Finding $\int\left(\frac{x}{\cos\left(x\right)+x\csc x}\right)^2\,dx$

Question

Consider the integral

$$\int\left(\dfrac{x}{\cos\left(x\right)+x\csc x}\right)^2\,dx$$

How to start integrating? Any hint would be appreciated.

Answer 1

You can write this as

$$ \int \dfrac{x^2 \sin^2(x)}{(x + \sin(x)\cos(x))^2} $$ Now since $$ \dfrac{d}{dx} \dfrac{1}{x + \sin(x) \cos(x)} = - 2 \dfrac{\cos^2 x}{(x+\sin(x)\cos(x))^2}$$ integrate by parts with $$u = \dfrac{x^2 \sin^2(x)}{\cos^2(x)},\ dv = \dfrac{\cos^2(x)\; dx}{(x + \sin(x)\cos(x))^2}$$ After a miraculous simplification, you end up looking at $$ \ldots + \int \dfrac{x \sin(x)\; dx}{\cos^3 (x)}$$ which yields to another integration by parts.