<s>[INST] trigonometric integral with tangent $$\int\biggl( \sqrt[3]{\frac{\sin x+1}{\cos x}}+\sqrt[3]\frac{\sin x-1}{\cos x}\biggr)\frac{1}{\cos^2x}\,dx ,\;x\in\Bigl(-\frac{\pi}{2},\frac{\pi}{2}\Bigr)$$ Can somebody give some tips about how can I evaluate this integral,please? I observed that $\frac{1}{\cos^2x}$ is the derivative of tangent, but I don't know how to do it using the substitution $t=\tan x$ because there will be $\frac{1}{\cos x}$ inside and I think there is another method.
 [/INST] I will use integration by parts twice. First we write
\begin{align}
I &= \int \left (\sqrt[3]{\frac{\sin x + 1}{\cos x}} + \sqrt[3]{\frac{\sin x - 1}{\cos x}} \right ) \sec^2 x \, dx\\
&= \int \big{(}\sqrt[3]{\tan x + \sec x} + \sqrt[3]{\tan x - \sec x} \big{)} \sec^2 x \, dx.
\end{align}
If we let
$$f(x) = \sqrt[3]{\tan x + \sec x} + \sqrt[3]{\tan x - \sec x}$$
and
$$g(x) = \sqrt[3]{\tan x + \sec x} - \sqrt[3]{\tan x - \sec x},$$
then obsereve that
$$f'(x) = \frac{1}{3} \sec x \cdot g(x) \quad \text{and} \quad g'(x) = \frac{1}{3} \sec x \cdot f(x).$$
So for the integral we have
\begin{align}
I &= \int \sec^2 x \cdot f(x) \, dx\\
&= f(x) \cdot \tan x - \int \tan x \cdot f'(x) \, dx \qquad \text{(by parts)}\\
&= f(x) \cdot \tan x - \frac{1}{3} \int \sec x \tan x \cdot g(x) \, dx \\
&= f(x) \cdot \tan x - \frac{1}{3} \sec x \cdot g(x) + \frac{1}{3} \int \sec x \cdot g'(x) \, dx \quad \text{(by parts)}\\
&= f(x) \cdot \tan x - \frac{1}{3} \sec x \cdot g(x) + \frac{1}{9} \int \sec^2 x \cdot f(x) \, dx\\
&= f(x) \cdot \tan x - \frac{1}{3} \sec x \cdot g(x) + \frac{1}{9} I,
\end{align}
giving
$$I = \frac{9}{8} \tan x \cdot f(x) - \frac{3}{8} \sec x \cdot g(x) + C$$
or 
\begin{align}
I &= \frac{9}{8} \tan x \big{(} \sqrt[3]{\tan x + \sec x} + \sqrt[3]{\tan x - \sec x} \big{)}\\
& \qquad - \frac{3}{8} \sec x \big{(} \sqrt[3]{\tan x + \sec x} - \sqrt[3]{\tan x - \sec x} \big{)} + C.
\end{align}
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