sec(c + d*x)^6/(a + b*sec(c + d*x))^4, x, 9, -((4*a*atanh(sin(c + d*x)))/(b^5*d)) + (a^2*(8*a^6 - 28*a^4*b^2 + 35*a^2*b^4 - 20*b^6)*atanh((sqrt(a - b)*tan((1/2)*(c + d*x)))/sqrt(a + b)))/((a - b)^(7/2)*b^5*(a + b)^(7/2)*d) + ((12*a^4 - 23*a^2*b^2 + 6*b^4)*tan(c + d*x))/(6*b^4*(a^2 - b^2)^2*d) - (a^2*sec(c + d*x)^3*tan(c + d*x))/(3*b*(a^2 - b^2)*d*(a + b*sec(c + d*x))^3) - (a^2*(4*a^2 - 9*b^2)*sec(c + d*x)^2*tan(c + d*x))/(6*b^2*(a^2 - b^2)^2*d*(a + b*sec(c + d*x))^2) + (a^3*(4*a^4 - 11*a^2*b^2 + 12*b^4)*tan(c + d*x))/(2*b^4*(a^2 - b^2)^3*d*(a + b*sec(c + d*x)))]
    @test_int [sec(c + d*x)^5/(a + b*sec(c + d*x))^4, x, 8, atanh(sin(c + d*x))/(b^4*d) - (a*(2*a^6 - 7*a^4*b^2 + 8*a^2*b^4 - 8*b^6)*atanh((sqrt(a - b)*tan((1/2)*(c + d*x)))/sqrt(a + b)))/((a - b)^(7/2)*b^4*(a + b)^(7/2)*d) - (a^2*sec(c + d*x)^2*tan(c + d*x))/(3*b*(a^2 - b^2)*d*(a + b*sec(c + d*x))^3) + (a^3*(3*a^2 - 8*b^2)*tan(c + d*x))/(6*b^3*(a^2 - b^2)^2*d*(a + b*sec(c + d*x))^2) - (a^2*(9*a^4 - 28*a^2*b^2 + 34*b^4)*tan(c + d*x))/(6*b^3*(a^2 - b^2)^3*d*(a + b*sec(c + d*x)))]
    @test_int [sec(c + d*x)^4/(a + b*sec(c + d*x))^4, x, 7, -((b*(3*a^2 + 2*b^2)*atanh((sqrt(a - b)*tan((1/2)*(c + d*x)))/sqrt(a + b)))/((a - b)^(7/2)*(a + b)^(7/2)*d)) - (a^2*sec(c + d*x)*tan(c + d*x))/(3*b*(a^2 - b^2)*d*(a + b*sec(c + d*x))^3) - (a^2*(2*a^2 - 7*b^2)*tan(c + d*x))/(6*b^2*(a^2 - b^2)^2*d*(a + b*sec(c + d*x))^2) + (a*(2*a^4 - 5*a^2*b^2 + 18*b^4)*tan(c + d*x))/(6*b^2*(a^2 - b^2)^3*d*(a + b*sec(c + d*x)))]
    @test_int [sec(c + d*x)^3/(a + b*sec(c + d*x))^4, x, 7, (a*(a^2 + 4*b^2)*atanh((sqrt(a - b)*tan((1/2)*(c + d*x)))/sqrt(a + b)))/((a - b)^(7/2)*(a + b)^(7/2)*d) - (a^2*tan(c + d*x))/(3*b*(a^2 - b^2)*d*(a + b*sec(c + d*x))^3) + (a*(a^2 - 6*b^2)*tan(c + d*x))/(6*b*(a^2 - b^2)^2*d*(a + b*sec(c + d*x))^2) + ((a^4 - 10*a^2*b^2 - 6*b^4)*tan(c + d*x))/(6*b*(a^2 - b^2)^3*d*(a + b*sec(c + d*x)))]
    @test_int [sec(c + d*x)^2/(a + b*sec(c + d*x))^4, x, 7, -((b*(4*a^2 + b^2)*atanh((sqrt(a - b)*tan((1/2)*(c + d*x)))/sqrt(a + b)))/((a - b)^(7/2)*(a + b)^(7/2)*d)) + (a*tan(c + d*x))/(3*(a^2 - b^2)*d*(a + b*sec(c + d*x))^3) + ((2*a^2 + 3*b^2)*tan(c + d*x))/(6*(a^2 - b^2)^2*d*(a + b*sec(c + d*x))^2) + (a*(2*a^2 + 13*b^2)*tan(c + d*x))/(6*(a^2 - b^2)^3*d*(a + b*sec(c + d*x)))]
    @test_int [sec(c + d*x)^1/(a + b*sec(c + d*x))^4, x, 7, (a*(2*a^2 + 3*b^2)*atanh((sqrt(a - b)*tan((1/2)*(c + d*x)))/sqrt(a + b)))/((a - b)^(7/2)*(a + b)^(7/2)*d) - (b*tan(c + d*x))/(3*(a^2 - b^2)*d*(a + b*sec(c + d*x))^3) - (5*a*b*tan(c + d*x))/(6*(a^2 - b^2)^2*d*(a + b*sec(c + d*x))^2) - (b*(11*a^2 + 4*b^2)*tan(c + d*x))/(6*(a^2 - b^2)^3*d*(a + b*sec(c + d*x)))]
    @test_int [sec(c + d*x)^0/(a + b*sec(c + d*x))^4, x, 7, x/a^4 - (b*(8*a^6 - 8*a^4*b^2 + 7*a^2*b^4 - 2*b^6)*atanh((sqrt(a - b)*tan((1/2)*(c + d*x)))/sqrt(a + b)))/(a^4*(a - b)^(7/2)*(a + b)^(7/2)*d) + (b^2*tan(c + d*x))/(3*a*(a^2 - b^2)*d*(a + b*sec(c + d*x))^3) + (b^2*(8*a^2 - 3*b^2)*tan(c + d*x))/(6*a^2*(a^2 - b^2)^2*d*(a + b*sec(c + d*x))^2) + (b^2*(26*a^4 - 17*a^2*b^2 + 6*b^4)*tan(c + d*x))/(6*a^3*(a^2 - b^2)^3*d*(a + b*sec(c + d*x)))]
    @test_int [cos(c + d*x)^1/(a + b*sec(c + d*x))^4, x, 8, -((4*b*x)/a^5) + (b^2*(20*a^6 - 35*a^4*b^2 + 28*a^2*b^4 - 8*b^6)*atanh((sqrt(a - b)*tan((1/2)*(c + d*x)))/sqrt(a + b)))/(a^5*(a - b)^(7/2)*(a + b)^(7/2)*d) + ((6*a^6 - 65*a^4*b^2 + 68*a^2*b^4 - 24*b^6)*sin(c + d*x))/(6*a^4*(a^2 - b^2)^3*d) + (b^2*sin(c + d*x))/(3*a*(a^2 - b^2)*d*(a + b*sec(c + d*x))^3) + (b^2*(9*a^2 - 4*b^2)*sin(c + d*x))/(6*a^2*(a^2 - b^2)^2*d*(a + b*sec(c + d*x))^2) + (b^2*(12*a^4 - 11*a^2*b^2 + 4*b^4)*sin(c + d*x))/(2*a^3*(a^2 - b^2)^3*d*(a + b*sec(c + d*x)))]
    @test_int [cos(c + d*x)^2/(a + b*sec(c + d*x))^4, x, 9, ((a^2 + 20*b^2)*x)/(2*a^6) - (b^3*(40*a^6 - 84*a^4*b^2 + 69*a^2*b^4 - 20*b^6)*atanh((sqrt(a - b)*tan((1/2)*(c + d*x)))/sqrt(a + b)))/(a^6*(a - b)^(7/2)*(a + b)^(7/2)*d) - (b*(24*a^6 - 146*a^4*b^2 + 167*a^2*b^4 - 60*b^6)*sin(c + d*x))/(6*a^5*(a^2 - b^2)^3*d) + ((a^6 - 23*a^4*b^2 + 27*a^2*b^4 - 10*b^6)*cos(c + d*x)*sin(c + d*x))/(2*a^4*(a^2 - b^2)^3*d) + (b^2*cos(c + d*x)*sin(c + d*x))/(3*a*(a^2 - b^2)*d*(a + b*sec(c + d*x))^3) + (5*b^2*(2*a^2 - b^2)*cos(c + d*x)*sin(c + d*x))/(6*a^2*(a^2 - b^2)^2*d*(a + b*sec(c + d*x))^2) + (b^2*(48*a^4 - 53*a^2*b^2 + 20*b^4)*cos(c + d*x)*sin(c + d*x))/(6*a^3*(a^2 - b^2)^3*d*(a + b*sec(c + d*x)))]


    @test_int [1/(3 + 5*sec(c + d*x)), x, 2, -(x/12) + (5*atan(sin(c + d*x)/(3 + cos(c + d*x))))/(6*d)]
    @test_int [1/(3 + 5*sec(c + d*x))^2, x, 4, (29*x)/576 + (35*atan(sin(c + d*x)/(3 + cos(c + d*x))))/(288*d) - (25*tan(c + d*x))/(48*d*(3 + 5*sec(c + d*x)))]
    @test_int [1/(3 + 5*sec(c + d*x))^3, x, 5, -((1007*x)/55296) + (3055*atan(sin(c + d*x)/(3 + cos(c + d*x))))/(27648*d) - (25*tan(c + d*x))/(96*d*(3 + 5*sec(c + d*x))^2) - (125*tan(c + d*x))/(4608*d*(3 + 5*sec(c + d*x)))]
    @test_int [1/(3 + 5*sec(c + d*x))^4, x, 6, (21553*x)/2654208 + (11215*atan(sin(c + d*x)/(3 + cos(c + d*x))))/(1327104*d) - (25*tan(c + d*x))/(144*d*(3 + 5*sec(c + d*x))^3) - (25*tan(c + d*x))/(4608*d*(3 + 5*sec(c + d*x))^2) - (16925*tan(c + d*x))/(221184*d*(3 + 5*sec(c + d*x)))]

    @test_int [1/(5 + 3*sec(c + d*x)), x, 3, x/5 + (3*log(2*cos((1/2)*(c + d*x)) - sin((1/2)*(c + d*x))))/(20*d) - (3*log(2*cos((1/2)*(c + d*x)) + sin((1/2)*(c + d*x))))/(20*d)]
    @test_int [1/(5 + 3*sec(c + d*x))^2, x, 5, x/25 + (123*log(2*cos((1/2)*(c + d*x)) - sin((1/2)*(c + d*x))))/(1600*d) - (123*log(2*cos((1/2)*(c + d*x)) + sin((1/2)*(c + d*x))))/(1600*d) + (9*tan(c + d*x))/(80*d*(5 + 3*sec(c + d*x)))]
    @test_int [1/(5 + 3*sec(c + d*x))^3, x, 6, x/125 + (8361*log(2*cos((1/2)*(c + d*x)) - sin((1/2)*(c + d*x))))/(256000*d) - (8361*log(2*cos((1/2)*(c + d*x)) + sin((1/2)*(c + d*x))))/(256000*d) + (9*tan(c + d*x))/(160*d*(5 + 3*sec(c + d*x))^2) + (963*tan(c + d*x))/(12800*d*(5 + 3*sec(c + d*x)))]
    @test_int [1/(5 + 3*sec(c + d*x))^4, x, 7, x/625 + (278151*log(2*cos((1/2)*(c + d*x)) - sin((1/2)*(c + d*x))))/(20480000*d) - (278151*log(2*cos((1/2)*(c + d*x)) + sin((1/2)*(c + d*x))))/(20480000*d) + (3*tan(c + d*x))/(80*d*(5 + 3*sec(c + d*x))^3) + (519*tan(c + d*x))/(12800*d*(5 + 3*sec(c + d*x))^2) + (38733*tan(c + d*x))/(1024000*d*(5 + 3*sec(c + d*x)))]


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


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


    @test_int [sec(c + d*x)^3*sqrt(a + b*sec(c + d*x)), x, 5, (2*(a - b)*sqrt(a + b)*(2*a^2 - 9*b^2)*cot(c + d*x)*Elliptic.E(asin(sqrt(a + b*sec(c + d*x))/sqrt(a + b)), (a + b)/(a - b))*sqrt((b*(1 - sec(c + d*x)))/(a + b))*sqrt(-((b*(1 + sec(c + d*x)))/(a - b))))/(15*b^3*d) + (2*(a - b)*sqrt(a + b)*(2*a + 9*b)*cot(c + d*x)*Elliptic.F(asin(sqrt(a + b*sec(c + d*x))/sqrt(a + b)), (a + b)/(a - b))*sqrt((b*(1 - sec(c + d*x)))/(a + b))*sqrt(-((b*(1 + sec(c + d*x)))/(a - b))))/(15*b^2*d) - (4*a*sqrt(a + b*sec(c + d*x))*tan(c + d*x))/(15*b*d) + (2*(a + b*sec(c + d*x))^(3/2)*tan(c + d*x))/(5*b*d)]
    @test_int [sec(c + d*x)^2*sqrt(a + b*sec(c + d*x)), x, 4, (-2*a*(a - b)*sqrt(a + b)*cot(c + d*x)*Elliptic.E(asin(sqrt(a + b*sec(c + d*x))/sqrt(a + b)), (a + b)/(a - b))*sqrt((b*(1 - sec(c + d*x)))/(a + b))*sqrt(-((b*(1 + sec(c + d*x)))/(a - b))))/(3*b^2*d) - (2*(a - b)*sqrt(a + b)*cot(c + d*x)*Elliptic.F(asin(sqrt(a + b*sec(c + d*x))/sqrt(a + b)), (a + b)/(a - b))*sqrt((b*(1 - sec(c + d*x)))/(a + b))*sqrt(-((b*(1 + sec(c + d*x)))/(a - b))))/(3*b*d) + (2*sqrt(a + b*sec(c + d*x))*tan(c + d*x))/(3*d)]
    @test_int [sec(c + d*x)^1*sqrt(a + b*sec(c + d*x)), x, 3, (-2*(a - b)*sqrt(a + b)*cot(c + d*x)*Elliptic.E(asin(sqrt(a + b*sec(c + d*x))/sqrt(a + b)), (a + b)/(a - b))*sqrt((b*(1 - sec(c + d*x)))/(a + b))*sqrt(-((b*(1 + sec(c + d*x)))/(a - b))))/(b*d) + (2*(a - b)*sqrt(a + b)*cot(c + d*x)*Elliptic.F(asin(sqrt(a + b*sec(c + d*x))/sqrt(a + b)), (a + b)/(a - b))*sqrt((b*(1 - sec(c + d*x)))/(a + b))*sqrt(-((b*(1 + sec(c + d*x)))/(a - b))))/(b*d)]
    @test_int [sec(c + d*x)^0*sqrt(a + b*sec(c + d*x)), x, 1, (-2*cot(c + d*x)*Elliptic.Pi(a/(a + b), asin(sqrt(a + b)/sqrt(a + b*sec(c + d*x))), (a - b)/(a + b))*sqrt(-((b*(1 - sec(c + d*x)))/(a + b*sec(c + d*x))))*sqrt((b*(1 + sec(c + d*x)))/(a + b*sec(c + d*x)))*(a + b*sec(c + d*x)))/(sqrt(a + b)*d)]
    @test_int [cos(c + d*x)^1*sqrt(a + b*sec(c + d*x)), x, 6, ((a - b)*sqrt(a + b)*cot(c + d*x)*Elliptic.E(asin(sqrt(a + b*sec(c + d*x))/sqrt(a + b)), (a + b)/(a - b))*sqrt((b*(1 - sec(c + d*x)))/(a + b))*sqrt(-((b*(1 + sec(c + d*x)))/(a - b))))/(b*d) + (sqrt(a + b)*cot(c + d*x)*Elliptic.F(asin(sqrt(a + b*sec(c + d*x))/sqrt(a + b)), (a + b)/(a - b))*sqrt((b*(1 - sec(c + d*x)))/(a + b))*sqrt(-((b*(1 + sec(c + d*x)))/(a - b))))/d - (b*sqrt(a + b)*cot(c + d*x)*Elliptic.Pi((a + b)/a, asin(sqrt(a + b*sec(c + d*x))/sqrt(a + b)), (a + b)/(a - b))*sqrt((b*(1 - sec(c + d*x)))/(a + b))*sqrt(-((b*(1 + sec(c + d*x)))/(a - b))))/(a*d) + (sqrt(a + b*sec(c + d*x))*sin(c + d*x))/d]
    @test_int [cos(c + d*x)^2*sqrt(a + b*sec(c + d*x)), x, 7, ((a - b)*sqrt(a + b)*cot(c + d*x)*Elliptic.E(asin(sqrt(a + b*sec(c + d*x))/sqrt(a + b)), (a + b)/(a - b))*sqrt((b*(1 - sec(c + d*x)))/(a + b))*sqrt(-((b*(1 + sec(c + d*x)))/(a - b))))/(4*a*d) + (sqrt(a + b)*(2*a + b)*cot(c + d*x)*Elliptic.F(asin(sqrt(a + b*sec(c + d*x))/sqrt(a + b)), (a + b)/(a - b))*sqrt((b*(1 - sec(c + d*x)))/(a + b))*sqrt(-((b*(1 + sec(c + d*x)))/(a - b))))/(4*a*d) - (sqrt(a + b)*(4*a^2 - b^2)*cot(c + d*x)*Elliptic.Pi((a + b)/a, asin(sqrt(a + b*sec(c + d*x))/sqrt(a + b)), (a + b)/(a - b))*sqrt((b*(1 - sec(c + d*x)))/(a + b))*sqrt(-((b*(1 + sec(c + d*x)))/(a - b))))/(4*a^2*d) + (b*sqrt(a + b*sec(c + d*x))*sin(c + d*x))/(4*a*d) + (cos(c + d*x)*sqrt(a + b*sec(c + d*x))*sin(c + d*x))/(2*d)]


    @test_int [sec(c + d*x)^4*(a + b*sec(c + d*x))^(3/2), x, 7, (-2*(a - b)*sqrt(a + b)*(8*a^4 + 33*a^2*b^2 + 147*b^4)*cot(c + d*x)*Elliptic.E(asin(sqrt(a + b*sec(c + d*x))/sqrt(a + b)), (a + b)/(a - b))*sqrt((b*(1 - sec(c + d*x)))/(a + b))*sqrt(-((b*(1 + sec(c + d*x)))/(a - b))))/(315*b^4*d) - (2*(a - b)*sqrt(a + b)*(8*a^3 + 6*a^2*b + 39*a*b^2 - 147*b^3)*cot(c + d*x)*Elliptic.F(asin(sqrt(a + b*sec(c + d*x))/sqrt(a + b)), (a + b)/(a - b))*sqrt((b*(1 - sec(c + d*x)))/(a + b))*sqrt(-((b*(1 + sec(c + d*x)))/(a - b))))/(315*b^3*d) + (2*a*(8*a^2 + 39*b^2)*sqrt(a + b*sec(c + d*x))*tan(c + d*x))/(315*b^2*d) + (2*(8*a^2 + 49*b^2)*(a + b*sec(c + d*x))^(3/2)*tan(c + d*x))/(315*b^2*d) - (8*a*(a + b*sec(c + d*x))^(5/2)*tan(c + d*x))/(63*b^2*d) + (2*sec(c + d*x)*(a + b*sec(c + d*x))^(5/2)*tan(c + d*x))/(9*b*d)]
    @test_int [sec(c + d*x)^3*(a + b*sec(c + d*x))^(3/2), x, 6, (4*a*(a - b)*sqrt(a + b)*(3*a^2 - 41*b^2)*cot(c + d*x)*Elliptic.E(asin(sqrt(a + b*sec(c + d*x))/sqrt(a + b)), (a + b)/(a - b))*sqrt((b*(1 - sec(c + d*x)))/(a + b))*sqrt(-((b*(1 + sec(c + d*x)))/(a - b))))/(105*b^3*d) + (2*(a - b)*sqrt(a + b)*(6*a^2 + 57*a*b - 25*b^2)*cot(c + d*x)*Elliptic.F(asin(sqrt(a + b*sec(c + d*x))/sqrt(a + b)), (a + b)/(a - b))*sqrt((b*(1 - sec(c + d*x)))/(a + b))*sqrt(-((b*(1 + sec(c + d*x)))/(a - b))))/(105*b^2*d) - (2*(6*a^2 - 25*b^2)*sqrt(a + b*sec(c + d*x))*tan(c + d*x))/(105*b*d) - (4*a*(a + b*sec(c + d*x))^(3/2)*tan(c + d*x))/(35*b*d) + (2*(a + b*sec(c + d*x))^(5/2)*tan(c + d*x))/(7*b*d)]
    @test_int [sec(c + d*x)^2*(a + b*sec(c + d*x))^(3/2), x, 5, (-2*(a - b)*sqrt(a + b)*(a^2 + 3*b^2)*cot(c + d*x)*Elliptic.E(asin(sqrt(a + b*sec(c + d*x))/sqrt(a + b)), (a + b)/(a - b))*sqrt((b*(1 - sec(c + d*x)))/(a + b))*sqrt(-((b*(1 + sec(c + d*x)))/(a - b))))/(5*b^2*d) - (2*(a - 3*b)*(a - b)*sqrt(a + b)*cot(c + d*x)*Elliptic.F(asin(sqrt(a + b*sec(c + d*x))/sqrt(a + b)), (a + b)/(a - b))*sqrt((b*(1 - sec(c + d*x)))/(a + b))*sqrt(-((b*(1 + sec(c + d*x)))/(a - b))))/(5*b*d) + (2*a*sqrt(a + b*sec(c + d*x))*tan(c + d*x))/(5*d) + (2*(a + b*sec(c + d*x))^(3/2)*tan(c + d*x))/(5*d)]
    @test_int [sec(c + d*x)^1*(a + b*sec(c + d*x))^(3/2), x, 4, (-8*a*(a - b)*sqrt(a + b)*cot(c + d*x)*Elliptic.E(asin(sqrt(a + b*sec(c + d*x))/sqrt(a + b)), (a + b)/(a - b))*sqrt((b*(1 - sec(c + d*x)))/(a + b))*sqrt(-((b*(1 + sec(c + d*x)))/(a - b))))/(3*b*d) + (2*(a - b)*(3*a - b)*sqrt(a + b)*cot(c + d*x)*Elliptic.F(asin(sqrt(a + b*sec(c + d*x))/sqrt(a + b)), (a + b)/(a - b))*sqrt((b*(1 - sec(c + d*x)))/(a + b))*sqrt(-((b*(1 + sec(c + d*x)))/(a - b))))/(3*b*d) + (2*b*sqrt(a + b*sec(c + d*x))*tan(c + d*x))/(3*d)]
    @test_int [sec(c + d*x)^0*(a + b*sec(c + d*x))^(3/2), x, 5, (-2*(a - b)*sqrt(a + b)*cot(c + d*x)*Elliptic.E(asin(sqrt(a + b*sec(c + d*x))/sqrt(a + b)), (a + b)/(a - b))*sqrt((b*(1 - sec(c + d*x)))/(a + b))*sqrt(-((b*(1 + sec(c + d*x)))/(a - b))))/d + (2*(2*a - b)*sqrt(a + b)*cot(c + d*x)*Elliptic.F(asin(sqrt(a + b*sec(c + d*x))/sqrt(a + b)), (a + b)/(a - b))*sqrt((b*(1 - sec(c + d*x)))/(a + b))*sqrt(-((b*(1 + sec(c + d*x)))/(a - b))))/d - (2*a*sqrt(a + b)*cot(c + d*x)*Elliptic.Pi((a + b)/a, asin(sqrt(a + b*sec(c + d*x))/sqrt(a + b)), (a + b)/(a - b))*sqrt((b*(1 - sec(c + d*x)))/(a + b))*sqrt(-((b*(1 + sec(c + d*x)))/(a - b))))/d]
    @test_int [cos(c + d*x)^1*(a + b*sec(c + d*x))^(3/2), x, 6, (a*(a - b)*sqrt(a + b)*cot(c + d*x)*Elliptic.E(asin(sqrt(a + b*sec(c + d*x))/sqrt(a + b)), (a + b)/(a - b))*sqrt((b*(1 - sec(c + d*x)))/(a + b))*sqrt(-((b*(1 + sec(c + d*x)))/(a - b))))/(b*d) + (sqrt(a + b)*(a + 2*b)*cot(c + d*x)*Elliptic.F(asin(sqrt(a + b*sec(c + d*x))/sqrt(a + b)), (a + b)/(a - b))*sqrt((b*(1 - sec(c + d*x)))/(a + b))*sqrt(-((b*(1 + sec(c + d*x)))/(a - b))))/d - (3*b*sqrt(a + b)*cot(c + d*x)*Elliptic.Pi((a + b)/a, asin(sqrt(a + b*sec(c + d*x))/sqrt(a + b)), (a + b)/(a - b))*sqrt((b*(1 - sec(c + d*x)))/(a + b))*sqrt(-((b*(1 + sec(c + d*x)))/(a - b))))/d + (a*sqrt(a + b*sec(c + d*x))*sin(c + d*x))/d]
    @test_int [cos(c + d*x)^2*(a + b*sec(c + d*x))^(3/2), x, 7, (5*(a - b)*sqrt(a + b)*cot(c + d*x)*Elliptic.E(asin(sqrt(a + b*sec(c + d*x))/sqrt(a + b)), (a + b)/(a - b))*sqrt((b*(1 - sec(c + d*x)))/(a + b))*sqrt(-((b*(1 + sec(c + d*x)))/(a - b))))/(4*d) + (sqrt(a + b)*(2*a + 5*b)*cot(c + d*x)*Elliptic.F(asin(sqrt(a + b*sec(c + d*x))/sqrt(a + b)), (a + b)/(a - b))*sqrt((b*(1 - sec(c + d*x)))/(a + b))*sqrt(-((b*(1 + sec(c + d*x)))/(a - b))))/(4*d) - (sqrt(a + b)*(4*a^2 + 3*b^2)*cot(c + d*x)*Elliptic.Pi((a + b)/a, asin(sqrt(a + b*sec(c + d*x))/sqrt(a + b)), (a + b)/(a - b))*sqrt((b*(1 - sec(c + d*x)))/(a + b))*sqrt(-((b*(1 + sec(c + d*x)))/(a - b))))/(4*a*d) + (5*b*sqrt(a + b*sec(c + d*x))*sin(c + d*x))/(4*d) + (a*cos(c + d*x)*sqrt(a + b*sec(c + d*x))*sin(c + d*x))/(2*d)]


    @test_int [sec(c + d*x)^4*(a + b*sec(c + d*x))^(5/2), x, 8, (-2*a*(a - b)*sqrt(a + b)*(8*a^4 + 51*a^2*b^2 + 741*b^4)*cot(c + d*x)*Elliptic.E(asin(sqrt(a + b*sec(c + d*x))/sqrt(a + b)), (a + b)/(a - b))*sqrt((b*(1 - sec(c + d*x)))/(a + b))*sqrt(-((b*(1 + sec(c + d*x)))/(a - b))))/(693*b^4*d) - (2*(a - b)*sqrt(a + b)*(8*a^4 + 6*a^3*b + 57*a^2*b^2 - 606*a*b^3 + 135*b^4)*cot(c + d*x)*Elliptic.F(asin(sqrt(a + b*sec(c + d*x))/sqrt(a + b)), (a + b)/(a - b))*sqrt((b*(1 - sec(c + d*x)))/(a + b))*sqrt(-((b*(1 + sec(c + d*x)))/(a - b))))/(693*b^3*d) + (2*(8*a^4 + 57*a^2*b^2 + 135*b^4)*sqrt(a + b*sec(c + d*x))*tan(c + d*x))/(693*b^2*d) + (2*a*(8*a^2 + 67*b^2)*(a + b*sec(c + d*x))^(3/2)*tan(c + d*x))/(693*b^2*d) + (2*(8*a^2 + 81*b^2)*(a + b*sec(c + d*x))^(5/2)*tan(c + d*x))/(693*b^2*d) - (8*a*(a + b*sec(c + d*x))^(7/2)*tan(c + d*x))/(99*b^2*d) + (2*sec(c + d*x)*(a + b*sec(c + d*x))^(7/2)*tan(c + d*x))/(11*b*d)]
    @test_int [sec(c + d*x)^3*(a + b*sec(c + d*x))^(5/2), x, 7, (2*(a - b)*sqrt(a + b)*(10*a^4 - 279*a^2*b^2 - 147*b^4)*cot(c + d*x)*Elliptic.E(asin(sqrt(a + b*sec(c + d*x))/sqrt(a + b)), (a + b)/(a - b))*sqrt((b*(1 - sec(c + d*x)))/(a + b))*sqrt(-((b*(1 + sec(c + d*x)))/(a - b))))/(315*b^3*d) + (2*(a - b)*sqrt(a + b)*(10*a^3 + 165*a^2*b - 114*a*b^2 + 147*b^3)*cot(c + d*x)*Elliptic.F(asin(sqrt(a + b*sec(c + d*x))/sqrt(a + b)), (a + b)/(a - b))*sqrt((b*(1 - sec(c + d*x)))/(a + b))*sqrt(-((b*(1 + sec(c + d*x)))/(a - b))))/(315*b^2*d) - (4*a*(5*a^2 - 57*b^2)*sqrt(a + b*sec(c + d*x))*tan(c + d*x))/(315*b*d) - (2*(10*a^2 - 49*b^2)*(a + b*sec(c + d*x))^(3/2)*tan(c + d*x))/(315*b*d) - (4*a*(a + b*sec(c + d*x))^(5/2)*tan(c + d*x))/(63*b*d) + (2*(a + b*sec(c + d*x))^(7/2)*tan(c + d*x))/(9*b*d)]
    @test_int [sec(c + d*x)^2*(a + b*sec(c + d*x))^(5/2), x, 6, (-2*a*(a - b)*sqrt(a + b)*(3*a^2 + 29*b^2)*cot(c + d*x)*Elliptic.E(asin(sqrt(a + b*sec(c + d*x))/sqrt(a + b)), (a + b)/(a - b))*sqrt((b*(1 - sec(c + d*x)))/(a + b))*sqrt(-((b*(1 + sec(c + d*x)))/(a - b))))/(21*b^2*d) - (2*(a - b)*sqrt(a + b)*(3*a^2 - 24*a*b + 5*b^2)*cot(c + d*x)*Elliptic.F(asin(sqrt(a + b*sec(c + d*x))/sqrt(a + b)), (a + b)/(a - b))*sqrt((b*(1 - sec(c + d*x)))/(a + b))*sqrt(-((b*(1 + sec(c + d*x)))/(a - b))))/(21*b*d) + (2*(3*a^2 + 5*b^2)*sqrt(a + b*sec(c + d*x))*tan(c + d*x))/(21*d) + (2*a*(a + b*sec(c + d*x))^(3/2)*tan(c + d*x))/(7*d) + (2*(a + b*sec(c + d*x))^(5/2)*tan(c + d*x))/(7*d)]
    @test_int [sec(c + d*x)^1*(a + b*sec(c + d*x))^(5/2), x, 5, (-2*(a - b)*sqrt(a + b)*(23*a^2 + 9*b^2)*cot(c + d*x)*Elliptic.E(asin(sqrt(a + b*sec(c + d*x))/sqrt(a + b)), (a + b)/(a - b))*sqrt((b*(1 - sec(c + d*x)))/(a + b))*sqrt(-((b*(1 + sec(c + d*x)))/(a - b))))/(15*b*d) + (2*(a - b)*sqrt(a + b)*(15*a^2 - 8*a*b + 9*b^2)*cot(c + d*x)*Elliptic.F(asin(sqrt(a + b*sec(c + d*x))/sqrt(a + b)), (a + b)/(a - b))*sqrt((b*(1 - sec(c + d*x)))/(a + b))*sqrt(-((b*(1 + sec(c + d*x)))/(a - b))))/(15*b*d) + (16*a*b*sqrt(a + b*sec(c + d*x))*tan(c + d*x))/(15*d) + (2*b*(a + b*sec(c + d*x))^(3/2)*tan(c + d*x))/(5*d)]
    @test_int [sec(c + d*x)^0*(a + b*sec(c + d*x))^(5/2), x, 6, (-14*a*(a - b)*sqrt(a + b)*cot(c + d*x)*Elliptic.E(asin(sqrt(a + b*sec(c + d*x))/sqrt(a + b)), (a + b)/(a - b))*sqrt((b*(1 - sec(c + d*x)))/(a + b))*sqrt(-((b*(1 + sec(c + d*x)))/(a - b))))/(3*d) + (2*sqrt(a + b)*(9*a^2 - 7*a*b + b^2)*cot(c + d*x)*Elliptic.F(asin(sqrt(a + b*sec(c + d*x))/sqrt(a + b)), (a + b)/(a - b))*sqrt((b*(1 - sec(c + d*x)))/(a + b))*sqrt(-((b*(1 + sec(c + d*x)))/(a - b))))/(3*d) - (2*a^2*sqrt(a + b)*cot(c + d*x)*Elliptic.Pi((a + b)/a, asin(sqrt(a + b*sec(c + d*x))/sqrt(a + b)), (a + b)/(a - b))*sqrt((b*(1 - sec(c + d*x)))/(a + b))*sqrt(-((b*(1 + sec(c + d*x)))/(a - b))))/d + (2*b^2*sqrt(a + b*sec(c + d*x))*tan(c + d*x))/(3*d)]
    @test_int [cos(c + d*x)^1*(a + b*sec(c + d*x))^(5/2), x, 6, ((a - b)*sqrt(a + b)*(a^2 - 2*b^2)*cot(c + d*x)*Elliptic.E(asin(sqrt(a + b*sec(c + d*x))/sqrt(a + b)), (a + b)/(a - b))*sqrt((b*(1 - sec(c + d*x)))/(a + b))*sqrt(-((b*(1 + sec(c + d*x)))/(a - b))))/(b*d) + (sqrt(a + b)*(a^2 + 6*a*b - 2*b^2)*cot(c + d*x)*Elliptic.F(asin(sqrt(a + b*sec(c + d*x))/sqrt(a + b)), (a + b)/(a - b))*sqrt((b*(1 - sec(c + d*x)))/(a + b))*sqrt(-((b*(1 + sec(c + d*x)))/(a - b))))/d - (5*a*b*sqrt(a + b)*cot(c + d*x)*Elliptic.Pi((a + b)/a, asin(sqrt(a + b*sec(c + d*x))/sqrt(a + b)), (a + b)/(a - b))*sqrt((b*(1 - sec(c + d*x)))/(a + b))*sqrt(-((b*(1 + sec(c + d*x)))/(a - b))))/d + (a^2*sqrt(a + b*sec(c + d*x))*sin(c + d*x))/d]
    @test_int [cos(c + d*x)^2*(a + b*sec(c + d*x))^(5/2), x, 7, (9*a*(a - b)*sqrt(a + b)*cot(c + d*x)*Elliptic.E(asin(sqrt(a + b*sec(c + d*x))/sqrt(a + b)), (a + b)/(a - b))*sqrt((b*(1 - sec(c + d*x)))/(a + b))*sqrt(-((b*(1 + sec(c + d*x)))/(a - b))))/(4*d) + (sqrt(a + b)*(2*a^2 + 9*a*b + 8*b^2)*cot(c + d*x)*Elliptic.F(asin(sqrt(a + b*sec(c + d*x))/sqrt(a + b)), (a + b)/(a - b))*sqrt((b*(1 - sec(c + d*x)))/(a + b))*sqrt(-((b*(1 + sec(c + d*x)))/(a - b))))/(4*d) - (sqrt(a + b)*(4*a^2 + 15*b^2)*cot(c + d*x)*Elliptic.Pi((a + b)/a, asin(sqrt(a + b*sec(c + d*x))/sqrt(a + b)), (a + b)/(a - b))*sqrt((b*(1 - sec(c + d*x)))/(a + b))*sqrt(-((b*(1 + sec(c + d*x)))/(a - b))))/(4*d) + (9*a*b*sqrt(a + b*sec(c + d*x))*sin(c + d*x))/(4*d) + (a^2*cos(c + d*x)*sqrt(a + b*sec(c + d*x))*sin(c + d*x))/(2*d)]
    @test_int [cos(c + d*x)^3*(a + b*sec(c + d*x))^(5/2), x, 8, ((a - b)*sqrt(a + b)*(16*a^2 + 33*b^2)*cot(c + d*x)*Elliptic.E(asin(sqrt(a + b*sec(c + d*x))/sqrt(a + b)), (a + b)/(a - b))*sqrt((b*(1 - sec(c + d*x)))/(a + b))*sqrt(-((b*(1 + sec(c + d*x)))/(a - b))))/(24*b*d) + (sqrt(a + b)*(16*a^2 + 26*a*b + 33*b^2)*cot(c + d*x)*Elliptic.F(asin(sqrt(a + b*sec(c + d*x))/sqrt(a + b)), (a + b)/(a - b))*sqrt((b*(1 - sec(c + d*x)))/(a + b))*sqrt(-((b*(1 + sec(c + d*x)))/(a - b))))/(24*d) - (5*b*sqrt(a + b)*(4*a^2 + b^2)*cot(c + d*x)*Elliptic.Pi((a + b)/a, asin(sqrt(a + b*sec(c + d*x))/sqrt(a + b)), (a + b)/(a - b))*sqrt((b*(1 - sec(c + d*x)))/(a + b))*sqrt(-((b*(1 + sec(c + d*x)))/(a - b))))/(8*a*d) + ((16*a^2 + 33*b^2)*sqrt(a + b*sec(c + d*x))*sin(c + d*x))/(24*d) + (13*a*b*cos(c + d*x)*sqrt(a + b*sec(c + d*x))*sin(c + d*x))/(12*d) + (a^2*cos(c + d*x)^2*sqrt(a + b*sec(c + d*x))*sin(c + d*x))/(3*d)]
    @test_int [cos(c + d*x)^4*(a + b*sec(c + d*x))^(5/2), x, 9, ((a - b)*sqrt(a + b)*(284*a^2 + 15*b^2)*cot(c + d*x)*Elliptic.E(asin(sqrt(a + b*sec(c + d*x))/sqrt(a + b)), (a + b)/(a - b))*sqrt((b*(1 - sec(c + d*x)))/(a + b))*sqrt(-((b*(1 + sec(c + d*x)))/(a - b))))/(192*a*d) + (sqrt(a + b)*(72*a^3 + 284*a^2*b + 118*a*b^2 + 15*b^3)*cot(c + d*x)*Elliptic.F(asin(sqrt(a + b*sec(c + d*x))/sqrt(a + b)), (a + b)/(a - b))*sqrt((b*(1 - sec(c + d*x)))/(a + b))*sqrt(-((b*(1 + sec(c + d*x)))/(a - b))))/(192*a*d) - (sqrt(a + b)*(48*a^4 + 120*a^2*b^2 - 5*b^4)*cot(c + d*x)*Elliptic.Pi((a + b)/a, asin(sqrt(a + b*sec(c + d*x))/sqrt(a + b)), (a + b)/(a - b))*sqrt((b*(1 - sec(c + d*x)))/(a + b))*sqrt(-((b*(1 + sec(c + d*x)))/(a - b))))/(64*a^2*d) + (b*(284*a^2 + 15*b^2)*sqrt(a + b*sec(c + d*x))*sin(c + d*x))/(192*a*d) + ((36*a^2 + 59*b^2)*cos(c + d*x)*sqrt(a + b*sec(c + d*x))*sin(c + d*x))/(96*d) + (17*a*b*cos(c + d*x)^2*sqrt(a + b*sec(c + d*x))*sin(c + d*x))/(24*d) + (a^2*cos(c + d*x)^3*sqrt(a + b*sec(c + d*x))*sin(c + d*x))/(4*d)]


