integral of a (cotan) rational function

Question

I'm trying to solve the following integral:

$$\int \frac{1}{1+\cot^3(x)}dx$$

While the solution can be found in Wolfram Alpha, I am not completely sure how to reduce the above integral to get the solution referenced. Pointers would be appreciated.

Answer 1

Let $u=\cot(x)$; then $dx = -\frac{1}{u^2+1}du$. Now the integral becomes

$$ I = - \int \frac{1}{(u^2+1)(u^3+1)} du $$

which can be resolved into partial fractions as:

$$ I =- \int \frac{1-2u}{3(u^2-u+1)} + \frac{u+1}{2(u^2+1)} + \frac{1}{6u+6} du $$

each sub-integral of which can be readily evaluated.

Answer 2

If you make the change of variables $ x=\arctan(t) $ you get

$$ \int \!{\frac {{t}^{3}}{ \left( 1+{t}^{3} \right) \left( 1+{t}^{2} \right) }}{dt}\,. $$