I'm working on an operator problem that requires solving the following complicated integral, involving the Gauss Hypergeometric function \begin{equation} \int_{-\infty}^{\infty} \int_{-\infty}^0 \frac{1}{\sinh^2(x)} f^*(\nu, k ,b, x, y) \, f(\nu, k, b, x,y)\, dx \, dy \end{equation} where \begin{equation} f(\nu, k, b, x, y) = e^{- i k y} e^{ x(k-b)} (1-e^{2x})^{\frac{1}{2}+ i \nu} \text{ }_2 F_1 \left(\frac{1}{2} +k + i \nu, \frac{1}{2} -b + i \nu, 1+k-b, e^{2x} \right) \end{equation} I've attempted to solve it with the aid of software to no avail, and have sought integral identities involving the Hypergeometric function that may soften this up, but so far I haven't found anything that could be of particular use.
Can someone more experienced with Hypergeometric functions help me solve this?