Does anyone know of a method by which to tackle the following rather nasty surface integral?
$$ \iint_{\theta\in[0,\pi], \phi\in[0,2\pi)} \left[\sigma^2[\left(x-y+R\sin{\theta} \right)\left( \cos{\phi}-\sin{\phi}\right)]^2 +[y+\left(z-\rho+R \cos{\theta}\right)\left(x+R\cos{\phi}\sin{\theta}\right)+R\sin{\theta}\sin{\phi}]^2 +[\beta(z+R\cos{\theta})-\left(x+R\cos{\phi}\sin{\theta}\right)\left(y+R\sin{\theta}\sin{\phi}\right)]^2\right]^{1/2}\cdot\sin{\theta}d\theta d\phi $$
I have a closed form for the integral without the square root, but I am thinking there may not be a solution with it. I'd love to be proved wrong however. If it's helpful, this equation is the magnitude of the Lorenz '63 equations, integrated on a sphere around some central point (x,y,z).
Even general suggestions for how to deal with difficult integrals of square roots would be appreciated.