    @test_int [(a + b*sec(c + d*x))^(7/2), x, 7, (-2*(a - b)*sqrt(a + b)*(58*a^2 + 9*b^2)*cot(c + d*x)*Elliptic.E(asin(sqrt(a + b*sec(c + d*x))/sqrt(a + b)), (a + b)/(a - b))*sqrt((b*(1 - sec(c + d*x)))/(a + b))*sqrt(-((b*(1 + sec(c + d*x)))/(a - b))))/(15*d) + (2*sqrt(a + b)*(60*a^3 - 58*a^2*b + 22*a*b^2 - 9*b^3)*cot(c + d*x)*Elliptic.F(asin(sqrt(a + b*sec(c + d*x))/sqrt(a + b)), (a + b)/(a - b))*sqrt((b*(1 - sec(c + d*x)))/(a + b))*sqrt(-((b*(1 + sec(c + d*x)))/(a - b))))/(15*d) - (2*a^3*sqrt(a + b)*cot(c + d*x)*Elliptic.Pi((a + b)/a, asin(sqrt(a + b*sec(c + d*x))/sqrt(a + b)), (a + b)/(a - b))*sqrt((b*(1 - sec(c + d*x)))/(a + b))*sqrt(-((b*(1 + sec(c + d*x)))/(a - b))))/d + (26*a*b^2*sqrt(a + b*sec(c + d*x))*tan(c + d*x))/(15*d) + (2*b^2*(a + b*sec(c + d*x))^(3/2)*tan(c + d*x))/(5*d)]


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


    @test_int [sec(c + d*x)^5/sqrt(a + b*sec(c + d*x)), x, 6, (8*a*(a - b)*sqrt(a + b)*(12*a^2 + 11*b^2)*cot(c + d*x)*Elliptic.E(asin(sqrt(a + b*sec(c + d*x))/sqrt(a + b)), (a + b)/(a - b))*sqrt((b*(1 - sec(c + d*x)))/(a + b))*sqrt(-((b*(1 + sec(c + d*x)))/(a - b))))/(105*b^5*d) + (2*sqrt(a + b)*(48*a^3 - 12*a^2*b + 44*a*b^2 + 25*b^3)*cot(c + d*x)*Elliptic.F(asin(sqrt(a + b*sec(c + d*x))/sqrt(a + b)), (a + b)/(a - b))*sqrt((b*(1 - sec(c + d*x)))/(a + b))*sqrt(-((b*(1 + sec(c + d*x)))/(a - b))))/(105*b^4*d) + (2*(24*a^2 + 25*b^2)*sqrt(a + b*sec(c + d*x))*tan(c + d*x))/(105*b^3*d) - (12*a*sec(c + d*x)*sqrt(a + b*sec(c + d*x))*tan(c + d*x))/(35*b^2*d) + (2*sec(c + d*x)^2*sqrt(a + b*sec(c + d*x))*tan(c + d*x))/(7*b*d)]
    @test_int [sec(c + d*x)^4/sqrt(a + b*sec(c + d*x)), x, 5, (-2*(a - b)*sqrt(a + b)*(8*a^2 + 9*b^2)*cot(c + d*x)*Elliptic.E(asin(sqrt(a + b*sec(c + d*x))/sqrt(a + b)), (a + b)/(a - b))*sqrt((b*(1 - sec(c + d*x)))/(a + b))*sqrt(-((b*(1 + sec(c + d*x)))/(a - b))))/(15*b^4*d) - (2*sqrt(a + b)*(8*a^2 - 2*a*b + 9*b^2)*cot(c + d*x)*Elliptic.F(asin(sqrt(a + b*sec(c + d*x))/sqrt(a + b)), (a + b)/(a - b))*sqrt((b*(1 - sec(c + d*x)))/(a + b))*sqrt(-((b*(1 + sec(c + d*x)))/(a - b))))/(15*b^3*d) - (8*a*sqrt(a + b*sec(c + d*x))*tan(c + d*x))/(15*b^2*d) + (2*sec(c + d*x)*sqrt(a + b*sec(c + d*x))*tan(c + d*x))/(5*b*d)]
    @test_int [sec(c + d*x)^3/sqrt(a + b*sec(c + d*x)), x, 4, (4*a*(a - b)*sqrt(a + b)*cot(c + d*x)*Elliptic.E(asin(sqrt(a + b*sec(c + d*x))/sqrt(a + b)), (a + b)/(a - b))*sqrt((b*(1 - sec(c + d*x)))/(a + b))*sqrt(-((b*(1 + sec(c + d*x)))/(a - b))))/(3*b^3*d) + (2*sqrt(a + b)*(2*a + b)*cot(c + d*x)*Elliptic.F(asin(sqrt(a + b*sec(c + d*x))/sqrt(a + b)), (a + b)/(a - b))*sqrt((b*(1 - sec(c + d*x)))/(a + b))*sqrt(-((b*(1 + sec(c + d*x)))/(a - b))))/(3*b^2*d) + (2*sqrt(a + b*sec(c + d*x))*tan(c + d*x))/(3*b*d)]
    @test_int [sec(c + d*x)^2/sqrt(a + b*sec(c + d*x)), x, 3, (-2*(a - b)*sqrt(a + b)*cot(c + d*x)*Elliptic.E(asin(sqrt(a + b*sec(c + d*x))/sqrt(a + b)), (a + b)/(a - b))*sqrt((b*(1 - sec(c + d*x)))/(a + b))*sqrt(-((b*(1 + sec(c + d*x)))/(a - b))))/(b^2*d) - (2*sqrt(a + b)*cot(c + d*x)*Elliptic.F(asin(sqrt(a + b*sec(c + d*x))/sqrt(a + b)), (a + b)/(a - b))*sqrt((b*(1 - sec(c + d*x)))/(a + b))*sqrt(-((b*(1 + sec(c + d*x)))/(a - b))))/(b*d)]
    @test_int [sec(c + d*x)^1/sqrt(a + b*sec(c + d*x)), x, 1, (2*sqrt(a + b)*cot(c + d*x)*Elliptic.F(asin(sqrt(a + b*sec(c + d*x))/sqrt(a + b)), (a + b)/(a - b))*sqrt((b*(1 - sec(c + d*x)))/(a + b))*sqrt(-((b*(1 + sec(c + d*x)))/(a - b))))/(b*d)]
    @test_int [sec(c + d*x)^0/sqrt(a + b*sec(c + d*x)), x, 1, (-2*sqrt(a + b)*cot(c + d*x)*Elliptic.Pi((a + b)/a, asin(sqrt(a + b*sec(c + d*x))/sqrt(a + b)), (a + b)/(a - b))*sqrt((b*(1 - sec(c + d*x)))/(a + b))*sqrt(-((b*(1 + sec(c + d*x)))/(a - b))))/(a*d)]
    @test_int [cos(c + d*x)^1/sqrt(a + b*sec(c + d*x)), x, 6, ((a - b)*sqrt(a + b)*cot(c + d*x)*Elliptic.E(asin(sqrt(a + b*sec(c + d*x))/sqrt(a + b)), (a + b)/(a - b))*sqrt((b*(1 - sec(c + d*x)))/(a + b))*sqrt(-((b*(1 + sec(c + d*x)))/(a - b))))/(a*b*d) + (sqrt(a + b)*cot(c + d*x)*Elliptic.F(asin(sqrt(a + b*sec(c + d*x))/sqrt(a + b)), (a + b)/(a - b))*sqrt((b*(1 - sec(c + d*x)))/(a + b))*sqrt(-((b*(1 + sec(c + d*x)))/(a - b))))/(a*d) + (b*sqrt(a + b)*cot(c + d*x)*Elliptic.Pi((a + b)/a, asin(sqrt(a + b*sec(c + d*x))/sqrt(a + b)), (a + b)/(a - b))*sqrt((b*(1 - sec(c + d*x)))/(a + b))*sqrt(-((b*(1 + sec(c + d*x)))/(a - b))))/(a^2*d) + (sqrt(a + b*sec(c + d*x))*sin(c + d*x))/(a*d)]
    @test_int [cos(c + d*x)^2/sqrt(a + b*sec(c + d*x)), x, 7, (-3*(a - b)*sqrt(a + b)*cot(c + d*x)*Elliptic.E(asin(sqrt(a + b*sec(c + d*x))/sqrt(a + b)), (a + b)/(a - b))*sqrt((b*(1 - sec(c + d*x)))/(a + b))*sqrt(-((b*(1 + sec(c + d*x)))/(a - b))))/(4*a^2*d) + ((2*a - 3*b)*sqrt(a + b)*cot(c + d*x)*Elliptic.F(asin(sqrt(a + b*sec(c + d*x))/sqrt(a + b)), (a + b)/(a - b))*sqrt((b*(1 - sec(c + d*x)))/(a + b))*sqrt(-((b*(1 + sec(c + d*x)))/(a - b))))/(4*a^2*d) - (sqrt(a + b)*(4*a^2 + 3*b^2)*cot(c + d*x)*Elliptic.Pi((a + b)/a, asin(sqrt(a + b*sec(c + d*x))/sqrt(a + b)), (a + b)/(a - b))*sqrt((b*(1 - sec(c + d*x)))/(a + b))*sqrt(-((b*(1 + sec(c + d*x)))/(a - b))))/(4*a^3*d) - (3*b*sqrt(a + b*sec(c + d*x))*sin(c + d*x))/(4*a^2*d) + (cos(c + d*x)*sqrt(a + b*sec(c + d*x))*sin(c + d*x))/(2*a*d)]


    @test_int [sec(c + d*x)^5/(a + b*sec(c + d*x))^(3/2), x, 6, (-2*(16*a^4 - 8*a^2*b^2 - 3*b^4)*cot(c + d*x)*Elliptic.E(asin(sqrt(a + b*sec(c + d*x))/sqrt(a + b)), (a + b)/(a - b))*sqrt((b*(1 - sec(c + d*x)))/(a + b))*sqrt(-((b*(1 + sec(c + d*x)))/(a - b))))/(5*b^5*sqrt(a + b)*d) - (2*(4*a + 3*b)*(4*a^2 + b^2)*cot(c + d*x)*Elliptic.F(asin(sqrt(a + b*sec(c + d*x))/sqrt(a + b)), (a + b)/(a - b))*sqrt((b*(1 - sec(c + d*x)))/(a + b))*sqrt(-((b*(1 + sec(c + d*x)))/(a - b))))/(5*b^4*sqrt(a + b)*d) - (2*a^2*sec(c + d*x)^2*tan(c + d*x))/(b*(a^2 - b^2)*d*sqrt(a + b*sec(c + d*x))) - (2*a*(8*a^2 - 3*b^2)*sqrt(a + b*sec(c + d*x))*tan(c + d*x))/(5*b^3*(a^2 - b^2)*d) + (2*(6*a^2 - b^2)*sec(c + d*x)*sqrt(a + b*sec(c + d*x))*tan(c + d*x))/(5*b^2*(a^2 - b^2)*d)]
    @test_int [sec(c + d*x)^4/(a + b*sec(c + d*x))^(3/2), x, 5, (2*a*(8*a^2 - 5*b^2)*cot(c + d*x)*Elliptic.E(asin(sqrt(a + b*sec(c + d*x))/sqrt(a + b)), (a + b)/(a - b))*sqrt((b*(1 - sec(c + d*x)))/(a + b))*sqrt(-((b*(1 + sec(c + d*x)))/(a - b))))/(3*b^4*sqrt(a + b)*d) + (2*(2*a + b)*(4*a + b)*cot(c + d*x)*Elliptic.F(asin(sqrt(a + b*sec(c + d*x))/sqrt(a + b)), (a + b)/(a - b))*sqrt((b*(1 - sec(c + d*x)))/(a + b))*sqrt(-((b*(1 + sec(c + d*x)))/(a - b))))/(3*b^3*sqrt(a + b)*d) - (2*a^2*sec(c + d*x)*tan(c + d*x))/(b*(a^2 - b^2)*d*sqrt(a + b*sec(c + d*x))) + (2*(4*a^2 - b^2)*sqrt(a + b*sec(c + d*x))*tan(c + d*x))/(3*b^2*(a^2 - b^2)*d)]
    @test_int [sec(c + d*x)^3/(a + b*sec(c + d*x))^(3/2), x, 4, (-2*(2*a^2 - b^2)*cot(c + d*x)*Elliptic.E(asin(sqrt(a + b*sec(c + d*x))/sqrt(a + b)), (a + b)/(a - b))*sqrt((b*(1 - sec(c + d*x)))/(a + b))*sqrt(-((b*(1 + sec(c + d*x)))/(a - b))))/(b^3*sqrt(a + b)*d) - (2*(2*a + b)*cot(c + d*x)*Elliptic.F(asin(sqrt(a + b*sec(c + d*x))/sqrt(a + b)), (a + b)/(a - b))*sqrt((b*(1 - sec(c + d*x)))/(a + b))*sqrt(-((b*(1 + sec(c + d*x)))/(a - b))))/(b^2*sqrt(a + b)*d) - (2*a^2*tan(c + d*x))/(b*(a^2 - b^2)*d*sqrt(a + b*sec(c + d*x)))]
    @test_int [sec(c + d*x)^2/(a + b*sec(c + d*x))^(3/2), x, 4, (2*a*cot(c + d*x)*Elliptic.E(asin(sqrt(a + b*sec(c + d*x))/sqrt(a + b)), (a + b)/(a - b))*sqrt((b*(1 - sec(c + d*x)))/(a + b))*sqrt(-((b*(1 + sec(c + d*x)))/(a - b))))/(b^2*sqrt(a + b)*d) + (2*cot(c + d*x)*Elliptic.F(asin(sqrt(a + b*sec(c + d*x))/sqrt(a + b)), (a + b)/(a - b))*sqrt((b*(1 - sec(c + d*x)))/(a + b))*sqrt(-((b*(1 + sec(c + d*x)))/(a - b))))/(b*sqrt(a + b)*d) + (2*a*tan(c + d*x))/((a^2 - b^2)*d*sqrt(a + b*sec(c + d*x)))]
    @test_int [sec(c + d*x)^1/(a + b*sec(c + d*x))^(3/2), x, 5, (-2*cot(c + d*x)*Elliptic.E(asin(sqrt(a + b*sec(c + d*x))/sqrt(a + b)), (a + b)/(a - b))*sqrt((b*(1 - sec(c + d*x)))/(a + b))*sqrt(-((b*(1 + sec(c + d*x)))/(a - b))))/(b*sqrt(a + b)*d) + (2*cot(c + d*x)*Elliptic.F(asin(sqrt(a + b*sec(c + d*x))/sqrt(a + b)), (a + b)/(a - b))*sqrt((b*(1 - sec(c + d*x)))/(a + b))*sqrt(-((b*(1 + sec(c + d*x)))/(a - b))))/(b*sqrt(a + b)*d) - (2*b*tan(c + d*x))/((a^2 - b^2)*d*sqrt(a + b*sec(c + d*x)))]
    @test_int [sec(c + d*x)^0/(a + b*sec(c + d*x))^(3/2), x, 6, (2*cot(c + d*x)*Elliptic.E(asin(sqrt(a + b*sec(c + d*x))/sqrt(a + b)), (a + b)/(a - b))*sqrt((b*(1 - sec(c + d*x)))/(a + b))*sqrt(-((b*(1 + sec(c + d*x)))/(a - b))))/(a*sqrt(a + b)*d) - (2*cot(c + d*x)*Elliptic.F(asin(sqrt(a + b*sec(c + d*x))/sqrt(a + b)), (a + b)/(a - b))*sqrt((b*(1 - sec(c + d*x)))/(a + b))*sqrt(-((b*(1 + sec(c + d*x)))/(a - b))))/(a*sqrt(a + b)*d) - (2*sqrt(a + b)*cot(c + d*x)*Elliptic.Pi((a + b)/a, asin(sqrt(a + b*sec(c + d*x))/sqrt(a + b)), (a + b)/(a - b))*sqrt((b*(1 - sec(c + d*x)))/(a + b))*sqrt(-((b*(1 + sec(c + d*x)))/(a - b))))/(a^2*d) + (2*b^2*tan(c + d*x))/(a*(a^2 - b^2)*d*sqrt(a + b*sec(c + d*x)))]
    @test_int [cos(c + d*x)^1/(a + b*sec(c + d*x))^(3/2), x, 7, ((a^2 - 3*b^2)*cot(c + d*x)*Elliptic.E(asin(sqrt(a + b*sec(c + d*x))/sqrt(a + b)), (a + b)/(a - b))*sqrt((b*(1 - sec(c + d*x)))/(a + b))*sqrt(-((b*(1 + sec(c + d*x)))/(a - b))))/(a^2*b*sqrt(a + b)*d) + ((a + 3*b)*cot(c + d*x)*Elliptic.F(asin(sqrt(a + b*sec(c + d*x))/sqrt(a + b)), (a + b)/(a - b))*sqrt((b*(1 - sec(c + d*x)))/(a + b))*sqrt(-((b*(1 + sec(c + d*x)))/(a - b))))/(a^2*sqrt(a + b)*d) + (3*b*sqrt(a + b)*cot(c + d*x)*Elliptic.Pi((a + b)/a, asin(sqrt(a + b*sec(c + d*x))/sqrt(a + b)), (a + b)/(a - b))*sqrt((b*(1 - sec(c + d*x)))/(a + b))*sqrt(-((b*(1 + sec(c + d*x)))/(a - b))))/(a^3*d) + sin(c + d*x)/(a*d*sqrt(a + b*sec(c + d*x))) + (b*(a^2 - 3*b^2)*tan(c + d*x))/(a^2*(a^2 - b^2)*d*sqrt(a + b*sec(c + d*x)))]
    @test_int [cos(c + d*x)^2/(a + b*sec(c + d*x))^(3/2), x, 8, -((7*a^2 - 15*b^2)*cot(c + d*x)*Elliptic.E(asin(sqrt(a + b*sec(c + d*x))/sqrt(a + b)), (a + b)/(a - b))*sqrt((b*(1 - sec(c + d*x)))/(a + b))*sqrt(-((b*(1 + sec(c + d*x)))/(a - b))))/(4*a^3*sqrt(a + b)*d) + ((2*a^2 - 5*a*b - 15*b^2)*cot(c + d*x)*Elliptic.F(asin(sqrt(a + b*sec(c + d*x))/sqrt(a + b)), (a + b)/(a - b))*sqrt((b*(1 - sec(c + d*x)))/(a + b))*sqrt(-((b*(1 + sec(c + d*x)))/(a - b))))/(4*a^3*sqrt(a + b)*d) - (sqrt(a + b)*(4*a^2 + 15*b^2)*cot(c + d*x)*Elliptic.Pi((a + b)/a, asin(sqrt(a + b*sec(c + d*x))/sqrt(a + b)), (a + b)/(a - b))*sqrt((b*(1 - sec(c + d*x)))/(a + b))*sqrt(-((b*(1 + sec(c + d*x)))/(a - b))))/(4*a^4*d) - (5*b*sin(c + d*x))/(4*a^2*d*sqrt(a + b*sec(c + d*x))) + (cos(c + d*x)*sin(c + d*x))/(2*a*d*sqrt(a + b*sec(c + d*x))) - (b^2*(7*a^2 - 15*b^2)*tan(c + d*x))/(4*a^3*(a^2 - b^2)*d*sqrt(a + b*sec(c + d*x)))]


    @test_int [sec(c + d*x)^5/(a + b*sec(c + d*x))^(5/2), x, 6, (8*a*(4*a^4 - 7*a^2*b^2 + 2*b^4)*cot(c + d*x)*Elliptic.E(asin(sqrt(a + b*sec(c + d*x))/sqrt(a + b)), (a + b)/(a - b))*sqrt((b*(1 - sec(c + d*x)))/(a + b))*sqrt(-((b*(1 + sec(c + d*x)))/(a - b))))/(3*(a - b)*b^5*(a + b)^(3/2)*d) + (2*(16*a^4 + 12*a^3*b - 16*a^2*b^2 - 9*a*b^3 - b^4)*cot(c + d*x)*Elliptic.F(asin(sqrt(a + b*sec(c + d*x))/sqrt(a + b)), (a + b)/(a - b))*sqrt((b*(1 - sec(c + d*x)))/(a + b))*sqrt(-((b*(1 + sec(c + d*x)))/(a - b))))/(3*(a - b)*b^4*(a + b)^(3/2)*d) - (2*a^2*sec(c + d*x)^2*tan(c + d*x))/(3*b*(a^2 - b^2)*d*(a + b*sec(c + d*x))^(3/2)) + (4*a^3*(3*a^2 - 5*b^2)*tan(c + d*x))/(3*b^3*(a^2 - b^2)^2*d*sqrt(a + b*sec(c + d*x))) + (2*(2*a^2 - b^2)*sqrt(a + b*sec(c + d*x))*tan(c + d*x))/(3*b^3*(a^2 - b^2)*d)]
    @test_int [sec(c + d*x)^4/(a + b*sec(c + d*x))^(5/2), x, 5, (-2*(8*a^4 - 15*a^2*b^2 + 3*b^4)*cot(c + d*x)*Elliptic.E(asin(sqrt(a + b*sec(c + d*x))/sqrt(a + b)), (a + b)/(a - b))*sqrt((b*(1 - sec(c + d*x)))/(a + b))*sqrt(-((b*(1 + sec(c + d*x)))/(a - b))))/(3*(a - b)*b^4*(a + b)^(3/2)*d) - (2*(8*a^3 + 6*a^2*b - 9*a*b^2 - 3*b^3)*cot(c + d*x)*Elliptic.F(asin(sqrt(a + b*sec(c + d*x))/sqrt(a + b)), (a + b)/(a - b))*sqrt((b*(1 - sec(c + d*x)))/(a + b))*sqrt(-((b*(1 + sec(c + d*x)))/(a - b))))/(3*(a - b)*b^3*(a + b)^(3/2)*d) - (2*a^2*sec(c + d*x)*tan(c + d*x))/(3*b*(a^2 - b^2)*d*(a + b*sec(c + d*x))^(3/2)) - (8*a^2*(a^2 - 2*b^2)*tan(c + d*x))/(3*b^2*(a^2 - b^2)^2*d*sqrt(a + b*sec(c + d*x)))]
    @test_int [sec(c + d*x)^3/(a + b*sec(c + d*x))^(5/2), x, 5, (4*a*(a^2 - 3*b^2)*cot(c + d*x)*Elliptic.E(asin(sqrt(a + b*sec(c + d*x))/sqrt(a + b)), (a + b)/(a - b))*sqrt((b*(1 - sec(c + d*x)))/(a + b))*sqrt(-((b*(1 + sec(c + d*x)))/(a - b))))/(3*(a - b)*b^3*(a + b)^(3/2)*d) + (2*(2*a^2 + 3*a*b - 3*b^2)*cot(c + d*x)*Elliptic.F(asin(sqrt(a + b*sec(c + d*x))/sqrt(a + b)), (a + b)/(a - b))*sqrt((b*(1 - sec(c + d*x)))/(a + b))*sqrt(-((b*(1 + sec(c + d*x)))/(a - b))))/(3*(a - b)*b^2*(a + b)^(3/2)*d) - (2*a^2*tan(c + d*x))/(3*b*(a^2 - b^2)*d*(a + b*sec(c + d*x))^(3/2)) + (4*a*(a^2 - 3*b^2)*tan(c + d*x))/(3*b*(a^2 - b^2)^2*d*sqrt(a + b*sec(c + d*x)))]
    @test_int [sec(c + d*x)^2/(a + b*sec(c + d*x))^(5/2), x, 5, (2*(a^2 + 3*b^2)*cot(c + d*x)*Elliptic.E(asin(sqrt(a + b*sec(c + d*x))/sqrt(a + b)), (a + b)/(a - b))*sqrt((b*(1 - sec(c + d*x)))/(a + b))*sqrt(-((b*(1 + sec(c + d*x)))/(a - b))))/(3*(a - b)*b^2*(a + b)^(3/2)*d) + (2*(a - 3*b)*cot(c + d*x)*Elliptic.F(asin(sqrt(a + b*sec(c + d*x))/sqrt(a + b)), (a + b)/(a - b))*sqrt((b*(1 - sec(c + d*x)))/(a + b))*sqrt(-((b*(1 + sec(c + d*x)))/(a - b))))/(3*(a - b)*b*(a + b)^(3/2)*d) + (2*a*tan(c + d*x))/(3*(a^2 - b^2)*d*(a + b*sec(c + d*x))^(3/2)) + (2*(a^2 + 3*b^2)*tan(c + d*x))/(3*(a^2 - b^2)^2*d*sqrt(a + b*sec(c + d*x)))]
    @test_int [sec(c + d*x)^1/(a + b*sec(c + d*x))^(5/2), x, 5, (-8*a*cot(c + d*x)*Elliptic.E(asin(sqrt(a + b*sec(c + d*x))/sqrt(a + b)), (a + b)/(a - b))*sqrt((b*(1 - sec(c + d*x)))/(a + b))*sqrt(-((b*(1 + sec(c + d*x)))/(a - b))))/(3*(a - b)*b*(a + b)^(3/2)*d) + (2*(3*a - b)*cot(c + d*x)*Elliptic.F(asin(sqrt(a + b*sec(c + d*x))/sqrt(a + b)), (a + b)/(a - b))*sqrt((b*(1 - sec(c + d*x)))/(a + b))*sqrt(-((b*(1 + sec(c + d*x)))/(a - b))))/(3*(a - b)*b*(a + b)^(3/2)*d) - (2*b*tan(c + d*x))/(3*(a^2 - b^2)*d*(a + b*sec(c + d*x))^(3/2)) - (8*a*b*tan(c + d*x))/(3*(a^2 - b^2)^2*d*sqrt(a + b*sec(c + d*x)))]
    @test_int [sec(c + d*x)^0/(a + b*sec(c + d*x))^(5/2), x, 7, (2*(7*a^2 - 3*b^2)*cot(c + d*x)*Elliptic.E(asin(sqrt(a + b*sec(c + d*x))/sqrt(a + b)), (a + b)/(a - b))*sqrt((b*(1 - sec(c + d*x)))/(a + b))*sqrt(-((b*(1 + sec(c + d*x)))/(a - b))))/(3*a^2*(a - b)*(a + b)^(3/2)*d) - (2*(6*a^2 - a*b - 3*b^2)*cot(c + d*x)*Elliptic.F(asin(sqrt(a + b*sec(c + d*x))/sqrt(a + b)), (a + b)/(a - b))*sqrt((b*(1 - sec(c + d*x)))/(a + b))*sqrt(-((b*(1 + sec(c + d*x)))/(a - b))))/(3*a^2*(a - b)*(a + b)^(3/2)*d) - (2*sqrt(a + b)*cot(c + d*x)*Elliptic.Pi((a + b)/a, asin(sqrt(a + b*sec(c + d*x))/sqrt(a + b)), (a + b)/(a - b))*sqrt((b*(1 - sec(c + d*x)))/(a + b))*sqrt(-((b*(1 + sec(c + d*x)))/(a - b))))/(a^3*d) + (2*b^2*tan(c + d*x))/(3*a*(a^2 - b^2)*d*(a + b*sec(c + d*x))^(3/2)) + (2*b^2*(7*a^2 - 3*b^2)*tan(c + d*x))/(3*a^2*(a^2 - b^2)^2*d*sqrt(a + b*sec(c + d*x)))]
    @test_int [cos(c + d*x)^1/(a + b*sec(c + d*x))^(5/2), x, 8, ((3*a^4 - 26*a^2*b^2 + 15*b^4)*cot(c + d*x)*Elliptic.E(asin(sqrt(a + b*sec(c + d*x))/sqrt(a + b)), (a + b)/(a - b))*sqrt((b*(1 - sec(c + d*x)))/(a + b))*sqrt(-((b*(1 + sec(c + d*x)))/(a - b))))/(3*a^3*(a - b)*b*(a + b)^(3/2)*d) + ((3*a^3 + 21*a^2*b - 5*a*b^2 - 15*b^3)*cot(c + d*x)*Elliptic.F(asin(sqrt(a + b*sec(c + d*x))/sqrt(a + b)), (a + b)/(a - b))*sqrt((b*(1 - sec(c + d*x)))/(a + b))*sqrt(-((b*(1 + sec(c + d*x)))/(a - b))))/(3*a^3*(a - b)*(a + b)^(3/2)*d) + (5*b*sqrt(a + b)*cot(c + d*x)*Elliptic.Pi((a + b)/a, asin(sqrt(a + b*sec(c + d*x))/sqrt(a + b)), (a + b)/(a - b))*sqrt((b*(1 - sec(c + d*x)))/(a + b))*sqrt(-((b*(1 + sec(c + d*x)))/(a - b))))/(a^4*d) + sin(c + d*x)/(a*d*(a + b*sec(c + d*x))^(3/2)) + (b*(3*a^2 - 5*b^2)*tan(c + d*x))/(3*a^2*(a^2 - b^2)*d*(a + b*sec(c + d*x))^(3/2)) + (b*(3*a^4 - 26*a^2*b^2 + 15*b^4)*tan(c + d*x))/(3*a^3*(a^2 - b^2)^2*d*sqrt(a + b*sec(c + d*x)))]
    @test_int [cos(c + d*x)^2/(a + b*sec(c + d*x))^(5/2), x, 9, -((33*a^4 - 170*a^2*b^2 + 105*b^4)*cot(c + d*x)*Elliptic.E(asin(sqrt(a + b*sec(c + d*x))/sqrt(a + b)), (a + b)/(a - b))*sqrt((b*(1 - sec(c + d*x)))/(a + b))*sqrt(-((b*(1 + sec(c + d*x)))/(a - b))))/(12*a^4*(a - b)*(a + b)^(3/2)*d) + ((a + 3*b)*(6*a^3 - 45*a^2*b + 35*b^3)*cot(c + d*x)*Elliptic.F(asin(sqrt(a + b*sec(c + d*x))/sqrt(a + b)), (a + b)/(a - b))*sqrt((b*(1 - sec(c + d*x)))/(a + b))*sqrt(-((b*(1 + sec(c + d*x)))/(a - b))))/(12*a^4*(a - b)*(a + b)^(3/2)*d) - (sqrt(a + b)*(4*a^2 + 35*b^2)*cot(c + d*x)*Elliptic.Pi((a + b)/a, asin(sqrt(a + b*sec(c + d*x))/sqrt(a + b)), (a + b)/(a - b))*sqrt((b*(1 - sec(c + d*x)))/(a + b))*sqrt(-((b*(1 + sec(c + d*x)))/(a - b))))/(4*a^5*d) - (7*b*sin(c + d*x))/(4*a^2*d*(a + b*sec(c + d*x))^(3/2)) + (cos(c + d*x)*sin(c + d*x))/(2*a*d*(a + b*sec(c + d*x))^(3/2)) - (b^2*(27*a^2 - 35*b^2)*tan(c + d*x))/(12*a^3*(a^2 - b^2)*d*(a + b*sec(c + d*x))^(3/2)) - (b^2*(33*a^4 - 170*a^2*b^2 + 105*b^4)*tan(c + d*x))/(12*a^4*(a^2 - b^2)^2*d*sqrt(a + b*sec(c + d*x)))]


