I am trying to perform a surface integral over kind of a weird shape. So the radius of the shape should be equal to the multiple of $3$ constants (one for each of the $x, y$ and $z$ directions) each to the power of the magnitude of the component of the unit vector in the corresponding $x, y$ or $z$ direction. Sorry if this is not too clear but here is the equation:
$$r = \mathrm{x}^{|\cos(\theta)|} \mathrm{y}^{\sin(\theta)|\cos(\phi)|} \mathrm{z}^{\sin(\theta)\sin(\phi)}\sin(\theta) $$
Where x, y, z are constants. So the surface integral should then be:
$$S = \int_{0}^{2\pi} \int_{0}^{\pi} \mathrm{x}^{2|\cos(\theta)|} \mathrm{y}^{2\sin(\theta)|\cos(\phi)|} \mathrm{z}^{2\sin(\theta)\sin(\phi)}\sin(\theta) \mathrm{d}\theta \mathrm{d}\phi $$
I have been trying for some time to solve this but have not found a successful approach. Does anyone have any ideas how to solve this?