I have the following surface in $$R^3:{(x,y,z),(x^2 + y^2 + z^2)^2 = a^2(x^2 - y^2) \ ,\ x,y >=0}.$$ I want to find it's surface area. I've tried using spherical coordinates but calculating the normal vector it's a messy and also in cartesian coordinates. In the second case the resulting integral contains terms such as $\sqrt{1+\sqrt{\cos(x)}}$ which are hard or impossible to integrate.
How can I find the surface area? Which is the best parameters to use?
Assume $a=1$ and use spherical coordinates. Compute $r(\phi,\theta)$ in order to obtain a parametric representation
$$S:\quad(\phi,\theta)\mapsto{\bf r}(\phi,\theta)=\bigl(r(\phi,\theta)\cos\theta\cos\phi,r(\phi,\theta)\cos\theta\sin\phi,r(\phi,\theta)\sin\theta\bigr)\qquad\left(-{\pi\over4}\leq\phi\leq{\pi\over4}, \ -{\pi\over2}\leq\theta\leq{\pi\over2}\right)\ .$$
Mathematica produced $|{\bf r}_\phi\times {\bf r}_\theta|=\cos^2\theta$, so that the end result was $${\rm area}(S)={\pi^2\over4}\ .$$
By the way, here is a picture of the surface:
