Evaluating $\int \frac{x\sin x-\cos x}{x\left(2\cos x+x-x\sin x\right)} {\rm d} x$ using elementary methods

Question

I know how to evaluate elementary integrals. However, I encountered the following integral and was unable to how to solve it.

$$\int \frac{x\sin x-\cos x}{x\left(2\cos x+x-x\sin x\right)} {\rm d} x$$

The derivative of denominator is not present in the numerator too. Can't think of any substitution either. Breaking about the numerator also fails. WolframAlpha also fails.

Can someone help me evaluate the integral above, preferably using elementary methods?

Answer 1

Check if denominator has sinx^2, cause taking a substitution xcosx = t, the denominator will be 2t +t^2, and it will be easy to solve, and the answer will be log(1+2/xcosx)^(1/2).