    @test_int [(a + b*sec(c + d*x))^(-7/2), x, 8, (2*(58*a^4 - 41*a^2*b^2 + 15*b^4)*cot(c + d*x)*Elliptic.E(asin(sqrt(a + b*sec(c + d*x))/sqrt(a + b)), (a + b)/(a - b))*sqrt((b*(1 - sec(c + d*x)))/(a + b))*sqrt(-((b*(1 + sec(c + d*x)))/(a - b))))/(15*a^3*(a - b)^2*(a + b)^(5/2)*d) - (2*(45*a^4 - 13*a^3*b - 36*a^2*b^2 + 5*a*b^3 + 15*b^4)*cot(c + d*x)*Elliptic.F(asin(sqrt(a + b*sec(c + d*x))/sqrt(a + b)), (a + b)/(a - b))*sqrt((b*(1 - sec(c + d*x)))/(a + b))*sqrt(-((b*(1 + sec(c + d*x)))/(a - b))))/(15*a^3*(a - b)^2*(a + b)^(5/2)*d) - (2*sqrt(a + b)*cot(c + d*x)*Elliptic.Pi((a + b)/a, asin(sqrt(a + b*sec(c + d*x))/sqrt(a + b)), (a + b)/(a - b))*sqrt((b*(1 - sec(c + d*x)))/(a + b))*sqrt(-((b*(1 + sec(c + d*x)))/(a - b))))/(a^4*d) + (2*b^2*tan(c + d*x))/(5*a*(a^2 - b^2)*d*(a + b*sec(c + d*x))^(5/2)) + (2*b^2*(13*a^2 - 5*b^2)*tan(c + d*x))/(15*a^2*(a^2 - b^2)^2*d*(a + b*sec(c + d*x))^(3/2)) + (2*b^2*(58*a^4 - 41*a^2*b^2 + 15*b^4)*tan(c + d*x))/(15*a^3*(a^2 - b^2)^3*d*sqrt(a + b*sec(c + d*x)))]


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


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


    @test_int [sec(c + d*x)^(5/2)*(a + b*sec(c + d*x)), x, 8, -((6*b*sqrt(cos(c + d*x))*Elliptic.E((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(5*d)) + (2*a*sqrt(cos(c + d*x))*Elliptic.F((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(3*d) + (6*b*sqrt(sec(c + d*x))*sin(c + d*x))/(5*d) + (2*a*sec(c + d*x)^(3/2)*sin(c + d*x))/(3*d) + (2*b*sec(c + d*x)^(5/2)*sin(c + d*x))/(5*d)]
    @test_int [sec(c + d*x)^(3/2)*(a + b*sec(c + d*x)), x, 7, -((2*a*sqrt(cos(c + d*x))*Elliptic.E((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/d) + (2*b*sqrt(cos(c + d*x))*Elliptic.F((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(3*d) + (2*a*sqrt(sec(c + d*x))*sin(c + d*x))/d + (2*b*sec(c + d*x)^(3/2)*sin(c + d*x))/(3*d)]
    @test_int [sec(c + d*x)^(1/2)*(a + b*sec(c + d*x)), x, 6, -((2*b*sqrt(cos(c + d*x))*Elliptic.E((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/d) + (2*a*sqrt(cos(c + d*x))*Elliptic.F((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/d + (2*b*sqrt(sec(c + d*x))*sin(c + d*x))/d]
    @test_int [sec(c + d*x)^(-1/2)*(a + b*sec(c + d*x)), x, 5, (2*a*sqrt(cos(c + d*x))*Elliptic.E((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/d + (2*b*sqrt(cos(c + d*x))*Elliptic.F((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/d]
    @test_int [sec(c + d*x)^(-3/2)*(a + b*sec(c + d*x)), x, 6, (2*b*sqrt(cos(c + d*x))*Elliptic.E((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/d + (2*a*sqrt(cos(c + d*x))*Elliptic.F((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(3*d) + (2*a*sin(c + d*x))/(3*d*sqrt(sec(c + d*x)))]
    @test_int [sec(c + d*x)^(-5/2)*(a + b*sec(c + d*x)), x, 7, (6*a*sqrt(cos(c + d*x))*Elliptic.E((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(5*d) + (2*b*sqrt(cos(c + d*x))*Elliptic.F((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(3*d) + (2*a*sin(c + d*x))/(5*d*sec(c + d*x)^(3/2)) + (2*b*sin(c + d*x))/(3*d*sqrt(sec(c + d*x)))]
    @test_int [sec(c + d*x)^(-7/2)*(a + b*sec(c + d*x)), x, 8, (6*b*sqrt(cos(c + d*x))*Elliptic.E((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(5*d) + (10*a*sqrt(cos(c + d*x))*Elliptic.F((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(21*d) + (2*a*sin(c + d*x))/(7*d*sec(c + d*x)^(5/2)) + (2*b*sin(c + d*x))/(5*d*sec(c + d*x)^(3/2)) + (10*a*sin(c + d*x))/(21*d*sqrt(sec(c + d*x)))]


    @test_int [sec(c + d*x)^(5/2)*(a + b*sec(c + d*x))^2, x, 9, -((12*a*b*sqrt(cos(c + d*x))*Elliptic.E((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(5*d)) + (2*(7*a^2 + 5*b^2)*sqrt(cos(c + d*x))*Elliptic.F((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(21*d) + (12*a*b*sqrt(sec(c + d*x))*sin(c + d*x))/(5*d) + (2*(7*a^2 + 5*b^2)*sec(c + d*x)^(3/2)*sin(c + d*x))/(21*d) + (4*a*b*sec(c + d*x)^(5/2)*sin(c + d*x))/(5*d) + (2*b^2*sec(c + d*x)^(7/2)*sin(c + d*x))/(7*d)]
    @test_int [sec(c + d*x)^(3/2)*(a + b*sec(c + d*x))^2, x, 8, -((2*(5*a^2 + 3*b^2)*sqrt(cos(c + d*x))*Elliptic.E((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(5*d)) + (4*a*b*sqrt(cos(c + d*x))*Elliptic.F((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(3*d) + (2*(5*a^2 + 3*b^2)*sqrt(sec(c + d*x))*sin(c + d*x))/(5*d) + (4*a*b*sec(c + d*x)^(3/2)*sin(c + d*x))/(3*d) + (2*b^2*sec(c + d*x)^(5/2)*sin(c + d*x))/(5*d)]
    @test_int [sec(c + d*x)^(1/2)*(a + b*sec(c + d*x))^2, x, 7, -((4*a*b*sqrt(cos(c + d*x))*Elliptic.E((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/d) + (2*(3*a^2 + b^2)*sqrt(cos(c + d*x))*Elliptic.F((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(3*d) + (4*a*b*sqrt(sec(c + d*x))*sin(c + d*x))/d + (2*b^2*sec(c + d*x)^(3/2)*sin(c + d*x))/(3*d)]
    @test_int [sec(c + d*x)^(-1/2)*(a + b*sec(c + d*x))^2, x, 6, (2*(a^2 - b^2)*sqrt(cos(c + d*x))*Elliptic.E((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/d + (4*a*b*sqrt(cos(c + d*x))*Elliptic.F((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/d + (2*b^2*sqrt(sec(c + d*x))*sin(c + d*x))/d]
    @test_int [sec(c + d*x)^(-3/2)*(a + b*sec(c + d*x))^2, x, 6, (4*a*b*sqrt(cos(c + d*x))*Elliptic.E((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/d + (2*(a^2 + 3*b^2)*sqrt(cos(c + d*x))*Elliptic.F((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(3*d) + (2*a^2*sin(c + d*x))/(3*d*sqrt(sec(c + d*x)))]
    @test_int [sec(c + d*x)^(-5/2)*(a + b*sec(c + d*x))^2, x, 7, (2*(3*a^2 + 5*b^2)*sqrt(cos(c + d*x))*Elliptic.E((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(5*d) + (4*a*b*sqrt(cos(c + d*x))*Elliptic.F((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(3*d) + (2*a^2*sin(c + d*x))/(5*d*sec(c + d*x)^(3/2)) + (4*a*b*sin(c + d*x))/(3*d*sqrt(sec(c + d*x)))]
    @test_int [sec(c + d*x)^(-7/2)*(a + b*sec(c + d*x))^2, x, 8, (12*a*b*sqrt(cos(c + d*x))*Elliptic.E((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(5*d) + (2*(5*a^2 + 7*b^2)*sqrt(cos(c + d*x))*Elliptic.F((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(21*d) + (2*a^2*sin(c + d*x))/(7*d*sec(c + d*x)^(5/2)) + (4*a*b*sin(c + d*x))/(5*d*sec(c + d*x)^(3/2)) + (2*(5*a^2 + 7*b^2)*sin(c + d*x))/(21*d*sqrt(sec(c + d*x)))]


    @test_int [sec(c + d*x)^(3/2)*(a + b*sec(c + d*x))^3, x, 9, -((2*a*(5*a^2 + 9*b^2)*sqrt(cos(c + d*x))*Elliptic.E((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(5*d)) + (2*b*(21*a^2 + 5*b^2)*sqrt(cos(c + d*x))*Elliptic.F((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(21*d) + (2*a*(5*a^2 + 9*b^2)*sqrt(sec(c + d*x))*sin(c + d*x))/(5*d) + (2*b*(21*a^2 + 5*b^2)*sec(c + d*x)^(3/2)*sin(c + d*x))/(21*d) + (32*a*b^2*sec(c + d*x)^(5/2)*sin(c + d*x))/(35*d) + (2*b^2*sec(c + d*x)^(5/2)*(a + b*sec(c + d*x))*sin(c + d*x))/(7*d)]
    @test_int [sec(c + d*x)^(1/2)*(a + b*sec(c + d*x))^3, x, 8, -((6*b*(5*a^2 + b^2)*sqrt(cos(c + d*x))*Elliptic.E((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(5*d)) + (2*a*(a^2 + b^2)*sqrt(cos(c + d*x))*Elliptic.F((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/d + (6*b*(5*a^2 + b^2)*sqrt(sec(c + d*x))*sin(c + d*x))/(5*d) + (8*a*b^2*sec(c + d*x)^(3/2)*sin(c + d*x))/(5*d) + (2*b^2*sec(c + d*x)^(3/2)*(a + b*sec(c + d*x))*sin(c + d*x))/(5*d)]
    @test_int [sec(c + d*x)^(-1/2)*(a + b*sec(c + d*x))^3, x, 7, (2*a*(a^2 - 3*b^2)*sqrt(cos(c + d*x))*Elliptic.E((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/d + (2*b*(9*a^2 + b^2)*sqrt(cos(c + d*x))*Elliptic.F((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(3*d) + (16*a*b^2*sqrt(sec(c + d*x))*sin(c + d*x))/(3*d) + (2*b^2*sqrt(sec(c + d*x))*(a + b*sec(c + d*x))*sin(c + d*x))/(3*d)]
    @test_int [sec(c + d*x)^(-3/2)*(a + b*sec(c + d*x))^3, x, 7, (2*b*(3*a^2 - b^2)*sqrt(cos(c + d*x))*Elliptic.E((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/d + (2*a*(a^2 + 9*b^2)*sqrt(cos(c + d*x))*Elliptic.F((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(3*d) - (2*b*(a^2 - 3*b^2)*sqrt(sec(c + d*x))*sin(c + d*x))/(3*d) + (2*a^2*(a + b*sec(c + d*x))*sin(c + d*x))/(3*d*sqrt(sec(c + d*x)))]
    @test_int [sec(c + d*x)^(-5/2)*(a + b*sec(c + d*x))^3, x, 7, (6*a*(a^2 + 5*b^2)*sqrt(cos(c + d*x))*Elliptic.E((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(5*d) + (2*b*(a^2 + b^2)*sqrt(cos(c + d*x))*Elliptic.F((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/d + (8*a^2*b*sin(c + d*x))/(5*d*sqrt(sec(c + d*x))) + (2*a^2*(a + b*sec(c + d*x))*sin(c + d*x))/(5*d*sec(c + d*x)^(3/2))]
    @test_int [sec(c + d*x)^(-7/2)*(a + b*sec(c + d*x))^3, x, 8, (2*b*(9*a^2 + 5*b^2)*sqrt(cos(c + d*x))*Elliptic.E((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(5*d) + (2*a*(5*a^2 + 21*b^2)*sqrt(cos(c + d*x))*Elliptic.F((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(21*d) + (32*a^2*b*sin(c + d*x))/(35*d*sec(c + d*x)^(3/2)) + (2*a*(5*a^2 + 21*b^2)*sin(c + d*x))/(21*d*sqrt(sec(c + d*x))) + (2*a^2*(a + b*sec(c + d*x))*sin(c + d*x))/(7*d*sec(c + d*x)^(5/2))]
    @test_int [sec(c + d*x)^(-9/2)*(a + b*sec(c + d*x))^3, x, 9, (2*a*(7*a^2 + 27*b^2)*sqrt(cos(c + d*x))*Elliptic.E((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(15*d) + (2*b*(15*a^2 + 7*b^2)*sqrt(cos(c + d*x))*Elliptic.F((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(21*d) + (40*a^2*b*sin(c + d*x))/(63*d*sec(c + d*x)^(5/2)) + (2*a*(7*a^2 + 27*b^2)*sin(c + d*x))/(45*d*sec(c + d*x)^(3/2)) + (2*b*(15*a^2 + 7*b^2)*sin(c + d*x))/(21*d*sqrt(sec(c + d*x))) + (2*a^2*(a + b*sec(c + d*x))*sin(c + d*x))/(9*d*sec(c + d*x)^(7/2))]


    @test_int [sec(c + d*x)^(3/2)*(a + b*sec(c + d*x))^4, x, 10, -((2*(15*a^4 + 54*a^2*b^2 + 7*b^4)*sqrt(cos(c + d*x))*Elliptic.E((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(15*d)) + (8*a*b*(7*a^2 + 5*b^2)*sqrt(cos(c + d*x))*Elliptic.F((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(21*d) + (2*(15*a^4 + 54*a^2*b^2 + 7*b^4)*sqrt(sec(c + d*x))*sin(c + d*x))/(15*d) + (8*a*b*(7*a^2 + 5*b^2)*sec(c + d*x)^(3/2)*sin(c + d*x))/(21*d) + (14*b^2*(7*a^2 + b^2)*sec(c + d*x)^(5/2)*sin(c + d*x))/(45*d) + (44*a*b^3*sec(c + d*x)^(7/2)*sin(c + d*x))/(63*d) + (2*b^2*sec(c + d*x)^(5/2)*(a + b*sec(c + d*x))^2*sin(c + d*x))/(9*d)]
    @test_int [sec(c + d*x)^(1/2)*(a + b*sec(c + d*x))^4, x, 9, -((8*a*b*(5*a^2 + 3*b^2)*sqrt(cos(c + d*x))*Elliptic.E((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(5*d)) + (2*(21*a^4 + 42*a^2*b^2 + 5*b^4)*sqrt(cos(c + d*x))*Elliptic.F((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(21*d) + (8*a*b*(5*a^2 + 3*b^2)*sqrt(sec(c + d*x))*sin(c + d*x))/(5*d) + (2*b^2*(39*a^2 + 5*b^2)*sec(c + d*x)^(3/2)*sin(c + d*x))/(21*d) + (36*a*b^3*sec(c + d*x)^(5/2)*sin(c + d*x))/(35*d) + (2*b^2*sec(c + d*x)^(3/2)*(a + b*sec(c + d*x))^2*sin(c + d*x))/(7*d)]
    @test_int [sec(c + d*x)^(-1/2)*(a + b*sec(c + d*x))^4, x, 8, (2*(5*a^4 - 30*a^2*b^2 - 3*b^4)*sqrt(cos(c + d*x))*Elliptic.E((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(5*d) + (8*a*b*(3*a^2 + b^2)*sqrt(cos(c + d*x))*Elliptic.F((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(3*d) + (2*b^2*(29*a^2 + 3*b^2)*sqrt(sec(c + d*x))*sin(c + d*x))/(5*d) + (28*a*b^3*sec(c + d*x)^(3/2)*sin(c + d*x))/(15*d) + (2*b^2*sqrt(sec(c + d*x))*(a + b*sec(c + d*x))^2*sin(c + d*x))/(5*d)]
    @test_int [sec(c + d*x)^(-3/2)*(a + b*sec(c + d*x))^4, x, 8, (8*a*b*(a^2 - b^2)*sqrt(cos(c + d*x))*Elliptic.E((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/d + (2*(a^4 + 18*a^2*b^2 + b^4)*sqrt(cos(c + d*x))*Elliptic.F((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(3*d) - (4*a*b*(a^2 - 6*b^2)*sqrt(sec(c + d*x))*sin(c + d*x))/(3*d) - (2*b^2*(a^2 - b^2)*sec(c + d*x)^(3/2)*sin(c + d*x))/(3*d) + (2*a^2*(a + b*sec(c + d*x))^2*sin(c + d*x))/(3*d*sqrt(sec(c + d*x)))]
    @test_int [sec(c + d*x)^(-5/2)*(a + b*sec(c + d*x))^4, x, 8, (2*(3*a^4 + 30*a^2*b^2 - 5*b^4)*sqrt(cos(c + d*x))*Elliptic.E((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(5*d) + (8*a*b*(a^2 + 3*b^2)*sqrt(cos(c + d*x))*Elliptic.F((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(3*d) + (28*a^3*b*sin(c + d*x))/(15*d*sqrt(sec(c + d*x))) - (2*b^2*(a^2 - 5*b^2)*sqrt(sec(c + d*x))*sin(c + d*x))/(5*d) + (2*a^2*(a + b*sec(c + d*x))^2*sin(c + d*x))/(5*d*sec(c + d*x)^(3/2))]
    @test_int [sec(c + d*x)^(-7/2)*(a + b*sec(c + d*x))^4, x, 8, (8*a*b*(3*a^2 + 5*b^2)*sqrt(cos(c + d*x))*Elliptic.E((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(5*d) + (2*(5*a^4 + 42*a^2*b^2 + 21*b^4)*sqrt(cos(c + d*x))*Elliptic.F((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(21*d) + (36*a^3*b*sin(c + d*x))/(35*d*sec(c + d*x)^(3/2)) + (2*a^2*(5*a^2 + 39*b^2)*sin(c + d*x))/(21*d*sqrt(sec(c + d*x))) + (2*a^2*(a + b*sec(c + d*x))^2*sin(c + d*x))/(7*d*sec(c + d*x)^(5/2))]
    @test_int [sec(c + d*x)^(-9/2)*(a + b*sec(c + d*x))^4, x, 9, (2*(7*a^4 + 54*a^2*b^2 + 15*b^4)*sqrt(cos(c + d*x))*Elliptic.E((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(15*d) + (8*a*b*(5*a^2 + 7*b^2)*sqrt(cos(c + d*x))*Elliptic.F((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(21*d) + (44*a^3*b*sin(c + d*x))/(63*d*sec(c + d*x)^(5/2)) + (14*a^2*(a^2 + 7*b^2)*sin(c + d*x))/(45*d*sec(c + d*x)^(3/2)) + (8*a*b*(5*a^2 + 7*b^2)*sin(c + d*x))/(21*d*sqrt(sec(c + d*x))) + (2*a^2*(a + b*sec(c + d*x))^2*sin(c + d*x))/(9*d*sec(c + d*x)^(7/2))]
    @test_int [sec(c + d*x)^(-11/2)*(a + b*sec(c + d*x))^4, x, 10, (8*a*b*(7*a^2 + 9*b^2)*sqrt(cos(c + d*x))*Elliptic.E((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(15*d) + (2*(45*a^4 + 330*a^2*b^2 + 77*b^4)*sqrt(cos(c + d*x))*Elliptic.F((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(231*d) + (52*a^3*b*sin(c + d*x))/(99*d*sec(c + d*x)^(7/2)) + (2*a^2*(9*a^2 + 59*b^2)*sin(c + d*x))/(77*d*sec(c + d*x)^(5/2)) + (8*a*b*(7*a^2 + 9*b^2)*sin(c + d*x))/(45*d*sec(c + d*x)^(3/2)) + (2*(45*a^4 + 330*a^2*b^2 + 77*b^4)*sin(c + d*x))/(231*d*sqrt(sec(c + d*x))) + (2*a^2*(a + b*sec(c + d*x))^2*sin(c + d*x))/(11*d*sec(c + d*x)^(9/2))]


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


    @test_int [sec(c + d*x)^(7/2)/(a + b*sec(c + d*x)), x, 10, (2*a*sqrt(cos(c + d*x))*Elliptic.E((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(b^2*d) + (2*sqrt(cos(c + d*x))*Elliptic.F((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(3*b*d) + (2*a^2*sqrt(cos(c + d*x))*Elliptic.Pi((2*a)/(a + b), (c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(b^2*(a + b)*d) - (2*a*sqrt(sec(c + d*x))*sin(c + d*x))/(b^2*d) + (2*sec(c + d*x)^(3/2)*sin(c + d*x))/(3*b*d)]
    @test_int [sec(c + d*x)^(5/2)/(a + b*sec(c + d*x)), x, 6, (-2*sqrt(cos(c + d*x))*Elliptic.E((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(b*d) - (2*a*sqrt(cos(c + d*x))*Elliptic.Pi((2*a)/(a + b), (c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(b*(a + b)*d) + (2*sqrt(sec(c + d*x))*sin(c + d*x))/(b*d)]
    @test_int [sec(c + d*x)^(3/2)/(a + b*sec(c + d*x)), x, 2, (2*sqrt(cos(c + d*x))*Elliptic.Pi((2*a)/(a + b), (c + d*x)/2, 2)*sqrt(sec(c + d*x)))/((a + b)*d)]
    @test_int [sqrt(sec(c + d*x))/(a + b*sec(c + d*x)), x, 4, (2*sqrt(cos(c + d*x))*Elliptic.F((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(a*d) - (2*b*sqrt(cos(c + d*x))*Elliptic.Pi((2*a)/(a + b), (c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(a*(a + b)*d)]
    @test_int [1/(sqrt(sec(c + d*x))*(a + b*sec(c + d*x))), x, 8, (2*sqrt(cos(c + d*x))*Elliptic.E((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(a*d) - (2*b*sqrt(cos(c + d*x))*Elliptic.F((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(a^2*d) + (2*b^2*sqrt(cos(c + d*x))*Elliptic.Pi((2*a)/(a + b), (c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(a^2*(a + b)*d)]
    @test_int [1/(sec(c + d*x)^(3/2)*(a + b*sec(c + d*x))), x, 9, (-2*b*sqrt(cos(c + d*x))*Elliptic.E((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(a^2*d) + (2*(a^2 + 3*b^2)*sqrt(cos(c + d*x))*Elliptic.F((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(3*a^3*d) - (2*b^3*sqrt(cos(c + d*x))*Elliptic.Pi((2*a)/(a + b), (c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(a^3*(a + b)*d) + (2*sin(c + d*x))/(3*a*d*sqrt(sec(c + d*x)))]


    @test_int [sec(c + d*x)^(9/2)/(a + b*sec(c + d*x))^2, x, 11, (a*(5*a^2 - 4*b^2)*sqrt(cos(c + d*x))*Elliptic.E((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(b^3*(a^2 - b^2)*d) + ((5*a^2 - 2*b^2)*sqrt(cos(c + d*x))*Elliptic.F((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(3*b^2*(a^2 - b^2)*d) + (a^2*(5*a^2 - 7*b^2)*sqrt(cos(c + d*x))*Elliptic.Pi((2*a)/(a + b), (c + d*x)/2, 2)*sqrt(sec(c + d*x)))/((a - b)*b^3*(a + b)^2*d) - (a*(5*a^2 - 4*b^2)*sqrt(sec(c + d*x))*sin(c + d*x))/(b^3*(a^2 - b^2)*d) + ((5*a^2 - 2*b^2)*sec(c + d*x)^(3/2)*sin(c + d*x))/(3*b^2*(a^2 - b^2)*d) - (a^2*sec(c + d*x)^(5/2)*sin(c + d*x))/(b*(a^2 - b^2)*d*(a + b*sec(c + d*x)))]
    @test_int [sec(c + d*x)^(7/2)/(a + b*sec(c + d*x))^2, x, 10, -(((3*a^2 - 2*b^2)*sqrt(cos(c + d*x))*Elliptic.E((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(b^2*(a^2 - b^2)*d)) - (a*sqrt(cos(c + d*x))*Elliptic.F((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(b*(a^2 - b^2)*d) - (a*(3*a^2 - 5*b^2)*sqrt(cos(c + d*x))*Elliptic.Pi((2*a)/(a + b), (c + d*x)/2, 2)*sqrt(sec(c + d*x)))/((a - b)*b^2*(a + b)^2*d) + ((3*a^2 - 2*b^2)*sqrt(sec(c + d*x))*sin(c + d*x))/(b^2*(a^2 - b^2)*d) - (a^2*sec(c + d*x)^(3/2)*sin(c + d*x))/(b*(a^2 - b^2)*d*(a + b*sec(c + d*x)))]
    @test_int [sec(c + d*x)^(5/2)/(a + b*sec(c + d*x))^2, x, 9, (a*sqrt(cos(c + d*x))*Elliptic.E((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(b*(a^2 - b^2)*d) + (sqrt(cos(c + d*x))*Elliptic.F((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/((a^2 - b^2)*d) + ((a^2 - 3*b^2)*sqrt(cos(c + d*x))*Elliptic.Pi((2*a)/(a + b), (c + d*x)/2, 2)*sqrt(sec(c + d*x)))/((a - b)*b*(a + b)^2*d) - (a^2*sqrt(sec(c + d*x))*sin(c + d*x))/(b*(a^2 - b^2)*d*(a + b*sec(c + d*x)))]
    @test_int [sec(c + d*x)^(3/2)/(a + b*sec(c + d*x))^2, x, 9, -((sqrt(cos(c + d*x))*Elliptic.E((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/((a^2 - b^2)*d)) - (b*sqrt(cos(c + d*x))*Elliptic.F((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(a*(a^2 - b^2)*d) + ((a^2 + b^2)*sqrt(cos(c + d*x))*Elliptic.Pi((2*a)/(a + b), (c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(a*(a - b)*(a + b)^2*d) + (a*sqrt(sec(c + d*x))*sin(c + d*x))/((a^2 - b^2)*d*(a + b*sec(c + d*x)))]
    @test_int [sqrt(sec(c + d*x))/(a + b*sec(c + d*x))^2, x, 9, (b*sqrt(cos(c + d*x))*Elliptic.E((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(a*(a^2 - b^2)*d) + ((2*a^2 - b^2)*sqrt(cos(c + d*x))*Elliptic.F((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(a^2*(a^2 - b^2)*d) - (b*(3*a^2 - b^2)*sqrt(cos(c + d*x))*Elliptic.Pi((2*a)/(a + b), (c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(a^2*(a - b)*(a + b)^2*d) - (b*sqrt(sec(c + d*x))*sin(c + d*x))/((a^2 - b^2)*d*(a + b*sec(c + d*x)))]
    @test_int [1/(sqrt(sec(c + d*x))*(a + b*sec(c + d*x))^2), x, 9, ((2*a^2 - 3*b^2)*sqrt(cos(c + d*x))*Elliptic.E((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(a^2*(a^2 - b^2)*d) - (b*(4*a^2 - 3*b^2)*sqrt(cos(c + d*x))*Elliptic.F((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(a^3*(a^2 - b^2)*d) + (b^2*(5*a^2 - 3*b^2)*sqrt(cos(c + d*x))*Elliptic.Pi((2*a)/(a + b), (c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(a^3*(a - b)*(a + b)^2*d) + (b^2*sqrt(sec(c + d*x))*sin(c + d*x))/(a*(a^2 - b^2)*d*(a + b*sec(c + d*x)))]
    @test_int [1/(sec(c + d*x)^(3/2)*(a + b*sec(c + d*x))^2), x, 10, -((b*(4*a^2 - 5*b^2)*sqrt(cos(c + d*x))*Elliptic.E((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(a^3*(a^2 - b^2)*d)) + ((2*a^4 + 16*a^2*b^2 - 15*b^4)*sqrt(cos(c + d*x))*Elliptic.F((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(3*a^4*(a^2 - b^2)*d) - (b^3*(7*a^2 - 5*b^2)*sqrt(cos(c + d*x))*Elliptic.Pi((2*a)/(a + b), (c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(a^4*(a - b)*(a + b)^2*d) + ((2*a^2 - 5*b^2)*sin(c + d*x))/(3*a^2*(a^2 - b^2)*d*sqrt(sec(c + d*x))) + (b^2*sin(c + d*x))/(a*(a^2 - b^2)*d*sqrt(sec(c + d*x))*(a + b*sec(c + d*x)))]


