Solving integral $\int\frac{\sin x}{1+x\cos x}dx$

Question

How I can find the anti-derivative? $$\int\frac{\sin x}{1+x\cos x}dx$$

Answer 1

Sorry for “cheating”, but it is as it seems: Wolfram|Alpha states it is not solvable “in terms of standard mathematical functions” (which should then be true).

http://www.wolframalpha.com/input/?i=%E2%88%ABsinx%2F%281%2Bxcosx%29dx

Are you sure this is the correct integral?

Answer 2

By using the formula http://upload.wikimedia.org/wikipedia/en/math/7/2/a/72a1058ad2087aec467af24bddcf9479.png, we have $\int\dfrac{\sin x}{1+x\cos x}dx=x\sum\limits_{n=1}^\infty\sum\limits_{m=1}^{2^n-1}\dfrac{(-1)^{m+1}\sin\dfrac{mx}{2^n}}{2^n+mx\cos\dfrac{mx}{2^n}}+C$