As part of some work I'm doing, I need to evaluate the normalisation integral for a modified student-t distribution shown below: $$ f(a, b, q, \nu) = \int\limits_{-\infty}^{\infty}\left( 1 + \frac{1}{\nu}\left(\frac{x}{\exp{(a \tanh{bx})}}\right)^{q} \right)^{-\frac{1}{2}(\nu+1)} \; \mathrm{d}x $$ where $q,\nu > 1$ and $b>0$.
I need to evaluate this function a huge number of times, and as far as I know, a closed-form doesn't exist.
I'd love to know whether it's possible to construct some converging series approximation of this integral, but I wouldn't know where to start.
Anyone have any ideas?
Thanks!