    @test_int [sec(c + d*x)^(9/2)/(a + b*sec(c + d*x))^3, x, 11, -((15*a^4 - 29*a^2*b^2 + 8*b^4)*sqrt(cos(c + d*x))*Elliptic.E((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(4*b^3*(a^2 - b^2)^2*d) - (a*(5*a^2 - 11*b^2)*sqrt(cos(c + d*x))*Elliptic.F((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(4*b^2*(a^2 - b^2)^2*d) - (a*(15*a^4 - 38*a^2*b^2 + 35*b^4)*sqrt(cos(c + d*x))*Elliptic.Pi((2*a)/(a + b), (c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(4*(a - b)^2*b^3*(a + b)^3*d) + ((15*a^4 - 29*a^2*b^2 + 8*b^4)*sqrt(sec(c + d*x))*sin(c + d*x))/(4*b^3*(a^2 - b^2)^2*d) - (a^2*sec(c + d*x)^(5/2)*sin(c + d*x))/(2*b*(a^2 - b^2)*d*(a + b*sec(c + d*x))^2) - (a^2*(5*a^2 - 11*b^2)*sec(c + d*x)^(3/2)*sin(c + d*x))/(4*b^2*(a^2 - b^2)^2*d*(a + b*sec(c + d*x)))]
    @test_int [sec(c + d*x)^(7/2)/(a + b*sec(c + d*x))^3, x, 10, (3*a*(a^2 - 3*b^2)*sqrt(cos(c + d*x))*Elliptic.E((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(4*b^2*(a^2 - b^2)^2*d) + ((a^2 - 7*b^2)*sqrt(cos(c + d*x))*Elliptic.F((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(4*b*(a^2 - b^2)^2*d) + (3*(a^4 - 2*a^2*b^2 + 5*b^4)*sqrt(cos(c + d*x))*Elliptic.Pi((2*a)/(a + b), (1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(4*(a - b)^2*b^2*(a + b)^3*d) - (a^2*sec(c + d*x)^(3/2)*sin(c + d*x))/(2*b*(a^2 - b^2)*d*(a + b*sec(c + d*x))^2) - (3*a^2*(a^2 - 3*b^2)*sqrt(sec(c + d*x))*sin(c + d*x))/(4*b^2*(a^2 - b^2)^2*d*(a + b*sec(c + d*x)))]
    @test_int [sec(c + d*x)^(5/2)/(a + b*sec(c + d*x))^3, x, 10, ((a^2 + 5*b^2)*sqrt(cos(c + d*x))*Elliptic.E((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(4*b*(a^2 - b^2)^2*d) + (3*(a^2 + b^2)*sqrt(cos(c + d*x))*Elliptic.F((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(4*a*(a^2 - b^2)^2*d) + ((a^4 - 10*a^2*b^2 - 3*b^4)*sqrt(cos(c + d*x))*Elliptic.Pi((2*a)/(a + b), (c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(4*a*(a - b)^2*b*(a + b)^3*d) - (a^2*sqrt(sec(c + d*x))*sin(c + d*x))/(2*b*(a^2 - b^2)*d*(a + b*sec(c + d*x))^2) + (a*(a^2 - 7*b^2)*sqrt(sec(c + d*x))*sin(c + d*x))/(4*b*(a^2 - b^2)^2*d*(a + b*sec(c + d*x)))]
    @test_int [sec(c + d*x)^(3/2)/(a + b*sec(c + d*x))^3, x, 10, -(((5*a^2 + b^2)*sqrt(cos(c + d*x))*Elliptic.E((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(4*a*(a^2 - b^2)^2*d)) - (b*(7*a^2 - b^2)*sqrt(cos(c + d*x))*Elliptic.F((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(4*a^2*(a^2 - b^2)^2*d) + ((3*a^4 + 10*a^2*b^2 - b^4)*sqrt(cos(c + d*x))*Elliptic.Pi((2*a)/(a + b), (1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(4*a^2*(a - b)^2*(a + b)^3*d) + (a*sqrt(sec(c + d*x))*sin(c + d*x))/(2*(a^2 - b^2)*d*(a + b*sec(c + d*x))^2) + (3*(a^2 + b^2)*sqrt(sec(c + d*x))*sin(c + d*x))/(4*(a^2 - b^2)^2*d*(a + b*sec(c + d*x)))]
    @test_int [sqrt(sec(c + d*x))/(a + b*sec(c + d*x))^3, x, 10, (3*b*(3*a^2 - b^2)*sqrt(cos(c + d*x))*Elliptic.E((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(4*a^2*(a^2 - b^2)^2*d) + ((8*a^4 - 5*a^2*b^2 + 3*b^4)*sqrt(cos(c + d*x))*Elliptic.F((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(4*a^3*(a^2 - b^2)^2*d) - (3*b*(5*a^4 - 2*a^2*b^2 + b^4)*sqrt(cos(c + d*x))*Elliptic.Pi((2*a)/(a + b), (c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(4*a^3*(a - b)^2*(a + b)^3*d) - (b*sqrt(sec(c + d*x))*sin(c + d*x))/(2*(a^2 - b^2)*d*(a + b*sec(c + d*x))^2) - (b*(7*a^2 - b^2)*sqrt(sec(c + d*x))*sin(c + d*x))/(4*a*(a^2 - b^2)^2*d*(a + b*sec(c + d*x)))]
    @test_int [1/(sqrt(sec(c + d*x))*(a + b*sec(c + d*x))^3), x, 10, ((8*a^4 - 29*a^2*b^2 + 15*b^4)*sqrt(cos(c + d*x))*Elliptic.E((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(4*a^3*(a^2 - b^2)^2*d) - (3*b*(8*a^4 - 11*a^2*b^2 + 5*b^4)*sqrt(cos(c + d*x))*Elliptic.F((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(4*a^4*(a^2 - b^2)^2*d) + (b^2*(35*a^4 - 38*a^2*b^2 + 15*b^4)*sqrt(cos(c + d*x))*Elliptic.Pi((2*a)/(a + b), (c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(4*a^4*(a - b)^2*(a + b)^3*d) + (b^2*sqrt(sec(c + d*x))*sin(c + d*x))/(2*a*(a^2 - b^2)*d*(a + b*sec(c + d*x))^2) + (b^2*(11*a^2 - 5*b^2)*sqrt(sec(c + d*x))*sin(c + d*x))/(4*a^2*(a^2 - b^2)^2*d*(a + b*sec(c + d*x)))]
    @test_int [1/(sec(c + d*x)^(3/2)*(a + b*sec(c + d*x))^3), x, 11, -(b*(24*a^4 - 65*a^2*b^2 + 35*b^4)*sqrt(cos(c + d*x))*Elliptic.E((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(4*a^4*(a^2 - b^2)^2*d) + ((8*a^6 + 128*a^4*b^2 - 223*a^2*b^4 + 105*b^6)*sqrt(cos(c + d*x))*Elliptic.F((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(12*a^5*(a^2 - b^2)^2*d) - (b^3*(63*a^4 - 86*a^2*b^2 + 35*b^4)*sqrt(cos(c + d*x))*Elliptic.Pi((2*a)/(a + b), (c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(4*a^5*(a - b)^2*(a + b)^3*d) + ((8*a^4 - 61*a^2*b^2 + 35*b^4)*sin(c + d*x))/(12*a^3*(a^2 - b^2)^2*d*sqrt(sec(c + d*x))) + (b^2*sin(c + d*x))/(2*a*(a^2 - b^2)*d*sqrt(sec(c + d*x))*(a + b*sec(c + d*x))^2) + (b^2*(13*a^2 - 7*b^2)*sin(c + d*x))/(4*a^2*(a^2 - b^2)^2*d*sqrt(sec(c + d*x))*(a + b*sec(c + d*x)))]


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


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


    @test_int [sec(c + d*x)^(3/2)*(a + b*sec(c + d*x))^(1/2), x, 12, (b*sqrt((b + a*cos(c + d*x))/(a + b))*Elliptic.F((1/2)*(c + d*x), (2*a)/(a + b))*sqrt(sec(c + d*x)))/(d*sqrt(a + b*sec(c + d*x))) + (a*sqrt((b + a*cos(c + d*x))/(a + b))*Elliptic.Pi(2, (1/2)*(c + d*x), (2*a)/(a + b))*sqrt(sec(c + d*x)))/(d*sqrt(a + b*sec(c + d*x))) - (Elliptic.E((1/2)*(c + d*x), (2*a)/(a + b))*sqrt(a + b*sec(c + d*x)))/(d*sqrt((b + a*cos(c + d*x))/(a + b))*sqrt(sec(c + d*x))) + (sqrt(sec(c + d*x))*sqrt(a + b*sec(c + d*x))*sin(c + d*x))/d]
    @test_int [sec(c + d*x)^(1/2)*(a + b*sec(c + d*x))^(1/2), x, 7, (2*a*Elliptic.F((1/2)*(c + d*x), (2*a)/(a + b))*sqrt(sec(c + d*x))*sqrt((b + a*cos(c + d*x))/(a + b)))/(d*sqrt(a + b*sec(c + d*x))) + (2*b*Elliptic.Pi(2, (1/2)*(c + d*x), (2*a)/(a + b))*sqrt(sec(c + d*x))*sqrt((b + a*cos(c + d*x))/(a + b)))/(d*sqrt(a + b*sec(c + d*x)))]
    @test_int [sec(c + d*x)^(-1/2)*(a + b*sec(c + d*x))^(1/2), x, 3, (2*Elliptic.E((1/2)*(c + d*x), (2*a)/(a + b))*sqrt(a + b*sec(c + d*x)))/(d*sqrt((b + a*cos(c + d*x))/(a + b))*sqrt(sec(c + d*x)))]
    @test_int [sec(c + d*x)^(-3/2)*(a + b*sec(c + d*x))^(1/2), x, 8, (2*(a^2 - b^2)*sqrt((b + a*cos(c + d*x))/(a + b))*Elliptic.F((1/2)*(c + d*x), (2*a)/(a + b))*sqrt(sec(c + d*x)))/(3*a*d*sqrt(a + b*sec(c + d*x))) + (2*b*Elliptic.E((1/2)*(c + d*x), (2*a)/(a + b))*sqrt(a + b*sec(c + d*x)))/(3*a*d*sqrt((b + a*cos(c + d*x))/(a + b))*sqrt(sec(c + d*x))) + (2*sqrt(a + b*sec(c + d*x))*sin(c + d*x))/(3*d*sqrt(sec(c + d*x)))]
    @test_int [sec(c + d*x)^(-5/2)*(a + b*sec(c + d*x))^(1/2), x, 9, -((4*b*(a^2 - b^2)*sqrt((b + a*cos(c + d*x))/(a + b))*Elliptic.F((1/2)*(c + d*x), (2*a)/(a + b))*sqrt(sec(c + d*x)))/(15*a^2*d*sqrt(a + b*sec(c + d*x)))) + (2*(9*a^2 - 2*b^2)*Elliptic.E((1/2)*(c + d*x), (2*a)/(a + b))*sqrt(a + b*sec(c + d*x)))/(15*a^2*d*sqrt((b + a*cos(c + d*x))/(a + b))*sqrt(sec(c + d*x))) + (2*sqrt(a + b*sec(c + d*x))*sin(c + d*x))/(5*d*sec(c + d*x)^(3/2)) + (2*b*sqrt(a + b*sec(c + d*x))*sin(c + d*x))/(15*a*d*sqrt(sec(c + d*x)))]
    @test_int [sec(c + d*x)^(-7/2)*(a + b*sec(c + d*x))^(1/2), x, 10, (2*(25*a^4 - 17*a^2*b^2 - 8*b^4)*sqrt((b + a*cos(c + d*x))/(a + b))*Elliptic.F((1/2)*(c + d*x), (2*a)/(a + b))*sqrt(sec(c + d*x)))/(105*a^3*d*sqrt(a + b*sec(c + d*x))) + (2*b*(19*a^2 + 8*b^2)*Elliptic.E((1/2)*(c + d*x), (2*a)/(a + b))*sqrt(a + b*sec(c + d*x)))/(105*a^3*d*sqrt((b + a*cos(c + d*x))/(a + b))*sqrt(sec(c + d*x))) + (2*sqrt(a + b*sec(c + d*x))*sin(c + d*x))/(7*d*sec(c + d*x)^(5/2)) + (2*b*sqrt(a + b*sec(c + d*x))*sin(c + d*x))/(35*a*d*sec(c + d*x)^(3/2)) + (2*(25*a^2 - 4*b^2)*sqrt(a + b*sec(c + d*x))*sin(c + d*x))/(105*a^2*d*sqrt(sec(c + d*x)))]


    @test_int [sec(c + d*x)^(3/2)*(a + b*sec(c + d*x))^(3/2), x, 13, (7*a*b*sqrt((b + a*cos(c + d*x))/(a + b))*Elliptic.F((1/2)*(c + d*x), (2*a)/(a + b))*sqrt(sec(c + d*x)))/(4*d*sqrt(a + b*sec(c + d*x))) + ((3*a^2 + 4*b^2)*sqrt((b + a*cos(c + d*x))/(a + b))*Elliptic.Pi(2, (1/2)*(c + d*x), (2*a)/(a + b))*sqrt(sec(c + d*x)))/(4*d*sqrt(a + b*sec(c + d*x))) - (5*a*Elliptic.E((1/2)*(c + d*x), (2*a)/(a + b))*sqrt(a + b*sec(c + d*x)))/(4*d*sqrt((b + a*cos(c + d*x))/(a + b))*sqrt(sec(c + d*x))) + (5*a*sqrt(sec(c + d*x))*sqrt(a + b*sec(c + d*x))*sin(c + d*x))/(4*d) + (b*sec(c + d*x)^(3/2)*sqrt(a + b*sec(c + d*x))*sin(c + d*x))/(2*d)]
    @test_int [sec(c + d*x)^(1/2)*(a + b*sec(c + d*x))^(3/2), x, 12, ((2*a^2 + b^2)*sqrt((b + a*cos(c + d*x))/(a + b))*Elliptic.F((1/2)*(c + d*x), (2*a)/(a + b))*sqrt(sec(c + d*x)))/(d*sqrt(a + b*sec(c + d*x))) + (3*a*b*sqrt((b + a*cos(c + d*x))/(a + b))*Elliptic.Pi(2, (1/2)*(c + d*x), (2*a)/(a + b))*sqrt(sec(c + d*x)))/(d*sqrt(a + b*sec(c + d*x))) - (b*Elliptic.E((1/2)*(c + d*x), (2*a)/(a + b))*sqrt(a + b*sec(c + d*x)))/(d*sqrt((b + a*cos(c + d*x))/(a + b))*sqrt(sec(c + d*x))) + (b*sqrt(sec(c + d*x))*sqrt(a + b*sec(c + d*x))*sin(c + d*x))/d]
    @test_int [sec(c + d*x)^(-1/2)*(a + b*sec(c + d*x))^(3/2), x, 11, (2*a*b*sqrt((b + a*cos(c + d*x))/(a + b))*Elliptic.F((1/2)*(c + d*x), (2*a)/(a + b))*sqrt(sec(c + d*x)))/(d*sqrt(a + b*sec(c + d*x))) + (2*b^2*sqrt((b + a*cos(c + d*x))/(a + b))*Elliptic.Pi(2, (1/2)*(c + d*x), (2*a)/(a + b))*sqrt(sec(c + d*x)))/(d*sqrt(a + b*sec(c + d*x))) + (2*a*Elliptic.E((1/2)*(c + d*x), (2*a)/(a + b))*sqrt(a + b*sec(c + d*x)))/(d*sqrt((b + a*cos(c + d*x))/(a + b))*sqrt(sec(c + d*x)))]
    @test_int [sec(c + d*x)^(-3/2)*(a + b*sec(c + d*x))^(3/2), x, 8, (2*(a^2 - b^2)*sqrt((b + a*cos(c + d*x))/(a + b))*Elliptic.F((1/2)*(c + d*x), (2*a)/(a + b))*sqrt(sec(c + d*x)))/(3*d*sqrt(a + b*sec(c + d*x))) + (8*b*Elliptic.E((1/2)*(c + d*x), (2*a)/(a + b))*sqrt(a + b*sec(c + d*x)))/(3*d*sqrt((b + a*cos(c + d*x))/(a + b))*sqrt(sec(c + d*x))) + (2*a*sqrt(a + b*sec(c + d*x))*sin(c + d*x))/(3*d*sqrt(sec(c + d*x)))]
    @test_int [sec(c + d*x)^(-5/2)*(a + b*sec(c + d*x))^(3/2), x, 9, (2*b*(a^2 - b^2)*sqrt((b + a*cos(c + d*x))/(a + b))*Elliptic.F((1/2)*(c + d*x), (2*a)/(a + b))*sqrt(sec(c + d*x)))/(5*a*d*sqrt(a + b*sec(c + d*x))) + (2*(3*a^2 + b^2)*Elliptic.E((1/2)*(c + d*x), (2*a)/(a + b))*sqrt(a + b*sec(c + d*x)))/(5*a*d*sqrt((b + a*cos(c + d*x))/(a + b))*sqrt(sec(c + d*x))) + (2*a*sqrt(a + b*sec(c + d*x))*sin(c + d*x))/(5*d*sec(c + d*x)^(3/2)) + (4*b*sqrt(a + b*sec(c + d*x))*sin(c + d*x))/(5*d*sqrt(sec(c + d*x)))]
    @test_int [sec(c + d*x)^(-7/2)*(a + b*sec(c + d*x))^(3/2), x, 10, (2*(25*a^4 - 31*a^2*b^2 + 6*b^4)*sqrt((b + a*cos(c + d*x))/(a + b))*Elliptic.F((1/2)*(c + d*x), (2*a)/(a + b))*sqrt(sec(c + d*x)))/(105*a^2*d*sqrt(a + b*sec(c + d*x))) + (4*b*(41*a^2 - 3*b^2)*Elliptic.E((1/2)*(c + d*x), (2*a)/(a + b))*sqrt(a + b*sec(c + d*x)))/(105*a^2*d*sqrt((b + a*cos(c + d*x))/(a + b))*sqrt(sec(c + d*x))) + (2*a*sqrt(a + b*sec(c + d*x))*sin(c + d*x))/(7*d*sec(c + d*x)^(5/2)) + (16*b*sqrt(a + b*sec(c + d*x))*sin(c + d*x))/(35*d*sec(c + d*x)^(3/2)) + (2*(25*a^2 + 3*b^2)*sqrt(a + b*sec(c + d*x))*sin(c + d*x))/(105*a*d*sqrt(sec(c + d*x)))]


    @test_int [sec(c + d*x)^(3/2)*(a + b*sec(c + d*x))^(5/2), x, 14, (b*(59*a^2 + 16*b^2)*sqrt((b + a*cos(c + d*x))/(a + b))*Elliptic.F((1/2)*(c + d*x), (2*a)/(a + b))*sqrt(sec(c + d*x)))/(24*d*sqrt(a + b*sec(c + d*x))) + (5*a*(a^2 + 4*b^2)*sqrt((b + a*cos(c + d*x))/(a + b))*Elliptic.Pi(2, (1/2)*(c + d*x), (2*a)/(a + b))*sqrt(sec(c + d*x)))/(8*d*sqrt(a + b*sec(c + d*x))) - ((33*a^2 + 16*b^2)*Elliptic.E((1/2)*(c + d*x), (2*a)/(a + b))*sqrt(a + b*sec(c + d*x)))/(24*d*sqrt((b + a*cos(c + d*x))/(a + b))*sqrt(sec(c + d*x))) + ((33*a^2 + 16*b^2)*sqrt(sec(c + d*x))*sqrt(a + b*sec(c + d*x))*sin(c + d*x))/(24*d) + (13*a*b*sec(c + d*x)^(3/2)*sqrt(a + b*sec(c + d*x))*sin(c + d*x))/(12*d) + (b^2*sec(c + d*x)^(5/2)*sqrt(a + b*sec(c + d*x))*sin(c + d*x))/(3*d)]
    @test_int [sec(c + d*x)^(1/2)*(a + b*sec(c + d*x))^(5/2), x, 13, (a*(8*a^2 + 11*b^2)*sqrt((b + a*cos(c + d*x))/(a + b))*Elliptic.F((1/2)*(c + d*x), (2*a)/(a + b))*sqrt(sec(c + d*x)))/(4*d*sqrt(a + b*sec(c + d*x))) + (b*(15*a^2 + 4*b^2)*sqrt((b + a*cos(c + d*x))/(a + b))*Elliptic.Pi(2, (1/2)*(c + d*x), (2*a)/(a + b))*sqrt(sec(c + d*x)))/(4*d*sqrt(a + b*sec(c + d*x))) - (9*a*b*Elliptic.E((1/2)*(c + d*x), (2*a)/(a + b))*sqrt(a + b*sec(c + d*x)))/(4*d*sqrt((b + a*cos(c + d*x))/(a + b))*sqrt(sec(c + d*x))) + (9*a*b*sqrt(sec(c + d*x))*sqrt(a + b*sec(c + d*x))*sin(c + d*x))/(4*d) + (b^2*sec(c + d*x)^(3/2)*sqrt(a + b*sec(c + d*x))*sin(c + d*x))/(2*d)]
    @test_int [sec(c + d*x)^(-1/2)*(a + b*sec(c + d*x))^(5/2), x, 12, (b*(4*a^2 + b^2)*sqrt((b + a*cos(c + d*x))/(a + b))*Elliptic.F((1/2)*(c + d*x), (2*a)/(a + b))*sqrt(sec(c + d*x)))/(d*sqrt(a + b*sec(c + d*x))) + (5*a*b^2*sqrt((b + a*cos(c + d*x))/(a + b))*Elliptic.Pi(2, (1/2)*(c + d*x), (2*a)/(a + b))*sqrt(sec(c + d*x)))/(d*sqrt(a + b*sec(c + d*x))) + ((2*a^2 - b^2)*Elliptic.E((1/2)*(c + d*x), (2*a)/(a + b))*sqrt(a + b*sec(c + d*x)))/(d*sqrt((b + a*cos(c + d*x))/(a + b))*sqrt(sec(c + d*x))) + (b^2*sqrt(sec(c + d*x))*sqrt(a + b*sec(c + d*x))*sin(c + d*x))/d]
    @test_int [sec(c + d*x)^(-3/2)*(a + b*sec(c + d*x))^(5/2), x, 12, (2*a*(a^2 + 2*b^2)*sqrt((b + a*cos(c + d*x))/(a + b))*Elliptic.F((1/2)*(c + d*x), (2*a)/(a + b))*sqrt(sec(c + d*x)))/(3*d*sqrt(a + b*sec(c + d*x))) + (2*b^3*sqrt((b + a*cos(c + d*x))/(a + b))*Elliptic.Pi(2, (1/2)*(c + d*x), (2*a)/(a + b))*sqrt(sec(c + d*x)))/(d*sqrt(a + b*sec(c + d*x))) + (14*a*b*Elliptic.E((1/2)*(c + d*x), (2*a)/(a + b))*sqrt(a + b*sec(c + d*x)))/(3*d*sqrt((b + a*cos(c + d*x))/(a + b))*sqrt(sec(c + d*x))) + (2*a^2*sqrt(a + b*sec(c + d*x))*sin(c + d*x))/(3*d*sqrt(sec(c + d*x)))]
    @test_int [sec(c + d*x)^(-5/2)*(a + b*sec(c + d*x))^(5/2), x, 9, (16*b*(a^2 - b^2)*sqrt((b + a*cos(c + d*x))/(a + b))*Elliptic.F((1/2)*(c + d*x), (2*a)/(a + b))*sqrt(sec(c + d*x)))/(15*d*sqrt(a + b*sec(c + d*x))) + (2*(9*a^2 + 23*b^2)*Elliptic.E((1/2)*(c + d*x), (2*a)/(a + b))*sqrt(a + b*sec(c + d*x)))/(15*d*sqrt((b + a*cos(c + d*x))/(a + b))*sqrt(sec(c + d*x))) + (2*a^2*sqrt(a + b*sec(c + d*x))*sin(c + d*x))/(5*d*sec(c + d*x)^(3/2)) + (22*a*b*sqrt(a + b*sec(c + d*x))*sin(c + d*x))/(15*d*sqrt(sec(c + d*x)))]
    @test_int [sec(c + d*x)^(-7/2)*(a + b*sec(c + d*x))^(5/2), x, 10, (2*(5*a^4 - 2*a^2*b^2 - 3*b^4)*sqrt((b + a*cos(c + d*x))/(a + b))*Elliptic.F((1/2)*(c + d*x), (2*a)/(a + b))*sqrt(sec(c + d*x)))/(21*a*d*sqrt(a + b*sec(c + d*x))) + (2*b*(29*a^2 + 3*b^2)*Elliptic.E((1/2)*(c + d*x), (2*a)/(a + b))*sqrt(a + b*sec(c + d*x)))/(21*a*d*sqrt((b + a*cos(c + d*x))/(a + b))*sqrt(sec(c + d*x))) + (2*a^2*sqrt(a + b*sec(c + d*x))*sin(c + d*x))/(7*d*sec(c + d*x)^(5/2)) + (6*a*b*sqrt(a + b*sec(c + d*x))*sin(c + d*x))/(7*d*sec(c + d*x)^(3/2)) + (2*(5*a^2 + 9*b^2)*sqrt(a + b*sec(c + d*x))*sin(c + d*x))/(21*d*sqrt(sec(c + d*x)))]
    @test_int [sec(c + d*x)^(-9/2)*(a + b*sec(c + d*x))^(5/2), x, 11, (4*b*(57*a^4 - 62*a^2*b^2 + 5*b^4)*sqrt((b + a*cos(c + d*x))/(a + b))*Elliptic.F((1/2)*(c + d*x), (2*a)/(a + b))*sqrt(sec(c + d*x)))/(315*a^2*d*sqrt(a + b*sec(c + d*x))) + (2*(147*a^4 + 279*a^2*b^2 - 10*b^4)*Elliptic.E((1/2)*(c + d*x), (2*a)/(a + b))*sqrt(a + b*sec(c + d*x)))/(315*a^2*d*sqrt((b + a*cos(c + d*x))/(a + b))*sqrt(sec(c + d*x))) + (2*a^2*sqrt(a + b*sec(c + d*x))*sin(c + d*x))/(9*d*sec(c + d*x)^(7/2)) + (38*a*b*sqrt(a + b*sec(c + d*x))*sin(c + d*x))/(63*d*sec(c + d*x)^(5/2)) + (2*(49*a^2 + 75*b^2)*sqrt(a + b*sec(c + d*x))*sin(c + d*x))/(315*d*sec(c + d*x)^(3/2)) + (2*b*(163*a^2 + 5*b^2)*sqrt(a + b*sec(c + d*x))*sin(c + d*x))/(315*a*d*sqrt(sec(c + d*x)))]


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


    @test_int [sec(c + d*x)^(7/2)/(a + b*sec(c + d*x))^(1/2), x, 13, -((a*sqrt((b + a*cos(c + d*x))/(a + b))*Elliptic.F((1/2)*(c + d*x), (2*a)/(a + b))*sqrt(sec(c + d*x)))/(4*b*d*sqrt(a + b*sec(c + d*x)))) + ((3*a^2 + 4*b^2)*sqrt((b + a*cos(c + d*x))/(a + b))*Elliptic.Pi(2, (1/2)*(c + d*x), (2*a)/(a + b))*sqrt(sec(c + d*x)))/(4*b^2*d*sqrt(a + b*sec(c + d*x))) + (3*a*Elliptic.E((1/2)*(c + d*x), (2*a)/(a + b))*sqrt(a + b*sec(c + d*x)))/(4*b^2*d*sqrt((b + a*cos(c + d*x))/(a + b))*sqrt(sec(c + d*x))) - (3*a*sqrt(sec(c + d*x))*sqrt(a + b*sec(c + d*x))*sin(c + d*x))/(4*b^2*d) + (sec(c + d*x)^(3/2)*sqrt(a + b*sec(c + d*x))*sin(c + d*x))/(2*b*d)]
    @test_int [sec(c + d*x)^(5/2)/(a + b*sec(c + d*x))^(1/2), x, 12, (sqrt((b + a*cos(c + d*x))/(a + b))*Elliptic.F((1/2)*(c + d*x), (2*a)/(a + b))*sqrt(sec(c + d*x)))/(d*sqrt(a + b*sec(c + d*x))) - (a*sqrt((b + a*cos(c + d*x))/(a + b))*Elliptic.Pi(2, (1/2)*(c + d*x), (2*a)/(a + b))*sqrt(sec(c + d*x)))/(b*d*sqrt(a + b*sec(c + d*x))) - (Elliptic.E((1/2)*(c + d*x), (2*a)/(a + b))*sqrt(a + b*sec(c + d*x)))/(b*d*sqrt((b + a*cos(c + d*x))/(a + b))*sqrt(sec(c + d*x))) + (sqrt(sec(c + d*x))*sqrt(a + b*sec(c + d*x))*sin(c + d*x))/(b*d)]
    @test_int [sec(c + d*x)^(3/2)/(a + b*sec(c + d*x))^(1/2), x, 3, (2*Elliptic.Pi(2, (1/2)*(c + d*x), (2*a)/(a + b))*sqrt(sec(c + d*x))*sqrt((b + a*cos(c + d*x))/(a + b)))/(d*sqrt(a + b*sec(c + d*x)))]
    @test_int [sec(c + d*x)^(1/2)/(a + b*sec(c + d*x))^(1/2), x, 3, (2*Elliptic.F((1/2)*(c + d*x), (2*a)/(a + b))*sqrt(sec(c + d*x))*sqrt((b + a*cos(c + d*x))/(a + b)))/(d*sqrt(a + b*sec(c + d*x)))]
    @test_int [sec(c + d*x)^(-1/2)/(a + b*sec(c + d*x))^(1/2), x, 7, -((2*b*sqrt((b + a*cos(c + d*x))/(a + b))*Elliptic.F((1/2)*(c + d*x), (2*a)/(a + b))*sqrt(sec(c + d*x)))/(a*d*sqrt(a + b*sec(c + d*x)))) + (2*Elliptic.E((1/2)*(c + d*x), (2*a)/(a + b))*sqrt(a + b*sec(c + d*x)))/(a*d*sqrt((b + a*cos(c + d*x))/(a + b))*sqrt(sec(c + d*x)))]
    @test_int [sec(c + d*x)^(-3/2)/(a + b*sec(c + d*x))^(1/2), x, 8, (2*(a^2 + 2*b^2)*sqrt((b + a*cos(c + d*x))/(a + b))*Elliptic.F((1/2)*(c + d*x), (2*a)/(a + b))*sqrt(sec(c + d*x)))/(3*a^2*d*sqrt(a + b*sec(c + d*x))) - (4*b*Elliptic.E((1/2)*(c + d*x), (2*a)/(a + b))*sqrt(a + b*sec(c + d*x)))/(3*a^2*d*sqrt((b + a*cos(c + d*x))/(a + b))*sqrt(sec(c + d*x))) + (2*sqrt(a + b*sec(c + d*x))*sin(c + d*x))/(3*a*d*sqrt(sec(c + d*x)))]
    @test_int [sec(c + d*x)^(-5/2)/(a + b*sec(c + d*x))^(1/2), x, 9, -((2*b*(7*a^2 + 8*b^2)*sqrt((b + a*cos(c + d*x))/(a + b))*Elliptic.F((1/2)*(c + d*x), (2*a)/(a + b))*sqrt(sec(c + d*x)))/(15*a^3*d*sqrt(a + b*sec(c + d*x)))) + (2*(9*a^2 + 8*b^2)*Elliptic.E((1/2)*(c + d*x), (2*a)/(a + b))*sqrt(a + b*sec(c + d*x)))/(15*a^3*d*sqrt((b + a*cos(c + d*x))/(a + b))*sqrt(sec(c + d*x))) + (2*sqrt(a + b*sec(c + d*x))*sin(c + d*x))/(5*a*d*sec(c + d*x)^(3/2)) - (8*b*sqrt(a + b*sec(c + d*x))*sin(c + d*x))/(15*a^2*d*sqrt(sec(c + d*x)))]


