# Series approximation for a definite integral of a modified student-t distribution

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!

• I wish you very pleasant times with the monster ! Cheers. – Claude Leibovici Aug 8 '17 at 11:20
• @ClaudeLeibovici hah thanks! That $\exp{\tanh{}}$ term really is awful... doesn't give me much hope. – CBowman Aug 8 '17 at 11:51
• It is not the only one which is awful ! – Claude Leibovici Aug 8 '17 at 12:01