I'm having difficulty re-deriving a result a calculation from a paper. The integral is $\int_0^{2\pi} \int_0^{2\pi} \frac{\sinh\eta}{(\cosh\eta-\cos\theta)^2}\left(1-c\sinh^2\eta\sin\phi\right)^\frac12d\theta d\phi,$ where $\eta$ and $c$ are parameters such that $\sinh^2\eta = 2$ and $c\sinh^2\eta < 1.$
From what I've read about hypergeometric functions, one could potentially evaluate this using elliptic integrals of the first and second kind, $\begin{align*} K(z) &= \int_0^{\pi/2} (1-z^2\sin^2 t)^{-1/2} dt \\ E(z) &= \int_0^{\pi/2} (1-z^2\sin^2 t)^{1/2} dt. \end{align*}$ These can be written in hypergeometric form as $\begin{align*} K(z) &= \frac{\pi}{2}F(\frac{1}{2},\frac{1}{2};1;z^2)\\ E(z) &= \frac{\pi}{2}F(-\frac{1}{2},\frac{1}{2};1;z^2). \end{align*}$ The expected result of the integral is $S = 8\pi^2 \frac{b^2}{a} G_1(c/a).$ where $a = \sqrt{c^2+b^2}$ and $G_1(x) = F(3/2,1/2,1;x^2)+x^2/2F(3/2,3/2,2;x^2).$
I've tried doing this, but I am inexperienced in these calculations and it's going to take me a while. I'll keep working at it, but I was wondering if anybody could take a look and tell me whether I am even going about this in the right way.