    @test_int [sec(c + d*x)^(7/2)/(a + b*sec(c + d*x))^(3/2), x, 13, (sqrt((b + a*cos(c + d*x))/(a + b))*Elliptic.F((1/2)*(c + d*x), (2*a)/(a + b))*sqrt(sec(c + d*x)))/(b*d*sqrt(a + b*sec(c + d*x))) - (3*a*sqrt((b + a*cos(c + d*x))/(a + b))*Elliptic.Pi(2, (1/2)*(c + d*x), (2*a)/(a + b))*sqrt(sec(c + d*x)))/(b^2*d*sqrt(a + b*sec(c + d*x))) - ((3*a^2 - b^2)*Elliptic.E((1/2)*(c + d*x), (2*a)/(a + b))*sqrt(a + b*sec(c + d*x)))/(b^2*(a^2 - b^2)*d*sqrt((b + a*cos(c + d*x))/(a + b))*sqrt(sec(c + d*x))) - (2*a^2*sec(c + d*x)^(3/2)*sin(c + d*x))/(b*(a^2 - b^2)*d*sqrt(a + b*sec(c + d*x))) + ((3*a^2 - b^2)*sqrt(sec(c + d*x))*sqrt(a + b*sec(c + d*x))*sin(c + d*x))/(b^2*(a^2 - b^2)*d)]
    @test_int [sec(c + d*x)^(5/2)/(a + b*sec(c + d*x))^(3/2), x, 9, (2*sqrt((b + a*cos(c + d*x))/(a + b))*Elliptic.Pi(2, (1/2)*(c + d*x), (2*a)/(a + b))*sqrt(sec(c + d*x)))/(b*d*sqrt(a + b*sec(c + d*x))) + (2*a*Elliptic.E((1/2)*(c + d*x), (2*a)/(a + b))*sqrt(a + b*sec(c + d*x)))/(b*(a^2 - b^2)*d*sqrt((b + a*cos(c + d*x))/(a + b))*sqrt(sec(c + d*x))) - (2*a^2*sqrt(sec(c + d*x))*sin(c + d*x))/(b*(a^2 - b^2)*d*sqrt(a + b*sec(c + d*x)))]
    @test_int [sec(c + d*x)^(3/2)/(a + b*sec(c + d*x))^(3/2), x, 5, -((2*Elliptic.E((1/2)*(c + d*x), (2*a)/(a + b))*sqrt(a + b*sec(c + d*x)))/((a^2 - b^2)*d*sqrt((b + a*cos(c + d*x))/(a + b))*sqrt(sec(c + d*x)))) + (2*a*sqrt(sec(c + d*x))*sin(c + d*x))/((a^2 - b^2)*d*sqrt(a + b*sec(c + d*x)))]
    @test_int [sec(c + d*x)^(1/2)/(a + b*sec(c + d*x))^(3/2), x, 8, (2*sqrt((b + a*cos(c + d*x))/(a + b))*Elliptic.F((1/2)*(c + d*x), (2*a)/(a + b))*sqrt(sec(c + d*x)))/(a*d*sqrt(a + b*sec(c + d*x))) + (2*b*Elliptic.E((1/2)*(c + d*x), (2*a)/(a + b))*sqrt(a + b*sec(c + d*x)))/(a*(a^2 - b^2)*d*sqrt((b + a*cos(c + d*x))/(a + b))*sqrt(sec(c + d*x))) - (2*b*sqrt(sec(c + d*x))*sin(c + d*x))/((a^2 - b^2)*d*sqrt(a + b*sec(c + d*x)))]
    @test_int [sec(c + d*x)^(-1/2)/(a + b*sec(c + d*x))^(3/2), x, 8, -((4*b*sqrt((b + a*cos(c + d*x))/(a + b))*Elliptic.F((1/2)*(c + d*x), (2*a)/(a + b))*sqrt(sec(c + d*x)))/(a^2*d*sqrt(a + b*sec(c + d*x)))) + (2*(a^2 - 2*b^2)*Elliptic.E((1/2)*(c + d*x), (2*a)/(a + b))*sqrt(a + b*sec(c + d*x)))/(a^2*(a^2 - b^2)*d*sqrt((b + a*cos(c + d*x))/(a + b))*sqrt(sec(c + d*x))) + (2*b^2*sqrt(sec(c + d*x))*sin(c + d*x))/(a*(a^2 - b^2)*d*sqrt(a + b*sec(c + d*x)))]
    @test_int [sec(c + d*x)^(-3/2)/(a + b*sec(c + d*x))^(3/2), x, 9, (2*(a^2 + 8*b^2)*sqrt((b + a*cos(c + d*x))/(a + b))*Elliptic.F((1/2)*(c + d*x), (2*a)/(a + b))*sqrt(sec(c + d*x)))/(3*a^3*d*sqrt(a + b*sec(c + d*x))) - (2*b*(5*a^2 - 8*b^2)*Elliptic.E((1/2)*(c + d*x), (2*a)/(a + b))*sqrt(a + b*sec(c + d*x)))/(3*a^3*(a^2 - b^2)*d*sqrt((b + a*cos(c + d*x))/(a + b))*sqrt(sec(c + d*x))) + (2*b^2*sin(c + d*x))/(a*(a^2 - b^2)*d*sqrt(sec(c + d*x))*sqrt(a + b*sec(c + d*x))) + (2*(a^2 - 4*b^2)*sqrt(a + b*sec(c + d*x))*sin(c + d*x))/(3*a^2*(a^2 - b^2)*d*sqrt(sec(c + d*x)))]
    @test_int [sec(c + d*x)^(-5/2)/(a + b*sec(c + d*x))^(3/2), x, 10, -((8*b*(a^2 + 4*b^2)*sqrt((b + a*cos(c + d*x))/(a + b))*Elliptic.F((1/2)*(c + d*x), (2*a)/(a + b))*sqrt(sec(c + d*x)))/(5*a^4*d*sqrt(a + b*sec(c + d*x)))) + (2*(3*a^4 + 8*a^2*b^2 - 16*b^4)*Elliptic.E((1/2)*(c + d*x), (2*a)/(a + b))*sqrt(a + b*sec(c + d*x)))/(5*a^4*(a^2 - b^2)*d*sqrt((b + a*cos(c + d*x))/(a + b))*sqrt(sec(c + d*x))) + (2*b^2*sin(c + d*x))/(a*(a^2 - b^2)*d*sec(c + d*x)^(3/2)*sqrt(a + b*sec(c + d*x))) + (2*(a^2 - 6*b^2)*sqrt(a + b*sec(c + d*x))*sin(c + d*x))/(5*a^2*(a^2 - b^2)*d*sec(c + d*x)^(3/2)) - (2*b*(3*a^2 - 8*b^2)*sqrt(a + b*sec(c + d*x))*sin(c + d*x))/(5*a^3*(a^2 - b^2)*d*sqrt(sec(c + d*x)))]


    @test_int [sec(c + d*x)^(9/2)/(a + b*sec(c + d*x))^(5/2), x, 14, ((5*a^2 - 3*b^2)*sqrt((b + a*cos(c + d*x))/(a + b))*Elliptic.F((1/2)*(c + d*x), (2*a)/(a + b))*sqrt(sec(c + d*x)))/(3*b^2*(a^2 - b^2)*d*sqrt(a + b*sec(c + d*x))) - (5*a*sqrt((b + a*cos(c + d*x))/(a + b))*Elliptic.Pi(2, (1/2)*(c + d*x), (2*a)/(a + b))*sqrt(sec(c + d*x)))/(b^3*d*sqrt(a + b*sec(c + d*x))) - ((15*a^4 - 26*a^2*b^2 + 3*b^4)*Elliptic.E((1/2)*(c + d*x), (2*a)/(a + b))*sqrt(a + b*sec(c + d*x)))/(3*b^3*(a^2 - b^2)^2*d*sqrt((b + a*cos(c + d*x))/(a + b))*sqrt(sec(c + d*x))) - (2*a^2*sec(c + d*x)^(5/2)*sin(c + d*x))/(3*b*(a^2 - b^2)*d*(a + b*sec(c + d*x))^(3/2)) - (2*a^2*(5*a^2 - 9*b^2)*sec(c + d*x)^(3/2)*sin(c + d*x))/(3*b^2*(a^2 - b^2)^2*d*sqrt(a + b*sec(c + d*x))) + ((15*a^4 - 26*a^2*b^2 + 3*b^4)*sqrt(sec(c + d*x))*sqrt(a + b*sec(c + d*x))*sin(c + d*x))/(3*b^3*(a^2 - b^2)^2*d)]
    @test_int [sec(c + d*x)^(7/2)/(a + b*sec(c + d*x))^(5/2), x, 13, -((2*a*sqrt((b + a*cos(c + d*x))/(a + b))*Elliptic.F((1/2)*(c + d*x), (2*a)/(a + b))*sqrt(sec(c + d*x)))/(3*b*(a^2 - b^2)*d*sqrt(a + b*sec(c + d*x)))) + (2*sqrt((b + a*cos(c + d*x))/(a + b))*Elliptic.Pi(2, (1/2)*(c + d*x), (2*a)/(a + b))*sqrt(sec(c + d*x)))/(b^2*d*sqrt(a + b*sec(c + d*x))) + (2*a*(3*a^2 - 7*b^2)*Elliptic.E((1/2)*(c + d*x), (2*a)/(a + b))*sqrt(a + b*sec(c + d*x)))/(3*b^2*(a^2 - b^2)^2*d*sqrt((b + a*cos(c + d*x))/(a + b))*sqrt(sec(c + d*x))) - (2*a^2*sec(c + d*x)^(3/2)*sin(c + d*x))/(3*b*(a^2 - b^2)*d*(a + b*sec(c + d*x))^(3/2)) - (2*a^2*(3*a^2 - 7*b^2)*sqrt(sec(c + d*x))*sin(c + d*x))/(3*b^2*(a^2 - b^2)^2*d*sqrt(a + b*sec(c + d*x)))]
    @test_int [sec(c + d*x)^(5/2)/(a + b*sec(c + d*x))^(5/2), x, 9, (2*sqrt((b + a*cos(c + d*x))/(a + b))*Elliptic.F((1/2)*(c + d*x), (2*a)/(a + b))*sqrt(sec(c + d*x)))/(3*(a^2 - b^2)*d*sqrt(a + b*sec(c + d*x))) + (8*b*Elliptic.E((1/2)*(c + d*x), (2*a)/(a + b))*sqrt(a + b*sec(c + d*x)))/(3*(a^2 - b^2)^2*d*sqrt((b + a*cos(c + d*x))/(a + b))*sqrt(sec(c + d*x))) - (2*a^2*sqrt(sec(c + d*x))*sin(c + d*x))/(3*b*(a^2 - b^2)*d*(a + b*sec(c + d*x))^(3/2)) + (2*a*(a^2 - 5*b^2)*sqrt(sec(c + d*x))*sin(c + d*x))/(3*b*(a^2 - b^2)^2*d*sqrt(a + b*sec(c + d*x)))]
    @test_int [sec(c + d*x)^(3/2)/(a + b*sec(c + d*x))^(5/2), x, 9, -((2*b*sqrt((b + a*cos(c + d*x))/(a + b))*Elliptic.F((1/2)*(c + d*x), (2*a)/(a + b))*sqrt(sec(c + d*x)))/(3*a*(a^2 - b^2)*d*sqrt(a + b*sec(c + d*x)))) - (2*(3*a^2 + b^2)*Elliptic.E((1/2)*(c + d*x), (2*a)/(a + b))*sqrt(a + b*sec(c + d*x)))/(3*a*(a^2 - b^2)^2*d*sqrt((b + a*cos(c + d*x))/(a + b))*sqrt(sec(c + d*x))) + (2*a*sqrt(sec(c + d*x))*sin(c + d*x))/(3*(a^2 - b^2)*d*(a + b*sec(c + d*x))^(3/2)) + (4*(a^2 + b^2)*sqrt(sec(c + d*x))*sin(c + d*x))/(3*(a^2 - b^2)^2*d*sqrt(a + b*sec(c + d*x)))]
    @test_int [sec(c + d*x)^(1/2)/(a + b*sec(c + d*x))^(5/2), x, 9, (2*(3*a^2 - 2*b^2)*sqrt((b + a*cos(c + d*x))/(a + b))*Elliptic.F((1/2)*(c + d*x), (2*a)/(a + b))*sqrt(sec(c + d*x)))/(3*a^2*(a^2 - b^2)*d*sqrt(a + b*sec(c + d*x))) + (4*b*(3*a^2 - b^2)*Elliptic.E((1/2)*(c + d*x), (2*a)/(a + b))*sqrt(a + b*sec(c + d*x)))/(3*a^2*(a^2 - b^2)^2*d*sqrt((b + a*cos(c + d*x))/(a + b))*sqrt(sec(c + d*x))) - (2*b*sqrt(sec(c + d*x))*sin(c + d*x))/(3*(a^2 - b^2)*d*(a + b*sec(c + d*x))^(3/2)) - (2*b*(5*a^2 - b^2)*sqrt(sec(c + d*x))*sin(c + d*x))/(3*a*(a^2 - b^2)^2*d*sqrt(a + b*sec(c + d*x)))]
    @test_int [sec(c + d*x)^(-1/2)/(a + b*sec(c + d*x))^(5/2), x, 9, -((2*b*(9*a^2 - 8*b^2)*sqrt((b + a*cos(c + d*x))/(a + b))*Elliptic.F((1/2)*(c + d*x), (2*a)/(a + b))*sqrt(sec(c + d*x)))/(3*a^3*(a^2 - b^2)*d*sqrt(a + b*sec(c + d*x)))) + (2*(3*a^4 - 15*a^2*b^2 + 8*b^4)*Elliptic.E((1/2)*(c + d*x), (2*a)/(a + b))*sqrt(a + b*sec(c + d*x)))/(3*a^3*(a^2 - b^2)^2*d*sqrt((b + a*cos(c + d*x))/(a + b))*sqrt(sec(c + d*x))) + (2*b^2*sqrt(sec(c + d*x))*sin(c + d*x))/(3*a*(a^2 - b^2)*d*(a + b*sec(c + d*x))^(3/2)) + (8*b^2*(2*a^2 - b^2)*sqrt(sec(c + d*x))*sin(c + d*x))/(3*a^2*(a^2 - b^2)^2*d*sqrt(a + b*sec(c + d*x)))]
    @test_int [sec(c + d*x)^(-3/2)/(a + b*sec(c + d*x))^(5/2), x, 10, (2*(a^4 + 16*a^2*b^2 - 16*b^4)*sqrt((b + a*cos(c + d*x))/(a + b))*Elliptic.F((1/2)*(c + d*x), (2*a)/(a + b))*sqrt(sec(c + d*x)))/(3*a^4*(a^2 - b^2)*d*sqrt(a + b*sec(c + d*x))) - (8*b*(2*a^4 - 7*a^2*b^2 + 4*b^4)*Elliptic.E((1/2)*(c + d*x), (2*a)/(a + b))*sqrt(a + b*sec(c + d*x)))/(3*a^4*(a^2 - b^2)^2*d*sqrt((b + a*cos(c + d*x))/(a + b))*sqrt(sec(c + d*x))) + (2*b^2*sin(c + d*x))/(3*a*(a^2 - b^2)*d*sqrt(sec(c + d*x))*(a + b*sec(c + d*x))^(3/2)) + (4*b^2*(5*a^2 - 3*b^2)*sin(c + d*x))/(3*a^2*(a^2 - b^2)^2*d*sqrt(sec(c + d*x))*sqrt(a + b*sec(c + d*x))) + (2*(a^4 - 13*a^2*b^2 + 8*b^4)*sqrt(a + b*sec(c + d*x))*sin(c + d*x))/(3*a^3*(a^2 - b^2)^2*d*sqrt(sec(c + d*x)))]
    @test_int [sec(c + d*x)^(-5/2)/(a + b*sec(c + d*x))^(5/2), x, 11, -((2*b*(17*a^4 + 116*a^2*b^2 - 128*b^4)*sqrt((b + a*cos(c + d*x))/(a + b))*Elliptic.F((1/2)*(c + d*x), (2*a)/(a + b))*sqrt(sec(c + d*x)))/(15*a^5*(a^2 - b^2)*d*sqrt(a + b*sec(c + d*x)))) + (2*(9*a^6 + 55*a^4*b^2 - 212*a^2*b^4 + 128*b^6)*Elliptic.E((1/2)*(c + d*x), (2*a)/(a + b))*sqrt(a + b*sec(c + d*x)))/(15*a^5*(a^2 - b^2)^2*d*sqrt((b + a*cos(c + d*x))/(a + b))*sqrt(sec(c + d*x))) + (2*b^2*sin(c + d*x))/(3*a*(a^2 - b^2)*d*sec(c + d*x)^(3/2)*(a + b*sec(c + d*x))^(3/2)) + (8*b^2*(3*a^2 - 2*b^2)*sin(c + d*x))/(3*a^2*(a^2 - b^2)^2*d*sec(c + d*x)^(3/2)*sqrt(a + b*sec(c + d*x))) + (2*(3*a^4 - 71*a^2*b^2 + 48*b^4)*sqrt(a + b*sec(c + d*x))*sin(c + d*x))/(15*a^3*(a^2 - b^2)^2*d*sec(c + d*x)^(3/2)) - (4*b*(7*a^4 - 49*a^2*b^2 + 32*b^4)*sqrt(a + b*sec(c + d*x))*sin(c + d*x))/(15*a^4*(a^2 - b^2)^2*d*sqrt(sec(c + d*x)))]


    @test_int [1/(sqrt(sec(c + d*x))*sqrt(2 + 3*sec(c + d*x))), x, 5, -((3*sqrt(3 + 2*cos(c + d*x))*Elliptic.F((1/2)*(c + d*x), 4/5)*sqrt(sec(c + d*x)))/(sqrt(5)*d*sqrt(2 + 3*sec(c + d*x)))) + (sqrt(5)*Elliptic.E((1/2)*(c + d*x), 4/5)*sqrt(2 + 3*sec(c + d*x)))/(d*sqrt(3 + 2*cos(c + d*x))*sqrt(sec(c + d*x)))]
    @test_int [1/(sqrt(sec(c + d*x))*sqrt(-2 + 3*sec(c + d*x))), x, 5, (3*sqrt(3 - 2*cos(c + d*x))*Elliptic.F((1/2)*(c + d*x), -4)*sqrt(sec(c + d*x)))/(d*sqrt(-2 + 3*sec(c + d*x))) - (Elliptic.E((1/2)*(c + d*x), -4)*sqrt(-2 + 3*sec(c + d*x)))/(d*sqrt(3 - 2*cos(c + d*x))*sqrt(sec(c + d*x)))]

    @test_int [1/(sqrt(sec(c + d*x))*sqrt(2 - 3*sec(c + d*x))), x, 7, (Elliptic.E((1/2)*(c + d*x), -4)*sqrt(2 - 3*sec(c + d*x)))/(d*sqrt(3 - 2*cos(c + d*x))*sqrt(sec(c + d*x))) + (3*sqrt(3 - 2*cos(c + d*x))*Elliptic.F((1/2)*(c + d*x), -4)*sqrt(sec(c + d*x)))/(d*sqrt(2 - 3*sec(c + d*x)))]
    @test_int [1/(sqrt(sec(c + d*x))*sqrt(-2 - 3*sec(c + d*x))), x, 7, -((sqrt(5)*Elliptic.E((1/2)*(c + d*x), 4/5)*sqrt(-2 - 3*sec(c + d*x)))/(d*sqrt(3 + 2*cos(c + d*x))*sqrt(sec(c + d*x)))) - (3*sqrt(3 + 2*cos(c + d*x))*Elliptic.F((1/2)*(c + d*x), 4/5)*sqrt(sec(c + d*x)))/(sqrt(5)*d*sqrt(-2 - 3*sec(c + d*x)))]

    @test_int [1/(sqrt(sec(c + d*x))*sqrt(3 + 2*sec(c + d*x))), x, 5, -((4*sqrt(2 + 3*cos(c + d*x))*Elliptic.F((1/2)*(c + d*x), 6/5)*sqrt(sec(c + d*x)))/(3*sqrt(5)*d*sqrt(3 + 2*sec(c + d*x)))) + (2*sqrt(5)*Elliptic.E((1/2)*(c + d*x), 6/5)*sqrt(3 + 2*sec(c + d*x)))/(3*d*sqrt(2 + 3*cos(c + d*x))*sqrt(sec(c + d*x)))]
    @test_int [1/(sqrt(sec(c + d*x))*sqrt(3 - 2*sec(c + d*x))), x, 5, (2*Elliptic.E((1/2)*(c + d*x), 6)*sqrt(3 - 2*sec(c + d*x)))/(3*d*sqrt(-2 + 3*cos(c + d*x))*sqrt(sec(c + d*x))) + (4*sqrt(-2 + 3*cos(c + d*x))*Elliptic.F((1/2)*(c + d*x), 6)*sqrt(sec(c + d*x)))/(3*d*sqrt(3 - 2*sec(c + d*x)))]

    @test_int [1/(sqrt(sec(c + d*x))*sqrt(-3 + 2*sec(c + d*x))), x, 5, (4*sqrt(2 - 3*cos(c + d*x))*Elliptic.F((1/2)*(c + pi + d*x), 6/5)*sqrt(sec(c + d*x)))/(3*sqrt(5)*d*sqrt(-3 + 2*sec(c + d*x))) - (2*sqrt(5)*Elliptic.E((1/2)*(c + pi + d*x), 6/5)*sqrt(-3 + 2*sec(c + d*x)))/(3*d*sqrt(2 - 3*cos(c + d*x))*sqrt(sec(c + d*x)))]
    @test_int [1/(sqrt(sec(c + d*x))*sqrt(-3 - 2*sec(c + d*x))), x, 5, -((2*Elliptic.E((1/2)*(c + pi + d*x), 6)*sqrt(-3 - 2*sec(c + d*x)))/(3*d*sqrt(-2 - 3*cos(c + d*x))*sqrt(sec(c + d*x)))) - (4*sqrt(-2 - 3*cos(c + d*x))*Elliptic.F((1/2)*(c + pi + d*x), 6)*sqrt(sec(c + d*x)))/(3*d*sqrt(-3 - 2*sec(c + d*x)))]


    @test_int [sqrt(sec(c + d*x))/sqrt(2 + 3*sec(c + d*x)), x, 2, (2*sqrt(3 + 2*cos(c + d*x))*Elliptic.F((1/2)*(c + d*x), 4/5)*sqrt(sec(c + d*x)))/(sqrt(5)*d*sqrt(2 + 3*sec(c + d*x)))]
    @test_int [sqrt(sec(c + d*x))/sqrt(-2 + 3*sec(c + d*x)), x, 2, (2*sqrt(3 - 2*cos(c + d*x))*Elliptic.F((1/2)*(c + d*x), -4)*sqrt(sec(c + d*x)))/(d*sqrt(-2 + 3*sec(c + d*x)))]

    @test_int [sqrt(sec(c + d*x))/sqrt(2 - 3*sec(c + d*x)), x, 3, (2*sqrt(3 - 2*cos(c + d*x))*Elliptic.F((1/2)*(c + d*x), -4)*sqrt(sec(c + d*x)))/(d*sqrt(2 - 3*sec(c + d*x)))]
    @test_int [sqrt(sec(c + d*x))/sqrt(-2 - 3*sec(c + d*x)), x, 3, (2*sqrt(3 + 2*cos(c + d*x))*Elliptic.F((1/2)*(c + d*x), 4/5)*sqrt(sec(c + d*x)))/(sqrt(5)*d*sqrt(-2 - 3*sec(c + d*x)))]

    @test_int [sqrt(sec(c + d*x))/sqrt(3 + 2*sec(c + d*x)), x, 2, (2*sqrt(2 + 3*cos(c + d*x))*Elliptic.F((1/2)*(c + d*x), 6/5)*sqrt(sec(c + d*x)))/(sqrt(5)*d*sqrt(3 + 2*sec(c + d*x)))]
    @test_int [sqrt(sec(c + d*x))/sqrt(3 - 2*sec(c + d*x)), x, 2, (2*sqrt(-2 + 3*cos(c + d*x))*Elliptic.F((1/2)*(c + d*x), 6)*sqrt(sec(c + d*x)))/(d*sqrt(3 - 2*sec(c + d*x)))]

    @test_int [sqrt(sec(c + d*x))/sqrt(-3 + 2*sec(c + d*x)), x, 2, (2*sqrt(2 - 3*cos(c + d*x))*Elliptic.F((1/2)*(c + pi + d*x), 6/5)*sqrt(sec(c + d*x)))/(sqrt(5)*d*sqrt(-3 + 2*sec(c + d*x)))]
    @test_int [sqrt(sec(c + d*x))/sqrt(-3 - 2*sec(c + d*x)), x, 2, (2*sqrt(-2 - 3*cos(c + d*x))*Elliptic.F((1/2)*(c + pi + d*x), 6)*sqrt(sec(c + d*x)))/(d*sqrt(-3 - 2*sec(c + d*x)))]


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


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


    @test_int [sec(c + d*x)^1*(a + b*sec(c + d*x))^(1/3), x, 3, (sqrt(2)*AppellF1(1/2, 1/2, -(1/3), 3/2, (1/2)*(1 - sec(c + d*x)), (b*(1 - sec(c + d*x)))/(a + b))*(a + b*sec(c + d*x))^(1/3)*tan(c + d*x))/(d*sqrt(1 + sec(c + d*x))*((a + b*sec(c + d*x))/(a + b))^(1/3))]
    @test_int [sec(c + d*x)^0*(a + b*sec(c + d*x))^(1/3), x, 0, Unintegrable((a + b*sec(c + d*x))^(1/3), x)]


    @test_int [sec(c + d*x)^4*(a + b*sec(c + d*x))^(2/3), x, 10, (3*(9*a^2 + 32*b^2)*(a + b*sec(c + d*x))^(2/3)*tan(c + d*x))/(220*b^2*d) - (9*a*(a + b*sec(c + d*x))^(5/3)*tan(c + d*x))/(44*b^2*d) + (3*sec(c + d*x)*(a + b*sec(c + d*x))^(5/3)*tan(c + d*x))/(11*b*d) + (a*(18*a^2 + 49*b^2)*AppellF1(1/2, 1/2, -(2/3), 3/2, (1/2)*(1 - sec(c + d*x)), (b*(1 - sec(c + d*x)))/(a + b))*(a + b*sec(c + d*x))^(2/3)*tan(c + d*x))/(110*sqrt(2)*b^3*d*sqrt(1 + sec(c + d*x))*((a + b*sec(c + d*x))/(a + b))^(2/3)) - ((9*a^4 + 23*a^2*b^2 - 32*b^4)*AppellF1(1/2, 1/2, 1/3, 3/2, (1/2)*(1 - sec(c + d*x)), (b*(1 - sec(c + d*x)))/(a + b))*((a + b*sec(c + d*x))/(a + b))^(1/3)*tan(c + d*x))/(55*sqrt(2)*b^3*d*sqrt(1 + sec(c + d*x))*(a + b*sec(c + d*x))^(1/3))]
    @test_int [sec(c + d*x)^3*(a + b*sec(c + d*x))^(2/3), x, 9, -((9*a*(a + b*sec(c + d*x))^(2/3)*tan(c + d*x))/(40*b*d)) + (3*(a + b*sec(c + d*x))^(5/3)*tan(c + d*x))/(8*b*d) - ((6*a^2 - 25*b^2)*AppellF1(1/2, 1/2, -(2/3), 3/2, (1/2)*(1 - sec(c + d*x)), (b*(1 - sec(c + d*x)))/(a + b))*(a + b*sec(c + d*x))^(2/3)*tan(c + d*x))/(20*sqrt(2)*b^2*d*sqrt(1 + sec(c + d*x))*((a + b*sec(c + d*x))/(a + b))^(2/3)) + (3*a*(a^2 - b^2)*AppellF1(1/2, 1/2, 1/3, 3/2, (1/2)*(1 - sec(c + d*x)), (b*(1 - sec(c + d*x)))/(a + b))*((a + b*sec(c + d*x))/(a + b))^(1/3)*tan(c + d*x))/(10*sqrt(2)*b^2*d*sqrt(1 + sec(c + d*x))*(a + b*sec(c + d*x))^(1/3))]
    @test_int [sec(c + d*x)^2*(a + b*sec(c + d*x))^(2/3), x, 8, (3*(a + b*sec(c + d*x))^(2/3)*tan(c + d*x))/(5*d) + (2*sqrt(2)*a*AppellF1(1/2, 1/2, -(2/3), 3/2, (1/2)*(1 - sec(c + d*x)), (b*(1 - sec(c + d*x)))/(a + b))*(a + b*sec(c + d*x))^(2/3)*tan(c + d*x))/(5*b*d*sqrt(1 + sec(c + d*x))*((a + b*sec(c + d*x))/(a + b))^(2/3)) - (2*sqrt(2)*(a^2 - b^2)*AppellF1(1/2, 1/2, 1/3, 3/2, (1/2)*(1 - sec(c + d*x)), (b*(1 - sec(c + d*x)))/(a + b))*((a + b*sec(c + d*x))/(a + b))^(1/3)*tan(c + d*x))/(5*b*d*sqrt(1 + sec(c + d*x))*(a + b*sec(c + d*x))^(1/3))]
    @test_int [sec(c + d*x)^1*(a + b*sec(c + d*x))^(2/3), x, 3, (sqrt(2)*AppellF1(1/2, 1/2, -(2/3), 3/2, (1/2)*(1 - sec(c + d*x)), (b*(1 - sec(c + d*x)))/(a + b))*(a + b*sec(c + d*x))^(2/3)*tan(c + d*x))/(d*sqrt(1 + sec(c + d*x))*((a + b*sec(c + d*x))/(a + b))^(2/3))]
    @test_int [sec(c + d*x)^0*(a + b*sec(c + d*x))^(2/3), x, 0, Unintegrable((a + b*sec(c + d*x))^(2/3), x)]


    @test_int [sec(c + d*x)^1*(a + b*sec(c + d*x))^(4/3), x, 3, (sqrt(2)*(a + b)*AppellF1(1/2, 1/2, -(4/3), 3/2, (1/2)*(1 - sec(c + d*x)), (b*(1 - sec(c + d*x)))/(a + b))*(a + b*sec(c + d*x))^(1/3)*tan(c + d*x))/(d*sqrt(1 + sec(c + d*x))*((a + b*sec(c + d*x))/(a + b))^(1/3))]
    @test_int [sec(c + d*x)^0*(a + b*sec(c + d*x))^(4/3), x, 0, Unintegrable((a + b*sec(c + d*x))^(4/3), x)]


    @test_int [sec(c + d*x)^4*(a + b*sec(c + d*x))^(5/3), x, 11, (3*a*(18*a^2 + 97*b^2)*(a + b*sec(c + d*x))^(2/3)*tan(c + d*x))/(1232*b^2*d) + (3*(18*a^2 + 121*b^2)*(a + b*sec(c + d*x))^(5/3)*tan(c + d*x))/(1232*b^2*d) - (9*a*(a + b*sec(c + d*x))^(8/3)*tan(c + d*x))/(77*b^2*d) + (3*sec(c + d*x)*(a + b*sec(c + d*x))^(8/3)*tan(c + d*x))/(14*b*d) + ((36*a^4 + 164*a^2*b^2 + 605*b^4)*AppellF1(1/2, 1/2, -(2/3), 3/2, (1/2)*(1 - sec(c + d*x)), (b*(1 - sec(c + d*x)))/(a + b))*(a + b*sec(c + d*x))^(2/3)*tan(c + d*x))/(616*sqrt(2)*b^3*d*sqrt(1 + sec(c + d*x))*((a + b*sec(c + d*x))/(a + b))^(2/3)) - (a*(18*a^4 + 79*a^2*b^2 - 97*b^4)*AppellF1(1/2, 1/2, 1/3, 3/2, (1/2)*(1 - sec(c + d*x)), (b*(1 - sec(c + d*x)))/(a + b))*((a + b*sec(c + d*x))/(a + b))^(1/3)*tan(c + d*x))/(308*sqrt(2)*b^3*d*sqrt(1 + sec(c + d*x))*(a + b*sec(c + d*x))^(1/3))]
    @test_int [sec(c + d*x)^3*(a + b*sec(c + d*x))^(5/3), x, 10, -((3*(15*a^2 - 64*b^2)*(a + b*sec(c + d*x))^(2/3)*tan(c + d*x))/(440*b*d)) - (9*a*(a + b*sec(c + d*x))^(5/3)*tan(c + d*x))/(88*b*d) + (3*(a + b*sec(c + d*x))^(8/3)*tan(c + d*x))/(11*b*d) - (a*(30*a^2 - 373*b^2)*AppellF1(1/2, 1/2, -(2/3), 3/2, (1/2)*(1 - sec(c + d*x)), (b*(1 - sec(c + d*x)))/(a + b))*(a + b*sec(c + d*x))^(2/3)*tan(c + d*x))/(220*sqrt(2)*b^2*d*sqrt(1 + sec(c + d*x))*((a + b*sec(c + d*x))/(a + b))^(2/3)) + ((15*a^4 - 79*a^2*b^2 + 64*b^4)*AppellF1(1/2, 1/2, 1/3, 3/2, (1/2)*(1 - sec(c + d*x)), (b*(1 - sec(c + d*x)))/(a + b))*((a + b*sec(c + d*x))/(a + b))^(1/3)*tan(c + d*x))/(110*sqrt(2)*b^2*d*sqrt(1 + sec(c + d*x))*(a + b*sec(c + d*x))^(1/3))]
    @test_int [sec(c + d*x)^2*(a + b*sec(c + d*x))^(5/3), x, 9, (3*a*(a + b*sec(c + d*x))^(2/3)*tan(c + d*x))/(8*d) + (3*(a + b*sec(c + d*x))^(5/3)*tan(c + d*x))/(8*d) + ((2*a^2 + 5*b^2)*AppellF1(1/2, 1/2, -(2/3), 3/2, (1/2)*(1 - sec(c + d*x)), (b*(1 - sec(c + d*x)))/(a + b))*(a + b*sec(c + d*x))^(2/3)*tan(c + d*x))/(4*sqrt(2)*b*d*sqrt(1 + sec(c + d*x))*((a + b*sec(c + d*x))/(a + b))^(2/3)) - (a*(a^2 - b^2)*AppellF1(1/2, 1/2, 1/3, 3/2, (1/2)*(1 - sec(c + d*x)), (b*(1 - sec(c + d*x)))/(a + b))*((a + b*sec(c + d*x))/(a + b))^(1/3)*tan(c + d*x))/(2*sqrt(2)*b*d*sqrt(1 + sec(c + d*x))*(a + b*sec(c + d*x))^(1/3))]
    @test_int [sec(c + d*x)^1*(a + b*sec(c + d*x))^(5/3), x, 3, (sqrt(2)*(a + b)*AppellF1(1/2, 1/2, -(5/3), 3/2, (1/2)*(1 - sec(c + d*x)), (b*(1 - sec(c + d*x)))/(a + b))*(a + b*sec(c + d*x))^(2/3)*tan(c + d*x))/(d*sqrt(1 + sec(c + d*x))*((a + b*sec(c + d*x))/(a + b))^(2/3))]
    @test_int [sec(c + d*x)^0*(a + b*sec(c + d*x))^(5/3), x, 0, Unintegrable((a + b*sec(c + d*x))^(5/3), x)]


