Can you help me solve this triple integral?

I've been trying to solve this integral $$\int^{-1}_{-2} \int^{-1}_{-2} \int^{-1}_{-2} \frac{x^2}{x^2+y^2+z^2}dxdydz$$ If I try some variable change like $$x=\sqrt{y^2+z^2} tan \theta $$ things complicated a lot. Wolfram alpha whit numerical method says that the integral is 0.33333 which makes me think that is possible to obtain an analytical solution. This problem is part of a example test for a master admission.

Im also thinking about change to spherical coordinates wich lead me to $$\int\int\int r^2 sin \theta cos \phi sin \phi drd\theta d\phi$$ but now I dont have clear my integration limits.

Solution 1:

Use symmetry. The integral will be the same with $y^2$ or $z^2$ in the numerator. But adding all three gives the integral of $1$ over the cube.