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


    @test_int [sec(c + d*x)^4/(a + b*sec(c + d*x))^(1/3), x, 9, -((9*a*(a + b*sec(c + d*x))^(2/3)*tan(c + d*x))/(20*b^2*d)) + (3*sec(c + d*x)*(a + b*sec(c + d*x))^(2/3)*tan(c + d*x))/(8*b*d) + ((18*a^2 + 25*b^2)*AppellF1(1/2, 1/2, -(2/3), 3/2, (1/2)*(1 - sec(c + d*x)), (b*(1 - sec(c + d*x)))/(a + b))*(a + b*sec(c + d*x))^(2/3)*tan(c + d*x))/(20*sqrt(2)*b^3*d*sqrt(1 + sec(c + d*x))*((a + b*sec(c + d*x))/(a + b))^(2/3)) - (a*(9*a^2 + 11*b^2)*AppellF1(1/2, 1/2, 1/3, 3/2, (1/2)*(1 - sec(c + d*x)), (b*(1 - sec(c + d*x)))/(a + b))*((a + b*sec(c + d*x))/(a + b))^(1/3)*tan(c + d*x))/(10*sqrt(2)*b^3*d*sqrt(1 + sec(c + d*x))*(a + b*sec(c + d*x))^(1/3))]
    @test_int [sec(c + d*x)^3/(a + b*sec(c + d*x))^(1/3), x, 8, (3*(a + b*sec(c + d*x))^(2/3)*tan(c + d*x))/(5*b*d) - (3*sqrt(2)*a*AppellF1(1/2, 1/2, -(2/3), 3/2, (1/2)*(1 - sec(c + d*x)), (b*(1 - sec(c + d*x)))/(a + b))*(a + b*sec(c + d*x))^(2/3)*tan(c + d*x))/(5*b^2*d*sqrt(1 + sec(c + d*x))*((a + b*sec(c + d*x))/(a + b))^(2/3)) + (sqrt(2)*(3*a^2 + 2*b^2)*AppellF1(1/2, 1/2, 1/3, 3/2, (1/2)*(1 - sec(c + d*x)), (b*(1 - sec(c + d*x)))/(a + b))*((a + b*sec(c + d*x))/(a + b))^(1/3)*tan(c + d*x))/(5*b^2*d*sqrt(1 + sec(c + d*x))*(a + b*sec(c + d*x))^(1/3))]
    @test_int [sec(c + d*x)^2/(a + b*sec(c + d*x))^(1/3), x, 7, (sqrt(2)*AppellF1(1/2, 1/2, -(2/3), 3/2, (1/2)*(1 - sec(c + d*x)), (b*(1 - sec(c + d*x)))/(a + b))*(a + b*sec(c + d*x))^(2/3)*tan(c + d*x))/(b*d*sqrt(1 + sec(c + d*x))*((a + b*sec(c + d*x))/(a + b))^(2/3)) - (sqrt(2)*a*AppellF1(1/2, 1/2, 1/3, 3/2, (1/2)*(1 - sec(c + d*x)), (b*(1 - sec(c + d*x)))/(a + b))*((a + b*sec(c + d*x))/(a + b))^(1/3)*tan(c + d*x))/(b*d*sqrt(1 + sec(c + d*x))*(a + b*sec(c + d*x))^(1/3))]
    @test_int [sec(c + d*x)^1/(a + b*sec(c + d*x))^(1/3), x, 3, (sqrt(2)*AppellF1(1/2, 1/2, 1/3, 3/2, (1/2)*(1 - sec(c + d*x)), (b*(1 - sec(c + d*x)))/(a + b))*((a + b*sec(c + d*x))/(a + b))^(1/3)*tan(c + d*x))/(d*sqrt(1 + sec(c + d*x))*(a + b*sec(c + d*x))^(1/3))]
    @test_int [sec(c + d*x)^0/(a + b*sec(c + d*x))^(1/3), x, 0, Unintegrable(1/(a + b*sec(c + d*x))^(1/3), x)]


    @test_int [sec(c + d*x)^1/(a + b*sec(c + d*x))^(2/3), x, 3, (sqrt(2)*AppellF1(1/2, 1/2, 2/3, 3/2, (1/2)*(1 - sec(c + d*x)), (b*(1 - sec(c + d*x)))/(a + b))*((a + b*sec(c + d*x))/(a + b))^(2/3)*tan(c + d*x))/(d*sqrt(1 + sec(c + d*x))*(a + b*sec(c + d*x))^(2/3))]
    @test_int [sec(c + d*x)^0/(a + b*sec(c + d*x))^(2/3), x, 0, Unintegrable(1/(a + b*sec(c + d*x))^(2/3), x)]


    @test_int [sec(c + d*x)^1/(a + b*sec(c + d*x))^(4/3), x, 3, (sqrt(2)*AppellF1(1/2, 1/2, 4/3, 3/2, (1/2)*(1 - sec(c + d*x)), (b*(1 - sec(c + d*x)))/(a + b))*((a + b*sec(c + d*x))/(a + b))^(1/3)*tan(c + d*x))/((a + b)*d*sqrt(1 + sec(c + d*x))*(a + b*sec(c + d*x))^(1/3))]
    @test_int [sec(c + d*x)^0/(a + b*sec(c + d*x))^(4/3), x, 0, Unintegrable(1/(a + b*sec(c + d*x))^(4/3), x)]


    @test_int [sec(c + d*x)^4/(a + b*sec(c + d*x))^(5/3), x, 9, -((3*a^2*sec(c + d*x)*tan(c + d*x))/(2*b*(a^2 - b^2)*d*(a + b*sec(c + d*x))^(2/3))) + (3*(3*a^2 - b^2)*(a + b*sec(c + d*x))^(1/3)*tan(c + d*x))/(4*b^2*(a^2 - b^2)*d) - (a*(9*a^2 - 7*b^2)*AppellF1(1/2, 1/2, -(1/3), 3/2, (1/2)*(1 - sec(c + d*x)), (b*(1 - sec(c + d*x)))/(a + b))*(a + b*sec(c + d*x))^(1/3)*tan(c + d*x))/(2*sqrt(2)*b^3*(a^2 - b^2)*d*sqrt(1 + sec(c + d*x))*((a + b*sec(c + d*x))/(a + b))^(1/3)) + ((9*a^4 - 10*a^2*b^2 - b^4)*AppellF1(1/2, 1/2, 2/3, 3/2, (1/2)*(1 - sec(c + d*x)), (b*(1 - sec(c + d*x)))/(a + b))*((a + b*sec(c + d*x))/(a + b))^(2/3)*tan(c + d*x))/(2*sqrt(2)*b^3*(a^2 - b^2)*d*sqrt(1 + sec(c + d*x))*(a + b*sec(c + d*x))^(2/3))]
    @test_int [sec(c + d*x)^3/(a + b*sec(c + d*x))^(5/3), x, 8, -((3*a^2*tan(c + d*x))/(2*b*(a^2 - b^2)*d*(a + b*sec(c + d*x))^(2/3))) + ((3*a^2 - 2*b^2)*AppellF1(1/2, 1/2, -(1/3), 3/2, (1/2)*(1 - sec(c + d*x)), (b*(1 - sec(c + d*x)))/(a + b))*(a + b*sec(c + d*x))^(1/3)*tan(c + d*x))/(sqrt(2)*b^2*(a^2 - b^2)*d*sqrt(1 + sec(c + d*x))*((a + b*sec(c + d*x))/(a + b))^(1/3)) - (a*(3*a^2 - 4*b^2)*AppellF1(1/2, 1/2, 2/3, 3/2, (1/2)*(1 - sec(c + d*x)), (b*(1 - sec(c + d*x)))/(a + b))*((a + b*sec(c + d*x))/(a + b))^(2/3)*tan(c + d*x))/(sqrt(2)*b^2*(a^2 - b^2)*d*sqrt(1 + sec(c + d*x))*(a + b*sec(c + d*x))^(2/3))]
    @test_int [sec(c + d*x)^2/(a + b*sec(c + d*x))^(5/3), x, 8, (3*a*tan(c + d*x))/(2*(a^2 - b^2)*d*(a + b*sec(c + d*x))^(2/3)) - (a*AppellF1(1/2, 1/2, -(1/3), 3/2, (1/2)*(1 - sec(c + d*x)), (b*(1 - sec(c + d*x)))/(a + b))*(a + b*sec(c + d*x))^(1/3)*tan(c + d*x))/(sqrt(2)*b*(a^2 - b^2)*d*sqrt(1 + sec(c + d*x))*((a + b*sec(c + d*x))/(a + b))^(1/3)) + ((a^2 - 2*b^2)*AppellF1(1/2, 1/2, 2/3, 3/2, (1/2)*(1 - sec(c + d*x)), (b*(1 - sec(c + d*x)))/(a + b))*((a + b*sec(c + d*x))/(a + b))^(2/3)*tan(c + d*x))/(sqrt(2)*b*(a^2 - b^2)*d*sqrt(1 + sec(c + d*x))*(a + b*sec(c + d*x))^(2/3))]
    @test_int [sec(c + d*x)^1/(a + b*sec(c + d*x))^(5/3), x, 3, (sqrt(2)*AppellF1(1/2, 1/2, 5/3, 3/2, (1/2)*(1 - sec(c + d*x)), (b*(1 - sec(c + d*x)))/(a + b))*((a + b*sec(c + d*x))/(a + b))^(2/3)*tan(c + d*x))/((a + b)*d*sqrt(1 + sec(c + d*x))*(a + b*sec(c + d*x))^(2/3))]
    @test_int [sec(c + d*x)^0/(a + b*sec(c + d*x))^(5/3), x, 0, Unintegrable(1/(a + b*sec(c + d*x))^(5/3), x)]


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


    @test_int [sec(c + d*x)^(2/3)/(a + b*sec(c + d*x)), x, 6, (a*AppellF1(1/2, -(1/6), 1, 3/2, sin(c + d*x)^2, (a^2*sin(c + d*x)^2)/(a^2 - b^2))*sin(c + d*x))/((a^2 - b^2)*d*(cos(c + d*x)^2)^(1/6)*sec(c + d*x)^(1/3)) - (b*AppellF1(1/2, 1/3, 1, 3/2, sin(c + d*x)^2, (a^2*sin(c + d*x)^2)/(a^2 - b^2))*(cos(c + d*x)^2)^(1/3)*sec(c + d*x)^(2/3)*sin(c + d*x))/((a^2 - b^2)*d)]
    @test_int [sec(c + d*x)^(1/3)/(a + b*sec(c + d*x)), x, 6, (a*AppellF1(1/2, -(1/3), 1, 3/2, sin(c + d*x)^2, (a^2*sin(c + d*x)^2)/(a^2 - b^2))*sin(c + d*x))/((a^2 - b^2)*d*(cos(c + d*x)^2)^(1/3)*sec(c + d*x)^(2/3)) - (b*AppellF1(1/2, 1/6, 1, 3/2, sin(c + d*x)^2, (a^2*sin(c + d*x)^2)/(a^2 - b^2))*(cos(c + d*x)^2)^(1/6)*sec(c + d*x)^(1/3)*sin(c + d*x))/((a^2 - b^2)*d)]
    @test_int [sec(c + d*x)^(-1/3)/(a + b*sec(c + d*x)), x, 6, -((b*AppellF1(1/2, -(1/6), 1, 3/2, sin(c + d*x)^2, (a^2*sin(c + d*x)^2)/(a^2 - b^2))*sin(c + d*x))/((a^2 - b^2)*d*(cos(c + d*x)^2)^(1/6)*sec(c + d*x)^(1/3))) + (a*AppellF1(1/2, -(2/3), 1, 3/2, sin(c + d*x)^2, (a^2*sin(c + d*x)^2)/(a^2 - b^2))*(cos(c + d*x)^2)^(1/3)*sec(c + d*x)^(2/3)*sin(c + d*x))/((a^2 - b^2)*d)]
    @test_int [sec(c + d*x)^(-2/3)/(a + b*sec(c + d*x)), x, 6, -((b*AppellF1(1/2, -(1/3), 1, 3/2, sin(c + d*x)^2, (a^2*sin(c + d*x)^2)/(a^2 - b^2))*sin(c + d*x))/((a^2 - b^2)*d*(cos(c + d*x)^2)^(1/3)*sec(c + d*x)^(2/3))) + (a*AppellF1(1/2, -(5/6), 1, 3/2, sin(c + d*x)^2, (a^2*sin(c + d*x)^2)/(a^2 - b^2))*(cos(c + d*x)^2)^(1/6)*sec(c + d*x)^(1/3)*sin(c + d*x))/((a^2 - b^2)*d)]


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


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


    @test_int [sec(c + d*x)^(7/3)*sqrt(a + b*sec(c + d*x)), x, 0, Unintegrable(sec(c + d*x)^(7/3)*sqrt(a + b*sec(c + d*x)), x)]
    @test_int [sec(c + d*x)^(5/3)*sqrt(a + b*sec(c + d*x)), x, 0, Unintegrable(sec(c + d*x)^(5/3)*sqrt(a + b*sec(c + d*x)), x)]
    @test_int [sec(c + d*x)^(4/3)*sqrt(a + b*sec(c + d*x)), x, 0, Unintegrable(sec(c + d*x)^(4/3)*sqrt(a + b*sec(c + d*x)), x)]
    @test_int [sec(c + d*x)^(2/3)*sqrt(a + b*sec(c + d*x)), x, 0, Unintegrable(sec(c + d*x)^(2/3)*sqrt(a + b*sec(c + d*x)), x)]
    @test_int [sec(c + d*x)^(1/3)*sqrt(a + b*sec(c + d*x)), x, 0, Unintegrable(sec(c + d*x)^(1/3)*sqrt(a + b*sec(c + d*x)), x)]
    @test_int [sqrt(a + b*sec(c + d*x))/sec(c + d*x)^(1/3), x, 0, Unintegrable(sqrt(a + b*sec(c + d*x))/sec(c + d*x)^(1/3), x)]
    @test_int [sqrt(a + b*sec(c + d*x))/sec(c + d*x)^(2/3), x, 0, Unintegrable(sqrt(a + b*sec(c + d*x))/sec(c + d*x)^(2/3), x)]
    @test_int [sqrt(a + b*sec(c + d*x))/sec(c + d*x)^(4/3), x, 0, Unintegrable(sqrt(a + b*sec(c + d*x))/sec(c + d*x)^(4/3), x)]
    @test_int [sqrt(a + b*sec(c + d*x))/sec(c + d*x)^(5/3), x, 0, Unintegrable(sqrt(a + b*sec(c + d*x))/sec(c + d*x)^(5/3), x)]
    @test_int [sqrt(a + b*sec(c + d*x))/sec(c + d*x)^(7/3), x, 0, Unintegrable(sqrt(a + b*sec(c + d*x))/sec(c + d*x)^(7/3), x)]


    @test_int [sec(c + d*x)^(7/3)*(a + b*sec(c + d*x))^(3/2), x, 0, Unintegrable(sec(c + d*x)^(7/3)*(a + b*sec(c + d*x))^(3/2), x)]
    @test_int [sec(c + d*x)^(5/3)*(a + b*sec(c + d*x))^(3/2), x, 0, Unintegrable(sec(c + d*x)^(5/3)*(a + b*sec(c + d*x))^(3/2), x)]
    @test_int [sec(c + d*x)^(4/3)*(a + b*sec(c + d*x))^(3/2), x, 0, Unintegrable(sec(c + d*x)^(4/3)*(a + b*sec(c + d*x))^(3/2), x)]
    @test_int [sec(c + d*x)^(2/3)*(a + b*sec(c + d*x))^(3/2), x, 0, Unintegrable(sec(c + d*x)^(2/3)*(a + b*sec(c + d*x))^(3/2), x)]
    @test_int [sec(c + d*x)^(1/3)*(a + b*sec(c + d*x))^(3/2), x, 0, Unintegrable(sec(c + d*x)^(1/3)*(a + b*sec(c + d*x))^(3/2), x)]
    @test_int [(a + b*sec(c + d*x))^(3/2)/sec(c + d*x)^(1/3), x, 0, Unintegrable((a + b*sec(c + d*x))^(3/2)/sec(c + d*x)^(1/3), x)]
    @test_int [(a + b*sec(c + d*x))^(3/2)/sec(c + d*x)^(2/3), x, 0, Unintegrable((a + b*sec(c + d*x))^(3/2)/sec(c + d*x)^(2/3), x)]
    @test_int [(a + b*sec(c + d*x))^(3/2)/sec(c + d*x)^(4/3), x, 0, Unintegrable((a + b*sec(c + d*x))^(3/2)/sec(c + d*x)^(4/3), x)]
    @test_int [(a + b*sec(c + d*x))^(3/2)/sec(c + d*x)^(5/3), x, 0, Unintegrable((a + b*sec(c + d*x))^(3/2)/sec(c + d*x)^(5/3), x)]
    @test_int [(a + b*sec(c + d*x))^(3/2)/sec(c + d*x)^(7/3), x, 0, Unintegrable((a + b*sec(c + d*x))^(3/2)/sec(c + d*x)^(7/3), x)]


    @test_int [sec(c + d*x)^(7/3)*(a + b*sec(c + d*x))^(5/2), x, 0, Unintegrable(sec(c + d*x)^(7/3)*(a + b*sec(c + d*x))^(5/2), x)]
    @test_int [sec(c + d*x)^(5/3)*(a + b*sec(c + d*x))^(5/2), x, 0, Unintegrable(sec(c + d*x)^(5/3)*(a + b*sec(c + d*x))^(5/2), x)]
    @test_int [sec(c + d*x)^(4/3)*(a + b*sec(c + d*x))^(5/2), x, 0, Unintegrable(sec(c + d*x)^(4/3)*(a + b*sec(c + d*x))^(5/2), x)]
    @test_int [sec(c + d*x)^(2/3)*(a + b*sec(c + d*x))^(5/2), x, 0, Unintegrable(sec(c + d*x)^(2/3)*(a + b*sec(c + d*x))^(5/2), x)]
    @test_int [sec(c + d*x)^(1/3)*(a + b*sec(c + d*x))^(5/2), x, 0, Unintegrable(sec(c + d*x)^(1/3)*(a + b*sec(c + d*x))^(5/2), x)]
    @test_int [(a + b*sec(c + d*x))^(5/2)/sec(c + d*x)^(1/3), x, 0, Unintegrable((a + b*sec(c + d*x))^(5/2)/sec(c + d*x)^(1/3), x)]
    @test_int [(a + b*sec(c + d*x))^(5/2)/sec(c + d*x)^(2/3), x, 0, Unintegrable((a + b*sec(c + d*x))^(5/2)/sec(c + d*x)^(2/3), x)]
    @test_int [(a + b*sec(c + d*x))^(5/2)/sec(c + d*x)^(4/3), x, 0, Unintegrable((a + b*sec(c + d*x))^(5/2)/sec(c + d*x)^(4/3), x)]
    @test_int [(a + b*sec(c + d*x))^(5/2)/sec(c + d*x)^(5/3), x, 0, Unintegrable((a + b*sec(c + d*x))^(5/2)/sec(c + d*x)^(5/3), x)]
    @test_int [(a + b*sec(c + d*x))^(5/2)/sec(c + d*x)^(7/3), x, 0, Unintegrable((a + b*sec(c + d*x))^(5/2)/sec(c + d*x)^(7/3), x)]


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


    @test_int [sec(c + d*x)^(7/3)/sqrt(a + b*sec(c + d*x)), x, 0, Unintegrable(sec(c + d*x)^(7/3)/sqrt(a + b*sec(c + d*x)), x)]
    @test_int [sec(c + d*x)^(5/3)/sqrt(a + b*sec(c + d*x)), x, 0, Unintegrable(sec(c + d*x)^(5/3)/sqrt(a + b*sec(c + d*x)), x)]
    @test_int [sec(c + d*x)^(4/3)/sqrt(a + b*sec(c + d*x)), x, 0, Unintegrable(sec(c + d*x)^(4/3)/sqrt(a + b*sec(c + d*x)), x)]
    @test_int [sec(c + d*x)^(2/3)/sqrt(a + b*sec(c + d*x)), x, 0, Unintegrable(sec(c + d*x)^(2/3)/sqrt(a + b*sec(c + d*x)), x)]
    @test_int [sec(c + d*x)^(1/3)/sqrt(a + b*sec(c + d*x)), x, 0, Unintegrable(sec(c + d*x)^(1/3)/sqrt(a + b*sec(c + d*x)), x)]
    @test_int [sec(c + d*x)^(-1/3)/sqrt(a + b*sec(c + d*x)), x, 0, Unintegrable(1/(sec(c + d*x)^(1/3)*sqrt(a + b*sec(c + d*x))), x)]
    @test_int [sec(c + d*x)^(-2/3)/sqrt(a + b*sec(c + d*x)), x, 0, Unintegrable(1/(sec(c + d*x)^(2/3)*sqrt(a + b*sec(c + d*x))), x)]
    @test_int [sec(c + d*x)^(-4/3)/sqrt(a + b*sec(c + d*x)), x, 0, Unintegrable(1/(sec(c + d*x)^(4/3)*sqrt(a + b*sec(c + d*x))), x)]
    @test_int [sec(c + d*x)^(-5/3)/sqrt(a + b*sec(c + d*x)), x, 0, Unintegrable(1/(sec(c + d*x)^(5/3)*sqrt(a + b*sec(c + d*x))), x)]
    @test_int [sec(c + d*x)^(-7/3)/sqrt(a + b*sec(c + d*x)), x, 0, Unintegrable(1/(sec(c + d*x)^(7/3)*sqrt(a + b*sec(c + d*x))), x)]


    @test_int [sec(c + d*x)^(7/3)/(a + b*sec(c + d*x))^(3/2), x, 0, Unintegrable(sec(c + d*x)^(7/3)/(a + b*sec(c + d*x))^(3/2), x)]
    @test_int [sec(c + d*x)^(5/3)/(a + b*sec(c + d*x))^(3/2), x, 0, Unintegrable(sec(c + d*x)^(5/3)/(a + b*sec(c + d*x))^(3/2), x)]
    @test_int [sec(c + d*x)^(4/3)/(a + b*sec(c + d*x))^(3/2), x, 0, Unintegrable(sec(c + d*x)^(4/3)/(a + b*sec(c + d*x))^(3/2), x)]
    @test_int [sec(c + d*x)^(2/3)/(a + b*sec(c + d*x))^(3/2), x, 0, Unintegrable(sec(c + d*x)^(2/3)/(a + b*sec(c + d*x))^(3/2), x)]
    @test_int [sec(c + d*x)^(1/3)/(a + b*sec(c + d*x))^(3/2), x, 0, Unintegrable(sec(c + d*x)^(1/3)/(a + b*sec(c + d*x))^(3/2), x)]
    @test_int [sec(c + d*x)^(-1/3)/(a + b*sec(c + d*x))^(3/2), x, 0, Unintegrable(1/(sec(c + d*x)^(1/3)*(a + b*sec(c + d*x))^(3/2)), x)]
    @test_int [sec(c + d*x)^(-2/3)/(a + b*sec(c + d*x))^(3/2), x, 0, Unintegrable(1/(sec(c + d*x)^(2/3)*(a + b*sec(c + d*x))^(3/2)), x)]
    @test_int [sec(c + d*x)^(-4/3)/(a + b*sec(c + d*x))^(3/2), x, 0, Unintegrable(1/(sec(c + d*x)^(4/3)*(a + b*sec(c + d*x))^(3/2)), x)]
    @test_int [sec(c + d*x)^(-5/3)/(a + b*sec(c + d*x))^(3/2), x, 0, Unintegrable(1/(sec(c + d*x)^(5/3)*(a + b*sec(c + d*x))^(3/2)), x)]
    @test_int [sec(c + d*x)^(-7/3)/(a + b*sec(c + d*x))^(3/2), x, 0, Unintegrable(1/(sec(c + d*x)^(7/3)*(a + b*sec(c + d*x))^(3/2)), x)]


    @test_int [sec(c + d*x)^(7/3)/(a + b*sec(c + d*x))^(5/2), x, 0, Unintegrable(sec(c + d*x)^(7/3)/(a + b*sec(c + d*x))^(5/2), x)]
    @test_int [sec(c + d*x)^(5/3)/(a + b*sec(c + d*x))^(5/2), x, 0, Unintegrable(sec(c + d*x)^(5/3)/(a + b*sec(c + d*x))^(5/2), x)]
    @test_int [sec(c + d*x)^(4/3)/(a + b*sec(c + d*x))^(5/2), x, 0, Unintegrable(sec(c + d*x)^(4/3)/(a + b*sec(c + d*x))^(5/2), x)]
    @test_int [sec(c + d*x)^(2/3)/(a + b*sec(c + d*x))^(5/2), x, 0, Unintegrable(sec(c + d*x)^(2/3)/(a + b*sec(c + d*x))^(5/2), x)]
    @test_int [sec(c + d*x)^(1/3)/(a + b*sec(c + d*x))^(5/2), x, 0, Unintegrable(sec(c + d*x)^(1/3)/(a + b*sec(c + d*x))^(5/2), x)]
    @test_int [sec(c + d*x)^(-1/3)/(a + b*sec(c + d*x))^(5/2), x, 0, Unintegrable(1/(sec(c + d*x)^(1/3)*(a + b*sec(c + d*x))^(5/2)), x)]
    @test_int [sec(c + d*x)^(-2/3)/(a + b*sec(c + d*x))^(5/2), x, 0, Unintegrable(1/(sec(c + d*x)^(2/3)*(a + b*sec(c + d*x))^(5/2)), x)]
    @test_int [sec(c + d*x)^(-4/3)/(a + b*sec(c + d*x))^(5/2), x, 0, Unintegrable(1/(sec(c + d*x)^(4/3)*(a + b*sec(c + d*x))^(5/2)), x)]
    @test_int [sec(c + d*x)^(-5/3)/(a + b*sec(c + d*x))^(5/2), x, 0, Unintegrable(1/(sec(c + d*x)^(5/3)*(a + b*sec(c + d*x))^(5/2)), x)]
    @test_int [sec(c + d*x)^(-7/3)/(a + b*sec(c + d*x))^(5/2), x, 0, Unintegrable(1/(sec(c + d*x)^(7/3)*(a + b*sec(c + d*x))^(5/2)), x)]


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


    @test_int [(d* sec(e + f*x))^n*(a + b*sec(e + f*x))^3, x, 7, -((a*d*(3*b^2*n + a^2*(1 + n))*HypergeometricFunctions._₂F₁(1/2, (1 - n)/2, (3 - n)/2, cos(e + f*x)^2)*(d*sec(e + f*x))^(-1 + n)*sin(e + f*x))/(f*(1 - n^2)*sqrt(sin(e + f*x)^2))) + (b*(b^2*(1 + n) + 3*a^2*(2 + n))*HypergeometricFunctions._₂F₁(1/2, -(n/2), (2 - n)/2, cos(e + f*x)^2)*(d*sec(e + f*x))^n*sin(e + f*x))/(f*n*(2 + n)*sqrt(sin(e + f*x)^2)) + (a*b^2*(5 + 2*n)*(d*sec(e + f*x))^n*tan(e + f*x))/(f*(1 + n)*(2 + n)) + (b^2*(d*sec(e + f*x))^n*(a + b*sec(e + f*x))*tan(e + f*x))/(f*(2 + n))]
    @test_int [(d* sec(e + f*x))^n*(a + b*sec(e + f*x))^2, x, 6, -((d*(b^2*n + a^2*(1 + n))*HypergeometricFunctions._₂F₁(1/2, (1 - n)/2, (3 - n)/2, cos(e + f*x)^2)*(d*sec(e + f*x))^(-1 + n)*sin(e + f*x))/(f*(1 - n^2)*sqrt(sin(e + f*x)^2))) + (2*a*b*HypergeometricFunctions._₂F₁(1/2, -(n/2), (2 - n)/2, cos(e + f*x)^2)*(d*sec(e + f*x))^n*sin(e + f*x))/(f*n*sqrt(sin(e + f*x)^2)) + (b^2*(d*sec(e + f*x))^n*tan(e + f*x))/(f*(1 + n))]
    @test_int [(d* sec(e + f*x))^n*(a + b*sec(e + f*x))^1, x, 5, -((a*d*HypergeometricFunctions._₂F₁(1/2, (1 - n)/2, (3 - n)/2, cos(e + f*x)^2)*(d*sec(e + f*x))^(-1 + n)*sin(e + f*x))/(f*(1 - n)*sqrt(sin(e + f*x)^2))) + (b*HypergeometricFunctions._₂F₁(1/2, -(n/2), (2 - n)/2, cos(e + f*x)^2)*(d*sec(e + f*x))^n*sin(e + f*x))/(f*n*sqrt(sin(e + f*x)^2))]
    @test_int [(d* sec(e + f*x))^n/(a + b*sec(e + f*x))^1, x, 6, (a*AppellF1(1/2, (1/2)*(-1 + n), 1, 3/2, sin(e + f*x)^2, (a^2*sin(e + f*x)^2)/(a^2 - b^2))*cos(e + f*x)*(cos(e + f*x)^2)^((1/2)*(-1 + n))*(d*sec(e + f*x))^n*sin(e + f*x))/((a^2 - b^2)*f) - (b*AppellF1(1/2, n/2, 1, 3/2, sin(e + f*x)^2, (a^2*sin(e + f*x)^2)/(a^2 - b^2))*(cos(e + f*x)^2)^(n/2)*(d*sec(e + f*x))^n*sin(e + f*x))/((a^2 - b^2)*f)]
    @test_int [(d* sec(e + f*x))^n/(a + b*sec(e + f*x))^2, x, 9, (a^2*AppellF1(1/2, (1/2)*(-3 + n), 2, 3/2, sin(e + f*x)^2, (a^2*sin(e + f*x)^2)/(a^2 - b^2))*cos(e + f*x)*(cos(e + f*x)^2)^((1/2)*(-1 + n))*(d*sec(e + f*x))^n*sin(e + f*x))/((a^2 - b^2)^2*f) + (b^2*AppellF1(1/2, (1/2)*(-1 + n), 2, 3/2, sin(e + f*x)^2, (a^2*sin(e + f*x)^2)/(a^2 - b^2))*cos(e + f*x)*(cos(e + f*x)^2)^((1/2)*(-1 + n))*(d*sec(e + f*x))^n*sin(e + f*x))/((a^2 - b^2)^2*f) - (2*a*b*AppellF1(1/2, (1/2)*(-2 + n), 2, 3/2, sin(e + f*x)^2, (a^2*sin(e + f*x)^2)/(a^2 - b^2))*(cos(e + f*x)^2)^(n/2)*(d*sec(e + f*x))^n*sin(e + f*x))/((a^2 - b^2)^2*f)]


    @test_int [(d* sec(e + f*x))^n*(a + b*sec(e + f*x))^(3/2), x, 0, Unintegrable((d*sec(e + f*x))^n*(a + b*sec(e + f*x))^(3/2), x)]
    @test_int [(d* sec(e + f*x))^n*(a + b*sec(e + f*x))^(1/2), x, 0, Unintegrable((d*sec(e + f*x))^n*sqrt(a + b*sec(e + f*x)), x)]
    @test_int [(d* sec(e + f*x))^n/(a + b*sec(e + f*x))^(1/2), x, 0, Unintegrable((d*sec(e + f*x))^n/sqrt(a + b*sec(e + f*x)), x)]
    @test_int [(d* sec(e + f*x))^n/(a + b*sec(e + f*x))^(3/2), x, 0, Unintegrable((d*sec(e + f*x))^n/(a + b*sec(e + f*x))^(3/2), x)]


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


    @test_int [sec(e + f*x)^n*(a + b*sec(e + f*x))^m, x, 0, Unintegrable(sec(e + f*x)^n*(a + b*sec(e + f*x))^m, x)]
    @test_int [(d* sec(e + f*x))^n*(a + b*sec(e + f*x))^m, x, 0, Unintegrable((d*sec(e + f*x))^n*(a + b*sec(e + f*x))^m, x)]


    @test_int [sec(e + f*x)^3*(a + b*sec(e + f*x))^m, x, 8, ((a + b*sec(e + f*x))^(1 + m)*tan(e + f*x))/(b*f*(2 + m)) - (sqrt(2)*a*(a + b)*AppellF1(1/2, 1/2, -1 - m, 3/2, (1/2)*(1 - sec(e + f*x)), (b*(1 - sec(e + f*x)))/(a + b))*(a + b*sec(e + f*x))^m*tan(e + f*x))/(((a + b*sec(e + f*x))/(a + b))^m*(b^2*f*(2 + m)*sqrt(1 + sec(e + f*x)))) + (sqrt(2)*(a^2 + b^2*(1 + m))*AppellF1(1/2, 1/2, -m, 3/2, (1/2)*(1 - sec(e + f*x)), (b*(1 - sec(e + f*x)))/(a + b))*(a + b*sec(e + f*x))^m*tan(e + f*x))/(((a + b*sec(e + f*x))/(a + b))^m*(b^2*f*(2 + m)*sqrt(1 + sec(e + f*x))))]
    @test_int [sec(e + f*x)^2*(a + b*sec(e + f*x))^m, x, 7, (sqrt(2)*(a + b)*AppellF1(1/2, 1/2, -1 - m, 3/2, (1/2)*(1 - sec(e + f*x)), (b*(1 - sec(e + f*x)))/(a + b))*(a + b*sec(e + f*x))^m*tan(e + f*x))/(((a + b*sec(e + f*x))/(a + b))^m*(b*f*sqrt(1 + sec(e + f*x)))) - (sqrt(2)*a*AppellF1(1/2, 1/2, -m, 3/2, (1/2)*(1 - sec(e + f*x)), (b*(1 - sec(e + f*x)))/(a + b))*(a + b*sec(e + f*x))^m*tan(e + f*x))/(((a + b*sec(e + f*x))/(a + b))^m*(b*f*sqrt(1 + sec(e + f*x))))]
    @test_int [sec(e + f*x)^1*(a + b*sec(e + f*x))^m, x, 3, (sqrt(2)*AppellF1(1/2, 1/2, -m, 3/2, (1/2)*(1 - sec(e + f*x)), (b*(1 - sec(e + f*x)))/(a + b))*(a + b*sec(e + f*x))^m*tan(e + f*x))/(((a + b*sec(e + f*x))/(a + b))^m*(f*sqrt(1 + sec(e + f*x))))]
    @test_int [sec(e + f*x)^0*(a + b*sec(e + f*x))^m, x, 0, Unintegrable((a + b*sec(e + f*x))^m, x)]
    @test_int [cos(e + f*x)^1*(a + b*sec(e + f*x))^m, x, 0, Unintegrable(cos(e + f*x)*(a + b*sec(e + f*x))^m, x)]
    @test_int [cos(e + f*x)^2*(a + b*sec(e + f*x))^m, x, 0, Unintegrable(cos(e + f*x)^2*(a + b*sec(e + f*x))^m, x)]


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


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


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


    @test_int [cos(c + d*x)^(9/2)*(a + b*sec(c + d*x)), x, 8, (14*a*Elliptic.E((1/2)*(c + d*x), 2))/(15*d) + (10*b*Elliptic.F((1/2)*(c + d*x), 2))/(21*d) + (10*b*sqrt(cos(c + d*x))*sin(c + d*x))/(21*d) + (14*a*cos(c + d*x)^(3/2)*sin(c + d*x))/(45*d) + (2*b*cos(c + d*x)^(5/2)*sin(c + d*x))/(7*d) + (2*a*cos(c + d*x)^(7/2)*sin(c + d*x))/(9*d)]
    @test_int [cos(c + d*x)^(7/2)*(a + b*sec(c + d*x)), x, 7, (6*b*Elliptic.E((1/2)*(c + d*x), 2))/(5*d) + (10*a*Elliptic.F((1/2)*(c + d*x), 2))/(21*d) + (10*a*sqrt(cos(c + d*x))*sin(c + d*x))/(21*d) + (2*b*cos(c + d*x)^(3/2)*sin(c + d*x))/(5*d) + (2*a*cos(c + d*x)^(5/2)*sin(c + d*x))/(7*d)]
    @test_int [cos(c + d*x)^(5/2)*(a + b*sec(c + d*x)), x, 6, (6*a*Elliptic.E((1/2)*(c + d*x), 2))/(5*d) + (2*b*Elliptic.F((1/2)*(c + d*x), 2))/(3*d) + (2*b*sqrt(cos(c + d*x))*sin(c + d*x))/(3*d) + (2*a*cos(c + d*x)^(3/2)*sin(c + d*x))/(5*d)]
    @test_int [cos(c + d*x)^(3/2)*(a + b*sec(c + d*x)), x, 5, (2*b*Elliptic.E((1/2)*(c + d*x), 2))/d + (2*a*Elliptic.F((1/2)*(c + d*x), 2))/(3*d) + (2*a*sqrt(cos(c + d*x))*sin(c + d*x))/(3*d)]
    @test_int [cos(c + d*x)^(1/2)*(a + b*sec(c + d*x)), x, 4, (2*a*Elliptic.E((1/2)*(c + d*x), 2))/d + (2*b*Elliptic.F((1/2)*(c + d*x), 2))/d]
    @test_int [cos(c + d*x)^(-1/2)*(a + b*sec(c + d*x)), x, 5, -((2*b*Elliptic.E((1/2)*(c + d*x), 2))/d) + (2*a*Elliptic.F((1/2)*(c + d*x), 2))/d + (2*b*sin(c + d*x))/(d*sqrt(cos(c + d*x)))]
    @test_int [cos(c + d*x)^(-3/2)*(a + b*sec(c + d*x)), x, 6, -((2*a*Elliptic.E((1/2)*(c + d*x), 2))/d) + (2*b*Elliptic.F((1/2)*(c + d*x), 2))/(3*d) + (2*b*sin(c + d*x))/(3*d*cos(c + d*x)^(3/2)) + (2*a*sin(c + d*x))/(d*sqrt(cos(c + d*x)))]
    @test_int [cos(c + d*x)^(-5/2)*(a + b*sec(c + d*x)), x, 7, -((6*b*Elliptic.E((1/2)*(c + d*x), 2))/(5*d)) + (2*a*Elliptic.F((1/2)*(c + d*x), 2))/(3*d) + (2*b*sin(c + d*x))/(5*d*cos(c + d*x)^(5/2)) + (2*a*sin(c + d*x))/(3*d*cos(c + d*x)^(3/2)) + (6*b*sin(c + d*x))/(5*d*sqrt(cos(c + d*x)))]


    @test_int [cos(c + d*x)^(9/2)*(a + b*sec(c + d*x))^2, x, 10, (2*(7*a^2 + 9*b^2)*Elliptic.E((1/2)*(c + d*x), 2))/(15*d) + (20*a*b*Elliptic.F((1/2)*(c + d*x), 2))/(21*d) + (20*a*b*sqrt(cos(c + d*x))*sin(c + d*x))/(21*d) + (2*(7*a^2 + 9*b^2)*cos(c + d*x)^(3/2)*sin(c + d*x))/(45*d) + (4*a*b*cos(c + d*x)^(5/2)*sin(c + d*x))/(7*d) + (2*a^2*cos(c + d*x)^(7/2)*sin(c + d*x))/(9*d)]
    @test_int [cos(c + d*x)^(7/2)*(a + b*sec(c + d*x))^2, x, 9, (12*a*b*Elliptic.E((1/2)*(c + d*x), 2))/(5*d) + (2*(5*a^2 + 7*b^2)*Elliptic.F((1/2)*(c + d*x), 2))/(21*d) + (2*(5*a^2 + 7*b^2)*sqrt(cos(c + d*x))*sin(c + d*x))/(21*d) + (4*a*b*cos(c + d*x)^(3/2)*sin(c + d*x))/(5*d) + (2*a^2*cos(c + d*x)^(5/2)*sin(c + d*x))/(7*d)]
    @test_int [cos(c + d*x)^(5/2)*(a + b*sec(c + d*x))^2, x, 8, (2*(3*a^2 + 5*b^2)*Elliptic.E((1/2)*(c + d*x), 2))/(5*d) + (4*a*b*Elliptic.F((1/2)*(c + d*x), 2))/(3*d) + (4*a*b*sqrt(cos(c + d*x))*sin(c + d*x))/(3*d) + (2*a^2*cos(c + d*x)^(3/2)*sin(c + d*x))/(5*d)]
    @test_int [cos(c + d*x)^(3/2)*(a + b*sec(c + d*x))^2, x, 7, (4*a*b*Elliptic.E((1/2)*(c + d*x), 2))/d + (2*(a^2 + 3*b^2)*Elliptic.F((1/2)*(c + d*x), 2))/(3*d) + (2*a^2*sqrt(cos(c + d*x))*sin(c + d*x))/(3*d)]
    @test_int [cos(c + d*x)^(1/2)*(a + b*sec(c + d*x))^2, x, 7, (2*(a^2 - b^2)*Elliptic.E((1/2)*(c + d*x), 2))/d + (4*a*b*Elliptic.F((1/2)*(c + d*x), 2))/d + (2*b^2*sin(c + d*x))/(d*sqrt(cos(c + d*x)))]
    @test_int [cos(c + d*x)^(-1/2)*(a + b*sec(c + d*x))^2, x, 8, -((4*a*b*Elliptic.E((1/2)*(c + d*x), 2))/d) + (2*(3*a^2 + b^2)*Elliptic.F((1/2)*(c + d*x), 2))/(3*d) + (2*b^2*sin(c + d*x))/(3*d*cos(c + d*x)^(3/2)) + (4*a*b*sin(c + d*x))/(d*sqrt(cos(c + d*x)))]
    @test_int [cos(c + d*x)^(-3/2)*(a + b*sec(c + d*x))^2, x, 9, -((2*(5*a^2 + 3*b^2)*Elliptic.E((1/2)*(c + d*x), 2))/(5*d)) + (4*a*b*Elliptic.F((1/2)*(c + d*x), 2))/(3*d) + (2*b^2*sin(c + d*x))/(5*d*cos(c + d*x)^(5/2)) + (4*a*b*sin(c + d*x))/(3*d*cos(c + d*x)^(3/2)) + (2*(5*a^2 + 3*b^2)*sin(c + d*x))/(5*d*sqrt(cos(c + d*x)))]
    @test_int [cos(c + d*x)^(-5/2)*(a + b*sec(c + d*x))^2, x, 10, -((12*a*b*Elliptic.E((1/2)*(c + d*x), 2))/(5*d)) + (2*(7*a^2 + 5*b^2)*Elliptic.F((1/2)*(c + d*x), 2))/(21*d) + (2*b^2*sin(c + d*x))/(7*d*cos(c + d*x)^(7/2)) + (4*a*b*sin(c + d*x))/(5*d*cos(c + d*x)^(5/2)) + (2*(7*a^2 + 5*b^2)*sin(c + d*x))/(21*d*cos(c + d*x)^(3/2)) + (12*a*b*sin(c + d*x))/(5*d*sqrt(cos(c + d*x)))]


    @test_int [cos(c + d*x)^(9/2)*(a + b*sec(c + d*x))^3, x, 10, (2*a*(7*a^2 + 27*b^2)*Elliptic.E((1/2)*(c + d*x), 2))/(15*d) + (2*b*(15*a^2 + 7*b^2)*Elliptic.F((1/2)*(c + d*x), 2))/(21*d) + (2*b*(15*a^2 + 7*b^2)*sqrt(cos(c + d*x))*sin(c + d*x))/(21*d) + (2*a*(7*a^2 + 27*b^2)*cos(c + d*x)^(3/2)*sin(c + d*x))/(45*d) + (40*a^2*b*cos(c + d*x)^(5/2)*sin(c + d*x))/(63*d) + (2*a^2*cos(c + d*x)^(7/2)*(a + b*sec(c + d*x))*sin(c + d*x))/(9*d)]
    @test_int [cos(c + d*x)^(7/2)*(a + b*sec(c + d*x))^3, x, 9, (2*b*(9*a^2 + 5*b^2)*Elliptic.E((1/2)*(c + d*x), 2))/(5*d) + (2*a*(5*a^2 + 21*b^2)*Elliptic.F((1/2)*(c + d*x), 2))/(21*d) + (2*a*(5*a^2 + 21*b^2)*sqrt(cos(c + d*x))*sin(c + d*x))/(21*d) + (32*a^2*b*cos(c + d*x)^(3/2)*sin(c + d*x))/(35*d) + (2*a^2*cos(c + d*x)^(5/2)*(a + b*sec(c + d*x))*sin(c + d*x))/(7*d)]
    @test_int [cos(c + d*x)^(5/2)*(a + b*sec(c + d*x))^3, x, 8, (6*a*(a^2 + 5*b^2)*Elliptic.E((1/2)*(c + d*x), 2))/(5*d) + (2*b*(a^2 + b^2)*Elliptic.F((1/2)*(c + d*x), 2))/d + (8*a^2*b*sqrt(cos(c + d*x))*sin(c + d*x))/(5*d) + (2*a^2*cos(c + d*x)^(3/2)*(a + b*sec(c + d*x))*sin(c + d*x))/(5*d)]
    @test_int [cos(c + d*x)^(3/2)*(a + b*sec(c + d*x))^3, x, 8, (2*b*(3*a^2 - b^2)*Elliptic.E((1/2)*(c + d*x), 2))/d + (2*a*(a^2 + 9*b^2)*Elliptic.F((1/2)*(c + d*x), 2))/(3*d) - (2*b*(a^2 - 3*b^2)*sin(c + d*x))/(3*d*sqrt(cos(c + d*x))) + (2*a^2*sqrt(cos(c + d*x))*(a + b*sec(c + d*x))*sin(c + d*x))/(3*d)]
    @test_int [cos(c + d*x)^(1/2)*(a + b*sec(c + d*x))^3, x, 8, (2*a*(a^2 - 3*b^2)*Elliptic.E((1/2)*(c + d*x), 2))/d + (2*b*(9*a^2 + b^2)*Elliptic.F((1/2)*(c + d*x), 2))/(3*d) + (16*a*b^2*sin(c + d*x))/(3*d*sqrt(cos(c + d*x))) + (2*b^2*(a + b*sec(c + d*x))*sin(c + d*x))/(3*d*sqrt(cos(c + d*x)))]
    @test_int [cos(c + d*x)^(-1/2)*(a + b*sec(c + d*x))^3, x, 9, -((6*b*(5*a^2 + b^2)*Elliptic.E((1/2)*(c + d*x), 2))/(5*d)) + (2*a*(a^2 + b^2)*Elliptic.F((1/2)*(c + d*x), 2))/d + (8*a*b^2*sin(c + d*x))/(5*d*cos(c + d*x)^(3/2)) + (6*b*(5*a^2 + b^2)*sin(c + d*x))/(5*d*sqrt(cos(c + d*x))) + (2*b^2*(a + b*sec(c + d*x))*sin(c + d*x))/(5*d*cos(c + d*x)^(3/2))]
    @test_int [cos(c + d*x)^(-3/2)*(a + b*sec(c + d*x))^3, x, 10, -((2*a*(5*a^2 + 9*b^2)*Elliptic.E((1/2)*(c + d*x), 2))/(5*d)) + (2*b*(21*a^2 + 5*b^2)*Elliptic.F((1/2)*(c + d*x), 2))/(21*d) + (32*a*b^2*sin(c + d*x))/(35*d*cos(c + d*x)^(5/2)) + (2*b*(21*a^2 + 5*b^2)*sin(c + d*x))/(21*d*cos(c + d*x)^(3/2)) + (2*a*(5*a^2 + 9*b^2)*sin(c + d*x))/(5*d*sqrt(cos(c + d*x))) + (2*b^2*(a + b*sec(c + d*x))*sin(c + d*x))/(7*d*cos(c + d*x)^(5/2))]


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


    @test_int [cos(c + d*x)^(5/2)/(a + b*sec(c + d*x)), x, 11, (2*(3*a^2 + 5*b^2)*Elliptic.E((c + d*x)/2, 2))/(5*a^3*d) - (2*b*(a^2 + 3*b^2)*Elliptic.F((c + d*x)/2, 2))/(3*a^4*d) + (2*b^4*Elliptic.Pi((2*a)/(a + b), (c + d*x)/2, 2))/(a^4*(a + b)*d) - (2*b*sqrt(cos(c + d*x))*sin(c + d*x))/(3*a^2*d) + (2*cos(c + d*x)^(3/2)*sin(c + d*x))/(5*a*d)]
    @test_int [cos(c + d*x)^(3/2)/(a + b*sec(c + d*x)), x, 10, (-2*b*Elliptic.E((c + d*x)/2, 2))/(a^2*d) + (2*(a^2 + 3*b^2)*Elliptic.F((c + d*x)/2, 2))/(3*a^3*d) - (2*b^3*Elliptic.Pi((2*a)/(a + b), (c + d*x)/2, 2))/(a^3*(a + b)*d) + (2*sqrt(cos(c + d*x))*sin(c + d*x))/(3*a*d)]
    @test_int [sqrt(cos(c + d*x))/(a + b*sec(c + d*x)), x, 9, (2*Elliptic.E((c + d*x)/2, 2))/(a*d) - (2*b*Elliptic.F((c + d*x)/2, 2))/(a^2*d) + (2*b^2*Elliptic.Pi((2*a)/(a + b), (c + d*x)/2, 2))/(a^2*(a + b)*d)]
    @test_int [1/(sqrt(cos(c + d*x))*(a + b*sec(c + d*x))), x, 5, (2*Elliptic.F((c + d*x)/2, 2))/(a*d) - (2*b*Elliptic.Pi((2*a)/(a + b), (c + d*x)/2, 2))/(a*(a + b)*d)]
    @test_int [1/(cos(c + d*x)^(3/2)*(a + b*sec(c + d*x))), x, 3, (2*Elliptic.Pi((2*a)/(a + b), (c + d*x)/2, 2))/((a + b)*d)]
    @test_int [1/(cos(c + d*x)^(5/2)*(a + b*sec(c + d*x))), x, 7, (-2*Elliptic.E((c + d*x)/2, 2))/(b*d) - (2*a*Elliptic.Pi((2*a)/(a + b), (c + d*x)/2, 2))/(b*(a + b)*d) + (2*sin(c + d*x))/(b*d*sqrt(cos(c + d*x)))]
    @test_int [1/(cos(c + d*x)^(7/2)*(a + b*sec(c + d*x))), x, 11, (2*a*Elliptic.E((c + d*x)/2, 2))/(b^2*d) + (2*Elliptic.F((c + d*x)/2, 2))/(3*b*d) + (2*a^2*Elliptic.Pi((2*a)/(a + b), (c + d*x)/2, 2))/(b^2*(a + b)*d) + (2*sin(c + d*x))/(3*b*d*cos(c + d*x)^(3/2)) - (2*a*sin(c + d*x))/(b^2*d*sqrt(cos(c + d*x)))]


    @test_int [cos(c + d*x)^(3/2)/(a + b*sec(c + d*x))^2, x, 11, -((b*(4*a^2 - 5*b^2)*Elliptic.E((1/2)*(c + d*x), 2))/(a^3*(a^2 - b^2)*d)) + ((2*a^4 + 16*a^2*b^2 - 15*b^4)*Elliptic.F((1/2)*(c + d*x), 2))/(3*a^4*(a^2 - b^2)*d) - (b^3*(7*a^2 - 5*b^2)*Elliptic.Pi((2*a)/(a + b), (1/2)*(c + d*x), 2))/(a^4*(a - b)*(a + b)^2*d) + ((2*a^2 - 5*b^2)*sqrt(cos(c + d*x))*sin(c + d*x))/(3*a^2*(a^2 - b^2)*d) + (b^2*sqrt(cos(c + d*x))*sin(c + d*x))/(a*(a^2 - b^2)*d*(a + b*sec(c + d*x)))]
    @test_int [sqrt(cos(c + d*x))/(a + b*sec(c + d*x))^2, x, 10, ((2*a^2 - 3*b^2)*Elliptic.E((1/2)*(c + d*x), 2))/(a^2*(a^2 - b^2)*d) - (b*(4*a^2 - 3*b^2)*Elliptic.F((1/2)*(c + d*x), 2))/(a^3*(a^2 - b^2)*d) + (b^2*(5*a^2 - 3*b^2)*Elliptic.Pi((2*a)/(a + b), (1/2)*(c + d*x), 2))/(a^3*(a - b)*(a + b)^2*d) + (b^2*sin(c + d*x))/(a*(a^2 - b^2)*d*sqrt(cos(c + d*x))*(a + b*sec(c + d*x)))]
    @test_int [1/(sqrt(cos(c + d*x))*(a + b*sec(c + d*x))^2), x, 10, (b*Elliptic.E((1/2)*(c + d*x), 2))/(a*(a^2 - b^2)*d) + ((2*a^2 - b^2)*Elliptic.F((1/2)*(c + d*x), 2))/(a^2*(a^2 - b^2)*d) - (b*(3*a^2 - b^2)*Elliptic.Pi((2*a)/(a + b), (1/2)*(c + d*x), 2))/(a^2*(a - b)*(a + b)^2*d) - (b*sin(c + d*x))/((a^2 - b^2)*d*sqrt(cos(c + d*x))*(a + b*sec(c + d*x)))]
    @test_int [1/(cos(c + d*x)^(3/2)*(a + b*sec(c + d*x))^2), x, 10, -(Elliptic.E((1/2)*(c + d*x), 2)/((a^2 - b^2)*d)) - (b*Elliptic.F((1/2)*(c + d*x), 2))/(a*(a^2 - b^2)*d) + ((a^2 + b^2)*Elliptic.Pi((2*a)/(a + b), (1/2)*(c + d*x), 2))/(a*(a - b)*(a + b)^2*d) + (a*sin(c + d*x))/((a^2 - b^2)*d*sqrt(cos(c + d*x))*(a + b*sec(c + d*x)))]
    @test_int [1/(cos(c + d*x)^(5/2)*(a + b*sec(c + d*x))^2), x, 10, (a*Elliptic.E((1/2)*(c + d*x), 2))/(b*(a^2 - b^2)*d) + Elliptic.F((1/2)*(c + d*x), 2)/((a^2 - b^2)*d) + ((a^2 - 3*b^2)*Elliptic.Pi((2*a)/(a + b), (1/2)*(c + d*x), 2))/((a - b)*b*(a + b)^2*d) - (a^2*sin(c + d*x))/(b*(a^2 - b^2)*d*sqrt(cos(c + d*x))*(a + b*sec(c + d*x)))]
    @test_int [1/(cos(c + d*x)^(7/2)*(a + b*sec(c + d*x))^2), x, 11, -(((3*a^2 - 2*b^2)*Elliptic.E((1/2)*(c + d*x), 2))/(b^2*(a^2 - b^2)*d)) - (a*Elliptic.F((1/2)*(c + d*x), 2))/(b*(a^2 - b^2)*d) - (a*(3*a^2 - 5*b^2)*Elliptic.Pi((2*a)/(a + b), (1/2)*(c + d*x), 2))/((a - b)*b^2*(a + b)^2*d) + ((3*a^2 - 2*b^2)*sin(c + d*x))/(b^2*(a^2 - b^2)*d*sqrt(cos(c + d*x))) - (a^2*sin(c + d*x))/(b*(a^2 - b^2)*d*cos(c + d*x)^(3/2)*(a + b*sec(c + d*x)))]


    @test_int [cos(c + d*x)^(3/2)/(a + b*sec(c + d*x))^3, x, 12, -((b*(24*a^4 - 65*a^2*b^2 + 35*b^4)*Elliptic.E((1/2)*(c + d*x), 2))/(4*a^4*(a^2 - b^2)^2*d)) + ((8*a^6 + 128*a^4*b^2 - 223*a^2*b^4 + 105*b^6)*Elliptic.F((1/2)*(c + d*x), 2))/(12*a^5*(a^2 - b^2)^2*d) - (b^3*(63*a^4 - 86*a^2*b^2 + 35*b^4)*Elliptic.Pi((2*a)/(a + b), (1/2)*(c + d*x), 2))/(4*a^5*(a - b)^2*(a + b)^3*d) + ((8*a^4 - 61*a^2*b^2 + 35*b^4)*sqrt(cos(c + d*x))*sin(c + d*x))/(12*a^3*(a^2 - b^2)^2*d) + (b^2*sqrt(cos(c + d*x))*sin(c + d*x))/(2*a*(a^2 - b^2)*d*(a + b*sec(c + d*x))^2) + (b^2*(13*a^2 - 7*b^2)*sqrt(cos(c + d*x))*sin(c + d*x))/(4*a^2*(a^2 - b^2)^2*d*(a + b*sec(c + d*x)))]
    @test_int [sqrt(cos(c + d*x))/(a + b*sec(c + d*x))^3, x, 11, ((8*a^4 - 29*a^2*b^2 + 15*b^4)*Elliptic.E((1/2)*(c + d*x), 2))/(4*a^3*(a^2 - b^2)^2*d) - (3*b*(8*a^4 - 11*a^2*b^2 + 5*b^4)*Elliptic.F((1/2)*(c + d*x), 2))/(4*a^4*(a^2 - b^2)^2*d) + (b^2*(35*a^4 - 38*a^2*b^2 + 15*b^4)*Elliptic.Pi((2*a)/(a + b), (1/2)*(c + d*x), 2))/(4*a^4*(a - b)^2*(a + b)^3*d) + (b^2*sin(c + d*x))/(2*a*(a^2 - b^2)*d*sqrt(cos(c + d*x))*(a + b*sec(c + d*x))^2) + (b^2*(11*a^2 - 5*b^2)*sin(c + d*x))/(4*a^2*(a^2 - b^2)^2*d*sqrt(cos(c + d*x))*(a + b*sec(c + d*x)))]
    @test_int [1/(sqrt(cos(c + d*x))*(a + b*sec(c + d*x))^3), x, 11, (3*b*(3*a^2 - b^2)*Elliptic.E((1/2)*(c + d*x), 2))/(4*a^2*(a^2 - b^2)^2*d) + ((8*a^4 - 5*a^2*b^2 + 3*b^4)*Elliptic.F((1/2)*(c + d*x), 2))/(4*a^3*(a^2 - b^2)^2*d) - (3*b*(5*a^4 - 2*a^2*b^2 + b^4)*Elliptic.Pi((2*a)/(a + b), (1/2)*(c + d*x), 2))/(4*a^3*(a - b)^2*(a + b)^3*d) - (b*sin(c + d*x))/(2*(a^2 - b^2)*d*sqrt(cos(c + d*x))*(a + b*sec(c + d*x))^2) - (b*(7*a^2 - b^2)*sin(c + d*x))/(4*a*(a^2 - b^2)^2*d*sqrt(cos(c + d*x))*(a + b*sec(c + d*x)))]
    @test_int [1/(cos(c + d*x)^(3/2)*(a + b*sec(c + d*x))^3), x, 11, -(((5*a^2 + b^2)*Elliptic.E((1/2)*(c + d*x), 2))/(4*a*(a^2 - b^2)^2*d)) - (b*(7*a^2 - b^2)*Elliptic.F((1/2)*(c + d*x), 2))/(4*a^2*(a^2 - b^2)^2*d) + ((3*a^4 + 10*a^2*b^2 - b^4)*Elliptic.Pi((2*a)/(a + b), (1/2)*(c + d*x), 2))/(4*a^2*(a - b)^2*(a + b)^3*d) + (a*sin(c + d*x))/(2*(a^2 - b^2)*d*sqrt(cos(c + d*x))*(a + b*sec(c + d*x))^2) + (3*(a^2 + b^2)*sin(c + d*x))/(4*(a^2 - b^2)^2*d*sqrt(cos(c + d*x))*(a + b*sec(c + d*x)))]
    @test_int [1/(cos(c + d*x)^(5/2)*(a + b*sec(c + d*x))^3), x, 11, ((a^2 + 5*b^2)*Elliptic.E((1/2)*(c + d*x), 2))/(4*b*(a^2 - b^2)^2*d) + (3*(a^2 + b^2)*Elliptic.F((1/2)*(c + d*x), 2))/(4*a*(a^2 - b^2)^2*d) + ((a^4 - 10*a^2*b^2 - 3*b^4)*Elliptic.Pi((2*a)/(a + b), (1/2)*(c + d*x), 2))/(4*a*(a - b)^2*b*(a + b)^3*d) - (a^2*sin(c + d*x))/(2*b*(a^2 - b^2)*d*sqrt(cos(c + d*x))*(a + b*sec(c + d*x))^2) + (a*(a^2 - 7*b^2)*sin(c + d*x))/(4*b*(a^2 - b^2)^2*d*sqrt(cos(c + d*x))*(a + b*sec(c + d*x)))]
    @test_int [1/(cos(c + d*x)^(7/2)*(a + b*sec(c + d*x))^3), x, 11, (3*a*(a^2 - 3*b^2)*Elliptic.E((1/2)*(c + d*x), 2))/(4*b^2*(a^2 - b^2)^2*d) + ((a^2 - 7*b^2)*Elliptic.F((1/2)*(c + d*x), 2))/(4*b*(a^2 - b^2)^2*d) + (3*(a^4 - 2*a^2*b^2 + 5*b^4)*Elliptic.Pi((2*a)/(a + b), (1/2)*(c + d*x), 2))/(4*(a - b)^2*b^2*(a + b)^3*d) - (a^2*sin(c + d*x))/(2*b*(a^2 - b^2)*d*cos(c + d*x)^(3/2)*(a + b*sec(c + d*x))^2) - (3*a^2*(a^2 - 3*b^2)*sin(c + d*x))/(4*b^2*(a^2 - b^2)^2*d*sqrt(cos(c + d*x))*(a + b*sec(c + d*x)))]
    @test_int [1/(cos(c + d*x)^(9/2)*(a + b*sec(c + d*x))^3), x, 12, -(((15*a^4 - 29*a^2*b^2 + 8*b^4)*Elliptic.E((1/2)*(c + d*x), 2))/(4*b^3*(a^2 - b^2)^2*d)) - (a*(5*a^2 - 11*b^2)*Elliptic.F((1/2)*(c + d*x), 2))/(4*b^2*(a^2 - b^2)^2*d) - (a*(15*a^4 - 38*a^2*b^2 + 35*b^4)*Elliptic.Pi((2*a)/(a + b), (1/2)*(c + d*x), 2))/(4*(a - b)^2*b^3*(a + b)^3*d) + ((15*a^4 - 29*a^2*b^2 + 8*b^4)*sin(c + d*x))/(4*b^3*(a^2 - b^2)^2*d*sqrt(cos(c + d*x))) - (a^2*sin(c + d*x))/(2*b*(a^2 - b^2)*d*cos(c + d*x)^(5/2)*(a + b*sec(c + d*x))^2) - (a^2*(5*a^2 - 11*b^2)*sin(c + d*x))/(4*b^2*(a^2 - b^2)^2*d*cos(c + d*x)^(3/2)*(a + b*sec(c + d*x)))]


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


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


    @test_int [cos(c + d*x)^(5/2)*sqrt(a + b*sec(c + d*x)), x, 10, (-4*b*(a^2 - b^2)*sqrt((b + a*cos(c + d*x))/(a + b))*Elliptic.F((c + d*x)/2, (2*a)/(a + b)))/(15*a^2*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + (2*(9*a^2 - 2*b^2)*sqrt(cos(c + d*x))*Elliptic.E((c + d*x)/2, (2*a)/(a + b))*sqrt(a + b*sec(c + d*x)))/(15*a^2*d*sqrt((b + a*cos(c + d*x))/(a + b))) + (2*b*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))*sin(c + d*x))/(15*a*d) + (2*cos(c + d*x)^(3/2)*sqrt(a + b*sec(c + d*x))*sin(c + d*x))/(5*d)]
    @test_int [cos(c + d*x)^(3/2)*sqrt(a + b*sec(c + d*x)), x, 9, (2*(a^2 - b^2)*sqrt((b + a*cos(c + d*x))/(a + b))*Elliptic.F((c + d*x)/2, (2*a)/(a + b)))/(3*a*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + (2*b*sqrt(cos(c + d*x))*Elliptic.E((c + d*x)/2, (2*a)/(a + b))*sqrt(a + b*sec(c + d*x)))/(3*a*d*sqrt((b + a*cos(c + d*x))/(a + b))) + (2*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))*sin(c + d*x))/(3*d)]
    @test_int [cos(c + d*x)^(1/2)*sqrt(a + b*sec(c + d*x)), x, 4, (2*sqrt(cos(c + d*x))*Elliptic.E((c + d*x)/2, (2*a)/(a + b))*sqrt(a + b*sec(c + d*x)))/(d*sqrt((b + a*cos(c + d*x))/(a + b)))]
    @test_int [sqrt(a + b*sec(c + d*x))/cos(c + d*x)^(1/2), x, 8, (2*a*sqrt((b + a*cos(c + d*x))/(a + b))*Elliptic.F((c + d*x)/2, (2*a)/(a + b)))/(d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + (2*b*sqrt((b + a*cos(c + d*x))/(a + b))*Elliptic.Pi(2, (c + d*x)/2, (2*a)/(a + b)))/(d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x)))]
    @test_int [sqrt(a + b*sec(c + d*x))/cos(c + d*x)^(3/2), x, 13, (b*sqrt((b + a*cos(c + d*x))/(a + b))*Elliptic.F((c + d*x)/2, (2*a)/(a + b)))/(d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + (a*sqrt((b + a*cos(c + d*x))/(a + b))*Elliptic.Pi(2, (c + d*x)/2, (2*a)/(a + b)))/(d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) - (sqrt(cos(c + d*x))*Elliptic.E((c + d*x)/2, (2*a)/(a + b))*sqrt(a + b*sec(c + d*x)))/(d*sqrt((b + a*cos(c + d*x))/(a + b))) + (sqrt(a + b*sec(c + d*x))*sin(c + d*x))/(d*sqrt(cos(c + d*x)))]


    @test_int [cos(c + d*x)^(7/2)*(a + b*sec(c + d*x))^(3/2), x, 11, (2*(25*a^4 - 31*a^2*b^2 + 6*b^4)*sqrt((b + a*cos(c + d*x))/(a + b))*Elliptic.F((c + d*x)/2, (2*a)/(a + b)))/(105*a^2*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + (4*b*(41*a^2 - 3*b^2)*sqrt(cos(c + d*x))*Elliptic.E((c + d*x)/2, (2*a)/(a + b))*sqrt(a + b*sec(c + d*x)))/(105*a^2*d*sqrt((b + a*cos(c + d*x))/(a + b))) + (2*(25*a^2 + 3*b^2)*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))*sin(c + d*x))/(105*a*d) + (16*b*cos(c + d*x)^(3/2)*sqrt(a + b*sec(c + d*x))*sin(c + d*x))/(35*d) + (2*a*cos(c + d*x)^(5/2)*sqrt(a + b*sec(c + d*x))*sin(c + d*x))/(7*d)]
    @test_int [cos(c + d*x)^(5/2)*(a + b*sec(c + d*x))^(3/2), x, 10, (2*b*(a^2 - b^2)*sqrt((b + a*cos(c + d*x))/(a + b))*Elliptic.F((1/2)*(c + d*x), (2*a)/(a + b)))/(5*a*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + (2*(3*a^2 + b^2)*sqrt(cos(c + d*x))*Elliptic.E((1/2)*(c + d*x), (2*a)/(a + b))*sqrt(a + b*sec(c + d*x)))/(5*a*d*sqrt((b + a*cos(c + d*x))/(a + b))) + (4*b*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))*sin(c + d*x))/(5*d) + (2*a*cos(c + d*x)^(3/2)*sqrt(a + b*sec(c + d*x))*sin(c + d*x))/(5*d)]
    @test_int [cos(c + d*x)^(3/2)*(a + b*sec(c + d*x))^(3/2), x, 9, (2*(a^2 - b^2)*sqrt((b + a*cos(c + d*x))/(a + b))*Elliptic.F((c + d*x)/2, (2*a)/(a + b)))/(3*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + (8*b*sqrt(cos(c + d*x))*Elliptic.E((c + d*x)/2, (2*a)/(a + b))*sqrt(a + b*sec(c + d*x)))/(3*d*sqrt((b + a*cos(c + d*x))/(a + b))) + (2*a*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))*sin(c + d*x))/(3*d)]
    @test_int [cos(c + d*x)^(1/2)*(a + b*sec(c + d*x))^(3/2), x, 12, (2*a*b*sqrt((b + a*cos(c + d*x))/(a + b))*Elliptic.F((c + d*x)/2, (2*a)/(a + b)))/(d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + (2*b^2*sqrt((b + a*cos(c + d*x))/(a + b))*Elliptic.Pi(2, (c + d*x)/2, (2*a)/(a + b)))/(d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + (2*a*sqrt(cos(c + d*x))*Elliptic.E((c + d*x)/2, (2*a)/(a + b))*sqrt(a + b*sec(c + d*x)))/(d*sqrt((b + a*cos(c + d*x))/(a + b)))]
    @test_int [(a + b*sec(c + d*x))^(3/2)/cos(c + d*x)^(1/2), x, 13, ((2*a^2 + b^2)*sqrt((b + a*cos(c + d*x))/(a + b))*Elliptic.F((c + d*x)/2, (2*a)/(a + b)))/(d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + (3*a*b*sqrt((b + a*cos(c + d*x))/(a + b))*Elliptic.Pi(2, (c + d*x)/2, (2*a)/(a + b)))/(d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) - (b*sqrt(cos(c + d*x))*Elliptic.E((c + d*x)/2, (2*a)/(a + b))*sqrt(a + b*sec(c + d*x)))/(d*sqrt((b + a*cos(c + d*x))/(a + b))) + (b*sqrt(a + b*sec(c + d*x))*sin(c + d*x))/(d*sqrt(cos(c + d*x)))]
    @test_int [(a + b*sec(c + d*x))^(3/2)/cos(c + d*x)^(3/2), x, 14, (7*a*b*sqrt((b + a*cos(c + d*x))/(a + b))*Elliptic.F((c + d*x)/2, (2*a)/(a + b)))/(4*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + ((3*a^2 + 4*b^2)*sqrt((b + a*cos(c + d*x))/(a + b))*Elliptic.Pi(2, (c + d*x)/2, (2*a)/(a + b)))/(4*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) - (5*a*sqrt(cos(c + d*x))*Elliptic.E((c + d*x)/2, (2*a)/(a + b))*sqrt(a + b*sec(c + d*x)))/(4*d*sqrt((b + a*cos(c + d*x))/(a + b))) + (b*sqrt(a + b*sec(c + d*x))*sin(c + d*x))/(2*d*cos(c + d*x)^(3/2)) + (5*a*sqrt(a + b*sec(c + d*x))*sin(c + d*x))/(4*d*sqrt(cos(c + d*x)))]


    @test_int [cos(c + d*x)^(9/2)*(a + b*sec(c + d*x))^(5/2), x, 12, (4*b*(57*a^4 - 62*a^2*b^2 + 5*b^4)*sqrt((b + a*cos(c + d*x))/(a + b))*Elliptic.F((c + d*x)/2, (2*a)/(a + b)))/(315*a^2*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + (2*(147*a^4 + 279*a^2*b^2 - 10*b^4)*sqrt(cos(c + d*x))*Elliptic.E((c + d*x)/2, (2*a)/(a + b))*sqrt(a + b*sec(c + d*x)))/(315*a^2*d*sqrt((b + a*cos(c + d*x))/(a + b))) + (2*b*(163*a^2 + 5*b^2)*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))*sin(c + d*x))/(315*a*d) + (2*(49*a^2 + 75*b^2)*cos(c + d*x)^(3/2)*sqrt(a + b*sec(c + d*x))*sin(c + d*x))/(315*d) + (38*a*b*cos(c + d*x)^(5/2)*sqrt(a + b*sec(c + d*x))*sin(c + d*x))/(63*d) + (2*a^2*cos(c + d*x)^(7/2)*sqrt(a + b*sec(c + d*x))*sin(c + d*x))/(9*d)]
    @test_int [cos(c + d*x)^(7/2)*(a + b*sec(c + d*x))^(5/2), x, 11, (2*(5*a^4 - 2*a^2*b^2 - 3*b^4)*sqrt((b + a*cos(c + d*x))/(a + b))*Elliptic.F((c + d*x)/2, (2*a)/(a + b)))/(21*a*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + (2*b*(29*a^2 + 3*b^2)*sqrt(cos(c + d*x))*Elliptic.E((c + d*x)/2, (2*a)/(a + b))*sqrt(a + b*sec(c + d*x)))/(21*a*d*sqrt((b + a*cos(c + d*x))/(a + b))) + (2*(5*a^2 + 9*b^2)*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))*sin(c + d*x))/(21*d) + (6*a*b*cos(c + d*x)^(3/2)*sqrt(a + b*sec(c + d*x))*sin(c + d*x))/(7*d) + (2*a^2*cos(c + d*x)^(5/2)*sqrt(a + b*sec(c + d*x))*sin(c + d*x))/(7*d)]
    @test_int [cos(c + d*x)^(5/2)*(a + b*sec(c + d*x))^(5/2), x, 10, (16*b*(a^2 - b^2)*sqrt((b + a*cos(c + d*x))/(a + b))*Elliptic.F((c + d*x)/2, (2*a)/(a + b)))/(15*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + (2*(9*a^2 + 23*b^2)*sqrt(cos(c + d*x))*Elliptic.E((c + d*x)/2, (2*a)/(a + b))*sqrt(a + b*sec(c + d*x)))/(15*d*sqrt((b + a*cos(c + d*x))/(a + b))) + (22*a*b*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))*sin(c + d*x))/(15*d) + (2*a^2*cos(c + d*x)^(3/2)*sqrt(a + b*sec(c + d*x))*sin(c + d*x))/(5*d)]
    @test_int [cos(c + d*x)^(3/2)*(a + b*sec(c + d*x))^(5/2), x, 13, (2*a*(a^2 + 2*b^2)*sqrt((b + a*cos(c + d*x))/(a + b))*Elliptic.F((c + d*x)/2, (2*a)/(a + b)))/(3*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + (2*b^3*sqrt((b + a*cos(c + d*x))/(a + b))*Elliptic.Pi(2, (c + d*x)/2, (2*a)/(a + b)))/(d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + (14*a*b*sqrt(cos(c + d*x))*Elliptic.E((c + d*x)/2, (2*a)/(a + b))*sqrt(a + b*sec(c + d*x)))/(3*d*sqrt((b + a*cos(c + d*x))/(a + b))) + (2*a^2*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))*sin(c + d*x))/(3*d)]
    @test_int [cos(c + d*x)^(1/2)*(a + b*sec(c + d*x))^(5/2), x, 13, (b*(4*a^2 + b^2)*sqrt((b + a*cos(c + d*x))/(a + b))*Elliptic.F((c + d*x)/2, (2*a)/(a + b)))/(d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + (5*a*b^2*sqrt((b + a*cos(c + d*x))/(a + b))*Elliptic.Pi(2, (c + d*x)/2, (2*a)/(a + b)))/(d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + ((2*a^2 - b^2)*sqrt(cos(c + d*x))*Elliptic.E((c + d*x)/2, (2*a)/(a + b))*sqrt(a + b*sec(c + d*x)))/(d*sqrt((b + a*cos(c + d*x))/(a + b))) + (b^2*sqrt(a + b*sec(c + d*x))*sin(c + d*x))/(d*sqrt(cos(c + d*x)))]
    @test_int [(a + b*sec(c + d*x))^(5/2)/cos(c + d*x)^(1/2), x, 14, (a*(8*a^2 + 11*b^2)*sqrt((b + a*cos(c + d*x))/(a + b))*Elliptic.F((c + d*x)/2, (2*a)/(a + b)))/(4*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + (b*(15*a^2 + 4*b^2)*sqrt((b + a*cos(c + d*x))/(a + b))*Elliptic.Pi(2, (c + d*x)/2, (2*a)/(a + b)))/(4*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) - (9*a*b*sqrt(cos(c + d*x))*Elliptic.E((c + d*x)/2, (2*a)/(a + b))*sqrt(a + b*sec(c + d*x)))/(4*d*sqrt((b + a*cos(c + d*x))/(a + b))) + (b^2*sqrt(a + b*sec(c + d*x))*sin(c + d*x))/(2*d*cos(c + d*x)^(3/2)) + (9*a*b*sqrt(a + b*sec(c + d*x))*sin(c + d*x))/(4*d*sqrt(cos(c + d*x)))]
    @test_int [(a + b*sec(c + d*x))^(5/2)/cos(c + d*x)^(3/2), x, 15, (b*(59*a^2 + 16*b^2)*sqrt((b + a*cos(c + d*x))/(a + b))*Elliptic.F((c + d*x)/2, (2*a)/(a + b)))/(24*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + (5*a*(a^2 + 4*b^2)*sqrt((b + a*cos(c + d*x))/(a + b))*Elliptic.Pi(2, (c + d*x)/2, (2*a)/(a + b)))/(8*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) - ((33*a^2 + 16*b^2)*sqrt(cos(c + d*x))*Elliptic.E((c + d*x)/2, (2*a)/(a + b))*sqrt(a + b*sec(c + d*x)))/(24*d*sqrt((b + a*cos(c + d*x))/(a + b))) + (b^2*sqrt(a + b*sec(c + d*x))*sin(c + d*x))/(3*d*cos(c + d*x)^(5/2)) + (13*a*b*sqrt(a + b*sec(c + d*x))*sin(c + d*x))/(12*d*cos(c + d*x)^(3/2)) + ((33*a^2 + 16*b^2)*sqrt(a + b*sec(c + d*x))*sin(c + d*x))/(24*d*sqrt(cos(c + d*x)))]


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


    @test_int [cos(c + d*x)^(5/2)/sqrt(a + b*sec(c + d*x)), x, 10, (-2*b*(7*a^2 + 8*b^2)*sqrt((b + a*cos(c + d*x))/(a + b))*Elliptic.F((c + d*x)/2, (2*a)/(a + b)))/(15*a^3*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + (2*(9*a^2 + 8*b^2)*sqrt(cos(c + d*x))*Elliptic.E((c + d*x)/2, (2*a)/(a + b))*sqrt(a + b*sec(c + d*x)))/(15*a^3*d*sqrt((b + a*cos(c + d*x))/(a + b))) - (8*b*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))*sin(c + d*x))/(15*a^2*d) + (2*cos(c + d*x)^(3/2)*sqrt(a + b*sec(c + d*x))*sin(c + d*x))/(5*a*d)]
    @test_int [cos(c + d*x)^(3/2)/sqrt(a + b*sec(c + d*x)), x, 9, (2*(a^2 + 2*b^2)*sqrt((b + a*cos(c + d*x))/(a + b))*Elliptic.F((c + d*x)/2, (2*a)/(a + b)))/(3*a^2*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) - (4*b*sqrt(cos(c + d*x))*Elliptic.E((c + d*x)/2, (2*a)/(a + b))*sqrt(a + b*sec(c + d*x)))/(3*a^2*d*sqrt((b + a*cos(c + d*x))/(a + b))) + (2*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))*sin(c + d*x))/(3*a*d)]
    @test_int [cos(c + d*x)^(1/2)/sqrt(a + b*sec(c + d*x)), x, 8, (-2*b*sqrt((b + a*cos(c + d*x))/(a + b))*Elliptic.F((c + d*x)/2, (2*a)/(a + b)))/(a*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + (2*sqrt(cos(c + d*x))*Elliptic.E((c + d*x)/2, (2*a)/(a + b))*sqrt(a + b*sec(c + d*x)))/(a*d*sqrt((b + a*cos(c + d*x))/(a + b)))]
    @test_int [1/(cos(c + d*x)^(1/2)*sqrt(a + b*sec(c + d*x))), x, 4, (2*sqrt((b + a*cos(c + d*x))/(a + b))*Elliptic.F((c + d*x)/2, (2*a)/(a + b)))/(d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x)))]
    @test_int [1/(cos(c + d*x)^(3/2)*sqrt(a + b*sec(c + d*x))), x, 4, (2*sqrt((b + a*cos(c + d*x))/(a + b))*Elliptic.Pi(2, (c + d*x)/2, (2*a)/(a + b)))/(d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x)))]
    @test_int [1/(cos(c + d*x)^(5/2)*sqrt(a + b*sec(c + d*x))), x, 13, (sqrt((b + a*cos(c + d*x))/(a + b))*Elliptic.F((c + d*x)/2, (2*a)/(a + b)))/(d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) - (a*sqrt((b + a*cos(c + d*x))/(a + b))*Elliptic.Pi(2, (c + d*x)/2, (2*a)/(a + b)))/(b*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) - (sqrt(cos(c + d*x))*Elliptic.E((c + d*x)/2, (2*a)/(a + b))*sqrt(a + b*sec(c + d*x)))/(b*d*sqrt((b + a*cos(c + d*x))/(a + b))) + (sqrt(a + b*sec(c + d*x))*sin(c + d*x))/(b*d*sqrt(cos(c + d*x)))]
    @test_int [1/(cos(c + d*x)^(7/2)*sqrt(a + b*sec(c + d*x))), x, 14, -(a*sqrt((b + a*cos(c + d*x))/(a + b))*Elliptic.F((c + d*x)/2, (2*a)/(a + b)))/(4*b*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + ((3*a^2 + 4*b^2)*sqrt((b + a*cos(c + d*x))/(a + b))*Elliptic.Pi(2, (c + d*x)/2, (2*a)/(a + b)))/(4*b^2*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + (3*a*sqrt(cos(c + d*x))*Elliptic.E((c + d*x)/2, (2*a)/(a + b))*sqrt(a + b*sec(c + d*x)))/(4*b^2*d*sqrt((b + a*cos(c + d*x))/(a + b))) + (sqrt(a + b*sec(c + d*x))*sin(c + d*x))/(2*b*d*cos(c + d*x)^(3/2)) - (3*a*sqrt(a + b*sec(c + d*x))*sin(c + d*x))/(4*b^2*d*sqrt(cos(c + d*x)))]


    @test_int [cos(c + d*x)^(5/2)/(a + b*sec(c + d*x))^(3/2), x, 11, -((8*b*(a^2 + 4*b^2)*sqrt((b + a*cos(c + d*x))/(a + b))*Elliptic.F((1/2)*(c + d*x), (2*a)/(a + b)))/(5*a^4*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x)))) + (2*(3*a^4 + 8*a^2*b^2 - 16*b^4)*sqrt(cos(c + d*x))*Elliptic.E((1/2)*(c + d*x), (2*a)/(a + b))*sqrt(a + b*sec(c + d*x)))/(5*a^4*(a^2 - b^2)*d*sqrt((b + a*cos(c + d*x))/(a + b))) + (2*b^2*cos(c + d*x)^(3/2)*sin(c + d*x))/(a*(a^2 - b^2)*d*sqrt(a + b*sec(c + d*x))) - (2*b*(3*a^2 - 8*b^2)*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))*sin(c + d*x))/(5*a^3*(a^2 - b^2)*d) + (2*(a^2 - 6*b^2)*cos(c + d*x)^(3/2)*sqrt(a + b*sec(c + d*x))*sin(c + d*x))/(5*a^2*(a^2 - b^2)*d)]
    @test_int [cos(c + d*x)^(3/2)/(a + b*sec(c + d*x))^(3/2), x, 10, (2*(a^2 + 8*b^2)*sqrt((b + a*cos(c + d*x))/(a + b))*Elliptic.F((1/2)*(c + d*x), (2*a)/(a + b)))/(3*a^3*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) - (2*b*(5*a^2 - 8*b^2)*sqrt(cos(c + d*x))*Elliptic.E((1/2)*(c + d*x), (2*a)/(a + b))*sqrt(a + b*sec(c + d*x)))/(3*a^3*(a^2 - b^2)*d*sqrt((b + a*cos(c + d*x))/(a + b))) + (2*b^2*sqrt(cos(c + d*x))*sin(c + d*x))/(a*(a^2 - b^2)*d*sqrt(a + b*sec(c + d*x))) + (2*(a^2 - 4*b^2)*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))*sin(c + d*x))/(3*a^2*(a^2 - b^2)*d)]
    @test_int [cos(c + d*x)^(1/2)/(a + b*sec(c + d*x))^(3/2), x, 9, (-4*b*sqrt((b + a*cos(c + d*x))/(a + b))*Elliptic.F((c + d*x)/2, (2*a)/(a + b)))/(a^2*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + (2*(a^2 - 2*b^2)*sqrt(cos(c + d*x))*Elliptic.E((c + d*x)/2, (2*a)/(a + b))*sqrt(a + b*sec(c + d*x)))/(a^2*(a^2 - b^2)*d*sqrt((b + a*cos(c + d*x))/(a + b))) + (2*b^2*sin(c + d*x))/(a*(a^2 - b^2)*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x)))]
    @test_int [1/(cos(c + d*x)^(1/2)*(a + b*sec(c + d*x))^(3/2)), x, 9, (2*sqrt((b + a*cos(c + d*x))/(a + b))*Elliptic.F((c + d*x)/2, (2*a)/(a + b)))/(a*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + (2*b*sqrt(cos(c + d*x))*Elliptic.E((c + d*x)/2, (2*a)/(a + b))*sqrt(a + b*sec(c + d*x)))/(a*(a^2 - b^2)*d*sqrt((b + a*cos(c + d*x))/(a + b))) - (2*b*sin(c + d*x))/((a^2 - b^2)*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x)))]
    @test_int [1/(cos(c + d*x)^(3/2)*(a + b*sec(c + d*x))^(3/2)), x, 6, (-2*sqrt(cos(c + d*x))*Elliptic.E((c + d*x)/2, (2*a)/(a + b))*sqrt(a + b*sec(c + d*x)))/((a^2 - b^2)*d*sqrt((b + a*cos(c + d*x))/(a + b))) + (2*a*sin(c + d*x))/((a^2 - b^2)*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x)))]
    @test_int [1/(cos(c + d*x)^(5/2)*(a + b*sec(c + d*x))^(3/2)), x, 10, (2*sqrt((b + a*cos(c + d*x))/(a + b))*Elliptic.Pi(2, (c + d*x)/2, (2*a)/(a + b)))/(b*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + (2*a*sqrt(cos(c + d*x))*Elliptic.E((c + d*x)/2, (2*a)/(a + b))*sqrt(a + b*sec(c + d*x)))/(b*(a^2 - b^2)*d*sqrt((b + a*cos(c + d*x))/(a + b))) - (2*a^2*sin(c + d*x))/(b*(a^2 - b^2)*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x)))]
    @test_int [1/(cos(c + d*x)^(7/2)*(a + b*sec(c + d*x))^(3/2)), x, 14, (sqrt((b + a*cos(c + d*x))/(a + b))*Elliptic.F((c + d*x)/2, (2*a)/(a + b)))/(b*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) - (3*a*sqrt((b + a*cos(c + d*x))/(a + b))*Elliptic.Pi(2, (c + d*x)/2, (2*a)/(a + b)))/(b^2*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) - ((3*a^2 - b^2)*sqrt(cos(c + d*x))*Elliptic.E((c + d*x)/2, (2*a)/(a + b))*sqrt(a + b*sec(c + d*x)))/(b^2*(a^2 - b^2)*d*sqrt((b + a*cos(c + d*x))/(a + b))) - (2*a^2*sin(c + d*x))/(b*(a^2 - b^2)*d*cos(c + d*x)^(3/2)*sqrt(a + b*sec(c + d*x))) + ((3*a^2 - b^2)*sqrt(a + b*sec(c + d*x))*sin(c + d*x))/(b^2*(a^2 - b^2)*d*sqrt(cos(c + d*x)))]


    @test_int [cos(c + d*x)^(3/2)/(a + b*sec(c + d*x))^(5/2), x, 11, (2*(a^4 + 16*a^2*b^2 - 16*b^4)*sqrt((b + a*cos(c + d*x))/(a + b))*Elliptic.F((1/2)*(c + d*x), (2*a)/(a + b)))/(3*a^4*(a^2 - b^2)*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) - (8*b*(2*a^4 - 7*a^2*b^2 + 4*b^4)*sqrt(cos(c + d*x))*Elliptic.E((1/2)*(c + d*x), (2*a)/(a + b))*sqrt(a + b*sec(c + d*x)))/(3*a^4*(a^2 - b^2)^2*d*sqrt((b + a*cos(c + d*x))/(a + b))) + (2*b^2*sqrt(cos(c + d*x))*sin(c + d*x))/(3*a*(a^2 - b^2)*d*(a + b*sec(c + d*x))^(3/2)) + (4*b^2*(5*a^2 - 3*b^2)*sqrt(cos(c + d*x))*sin(c + d*x))/(3*a^2*(a^2 - b^2)^2*d*sqrt(a + b*sec(c + d*x))) + (2*(a^4 - 13*a^2*b^2 + 8*b^4)*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))*sin(c + d*x))/(3*a^3*(a^2 - b^2)^2*d)]
    @test_int [cos(c + d*x)^(1/2)/(a + b*sec(c + d*x))^(5/2), x, 10, -((2*b*(9*a^2 - 8*b^2)*sqrt((b + a*cos(c + d*x))/(a + b))*Elliptic.F((1/2)*(c + d*x), (2*a)/(a + b)))/(3*a^3*(a^2 - b^2)*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x)))) + (2*(3*a^4 - 15*a^2*b^2 + 8*b^4)*sqrt(cos(c + d*x))*Elliptic.E((1/2)*(c + d*x), (2*a)/(a + b))*sqrt(a + b*sec(c + d*x)))/(3*a^3*(a^2 - b^2)^2*d*sqrt((b + a*cos(c + d*x))/(a + b))) + (2*b^2*sin(c + d*x))/(3*a*(a^2 - b^2)*d*sqrt(cos(c + d*x))*(a + b*sec(c + d*x))^(3/2)) + (8*b^2*(2*a^2 - b^2)*sin(c + d*x))/(3*a^2*(a^2 - b^2)^2*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x)))]
    @test_int [1/(cos(c + d*x)^(1/2)*(a + b*sec(c + d*x))^(5/2)), x, 10, (2*(3*a^2 - 2*b^2)*sqrt((b + a*cos(c + d*x))/(a + b))*Elliptic.F((1/2)*(c + d*x), (2*a)/(a + b)))/(3*a^2*(a^2 - b^2)*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + (4*b*(3*a^2 - b^2)*sqrt(cos(c + d*x))*Elliptic.E((1/2)*(c + d*x), (2*a)/(a + b))*sqrt(a + b*sec(c + d*x)))/(3*a^2*(a^2 - b^2)^2*d*sqrt((b + a*cos(c + d*x))/(a + b))) - (2*b*sin(c + d*x))/(3*(a^2 - b^2)*d*sqrt(cos(c + d*x))*(a + b*sec(c + d*x))^(3/2)) - (2*b*(5*a^2 - b^2)*sin(c + d*x))/(3*a*(a^2 - b^2)^2*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x)))]
    @test_int [1/(cos(c + d*x)^(3/2)*(a + b*sec(c + d*x))^(5/2)), x, 10, (-2*b*sqrt((b + a*cos(c + d*x))/(a + b))*Elliptic.F((c + d*x)/2, (2*a)/(a + b)))/(3*a*(a^2 - b^2)*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) - (2*(3*a^2 + b^2)*sqrt(cos(c + d*x))*Elliptic.E((c + d*x)/2, (2*a)/(a + b))*sqrt(a + b*sec(c + d*x)))/(3*a*(a^2 - b^2)^2*d*sqrt((b + a*cos(c + d*x))/(a + b))) + (2*a*sin(c + d*x))/(3*(a^2 - b^2)*d*sqrt(cos(c + d*x))*(a + b*sec(c + d*x))^(3/2)) + (4*(a^2 + b^2)*sin(c + d*x))/(3*(a^2 - b^2)^2*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x)))]
    @test_int [1/(cos(c + d*x)^(5/2)*(a + b*sec(c + d*x))^(5/2)), x, 10, (2*sqrt((b + a*cos(c + d*x))/(a + b))*Elliptic.F((c + d*x)/2, (2*a)/(a + b)))/(3*(a^2 - b^2)*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + (8*b*sqrt(cos(c + d*x))*Elliptic.E((c + d*x)/2, (2*a)/(a + b))*sqrt(a + b*sec(c + d*x)))/(3*(a^2 - b^2)^2*d*sqrt((b + a*cos(c + d*x))/(a + b))) - (2*a^2*sin(c + d*x))/(3*b*(a^2 - b^2)*d*sqrt(cos(c + d*x))*(a + b*sec(c + d*x))^(3/2)) + (2*a*(a^2 - 5*b^2)*sin(c + d*x))/(3*b*(a^2 - b^2)^2*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x)))]
    @test_int [1/(cos(c + d*x)^(7/2)*(a + b*sec(c + d*x))^(5/2)), x, 14, (-2*a*sqrt((b + a*cos(c + d*x))/(a + b))*Elliptic.F((c + d*x)/2, (2*a)/(a + b)))/(3*b*(a^2 - b^2)*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + (2*sqrt((b + a*cos(c + d*x))/(a + b))*Elliptic.Pi(2, (c + d*x)/2, (2*a)/(a + b)))/(b^2*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + (2*a*(3*a^2 - 7*b^2)*sqrt(cos(c + d*x))*Elliptic.E((c + d*x)/2, (2*a)/(a + b))*sqrt(a + b*sec(c + d*x)))/(3*b^2*(a^2 - b^2)^2*d*sqrt((b + a*cos(c + d*x))/(a + b))) - (2*a^2*sin(c + d*x))/(3*b*(a^2 - b^2)*d*cos(c + d*x)^(3/2)*(a + b*sec(c + d*x))^(3/2)) - (2*a^2*(3*a^2 - 7*b^2)*sin(c + d*x))/(3*b^2*(a^2 - b^2)^2*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x)))]


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


    @test_int [(d*cos(e + f*x))^n*(a + b*sec(e + f*x))^3, x, 8, If(13>=8, -((b*(b^2*(1 - n) + 3*a^2*(2 - n))*(d*cos(e + f*x))^n*HypergeometricFunctions._₂F₁(1/2, n/2, (2 + n)/2, cos(e + f*x)^2)*sin(e + f*x))/(f*(2 - n)*n*sqrt(sin(e + f*x)^2))) - (a*(a^2*(1 - n) - 3*b^2*n)*cos(e + f*x)*(d*cos(e + f*x))^n*HypergeometricFunctions._₂F₁(1/2, (1 + n)/2, (3 + n)/2, cos(e + f*x)^2)*sin(e + f*x))/(f*(1 - n)*(1 + n)*sqrt(sin(e + f*x)^2)) + (a*b^2*(5 - 2*n)*(d*cos(e + f*x))^n*tan(e + f*x))/(f*(1 - n)*(2 - n)) + (b^2*(d*cos(e + f*x))^n*(a + b*sec(e + f*x))*tan(e + f*x))/(f*(2 - n)), -((b*(b^2*(1 - n) + 3*a^2*(2 - n))*(d*cos(e + f*x))^n*HypergeometricFunctions._₂F₁(1/2, n/2, (2 + n)/2, cos(e + f*x)^2)*sin(e + f*x))/(f*(2 - n)*n*sqrt(sin(e + f*x)^2))) - (a*(a^2*(1 - n) - 3*b^2*n)*cos(e + f*x)*(d*cos(e + f*x))^n*HypergeometricFunctions._₂F₁(1/2, (1 + n)/2, (3 + n)/2, cos(e + f*x)^2)*sin(e + f*x))/(f*(1 - n^2)*sqrt(sin(e + f*x)^2)) + (a*b^2*(5 - 2*n)*(d*cos(e + f*x))^n*tan(e + f*x))/(f*(2 - 3*n + n^2)) + (b^2*(d*cos(e + f*x))^n*(a + b*sec(e + f*x))*tan(e + f*x))/(f*(2 - n)))]
    @test_int [(d*cos(e + f*x))^n*(a + b*sec(e + f*x))^2, x, 7, If(13>=8, -((2*a*b*(d*cos(e + f*x))^n*HypergeometricFunctions._₂F₁(1/2, n/2, (2 + n)/2, cos(e + f*x)^2)*sin(e + f*x))/(f*n*sqrt(sin(e + f*x)^2))) - ((a^2*(1 - n) - b^2*n)*cos(e + f*x)*(d*cos(e + f*x))^n*HypergeometricFunctions._₂F₁(1/2, (1 + n)/2, (3 + n)/2, cos(e + f*x)^2)*sin(e + f*x))/(f*(1 - n)*(1 + n)*sqrt(sin(e + f*x)^2)) + (b^2*(d*cos(e + f*x))^n*tan(e + f*x))/(f*(1 - n)), -((2*a*b*(d*cos(e + f*x))^n*HypergeometricFunctions._₂F₁(1/2, n/2, (2 + n)/2, cos(e + f*x)^2)*sin(e + f*x))/(f*n*sqrt(sin(e + f*x)^2))) - ((a^2*(1 - n) - b^2*n)*cos(e + f*x)*(d*cos(e + f*x))^n*HypergeometricFunctions._₂F₁(1/2, (1 + n)/2, (3 + n)/2, cos(e + f*x)^2)*sin(e + f*x))/(f*(1 - n^2)*sqrt(sin(e + f*x)^2)) + (b^2*(d*cos(e + f*x))^n*tan(e + f*x))/(f*(1 - n)))]
    @test_int [(d*cos(e + f*x))^n*(a + b*sec(e + f*x))^1, x, 5, -((b*(d*cos(e + f*x))^n*HypergeometricFunctions._₂F₁(1/2, n/2, (2 + n)/2, cos(e + f*x)^2)*sin(e + f*x))/(f*n*sqrt(sin(e + f*x)^2))) - (a*(d*cos(e + f*x))^(1 + n)*HypergeometricFunctions._₂F₁(1/2, (1 + n)/2, (3 + n)/2, cos(e + f*x)^2)*sin(e + f*x))/(d*f*(1 + n)*sqrt(sin(e + f*x)^2))]
    @test_int [(d*cos(e + f*x))^n/(a + b*sec(e + f*x))^1, x, 7, (a*AppellF1(1/2, (1/2)*(-1 - n), 1, 3/2, sin(e + f*x)^2, (a^2*sin(e + f*x)^2)/(a^2 - b^2))*cos(e + f*x)*(d*cos(e + f*x))^n*(cos(e + f*x)^2)^((1/2)*(-1 - n))*sin(e + f*x))/((a^2 - b^2)*f) - (b*AppellF1(1/2, -(n/2), 1, 3/2, sin(e + f*x)^2, (a^2*sin(e + f*x)^2)/(a^2 - b^2))*(d*cos(e + f*x))^n*sin(e + f*x))/((cos(e + f*x)^2)^(n/2)*((a^2 - b^2)*f))]
    @test_int [(d*cos(e + f*x))^n/(a + b*sec(e + f*x))^2, x, 10, (a^2*AppellF1(1/2, (1/2)*(-3 - n), 2, 3/2, sin(e + f*x)^2, (a^2*sin(e + f*x)^2)/(a^2 - b^2))*cos(e + f*x)*(d*cos(e + f*x))^n*(cos(e + f*x)^2)^((1/2)*(-1 - n))*sin(e + f*x))/((a^2 - b^2)^2*f) + (b^2*AppellF1(1/2, (1/2)*(-1 - n), 2, 3/2, sin(e + f*x)^2, (a^2*sin(e + f*x)^2)/(a^2 - b^2))*cos(e + f*x)*(d*cos(e + f*x))^n*(cos(e + f*x)^2)^((1/2)*(-1 - n))*sin(e + f*x))/((a^2 - b^2)^2*f) - (2*a*b*AppellF1(1/2, (1/2)*(-2 - n), 2, 3/2, sin(e + f*x)^2, (a^2*sin(e + f*x)^2)/(a^2 - b^2))*(d*cos(e + f*x))^n*sin(e + f*x))/((cos(e + f*x)^2)^(n/2)*((a^2 - b^2)^2*f))]
    end
