It would be very helpful if the following definite integral or a similar one had an analytic solution:
$\int\nolimits_{-\infty}^{\infty}\mathrm{sech}^2(x)\exp(-\alpha x^2)\,\mathrm dx,\qquad \alpha \geq 0$
I have attacked this problem from several directions now, including contour integration (the Gaussian blows up along Re=0), differentiation under the sign (no luck), and interpreting it as the expected value when sampling with a certain distribution (because both of these functions may be interpreted that way easily).
To me, it seems like this could have a nice formula because both functions separately do, there is so much symmetry, and the integrand seems like just a bunch of exponentials to me. I just wanted to know if anyone had a compelling reason why this won't have a 'nice' analytic solution - or even what the intuition of some more experienced people is concerning my probability of success or direction of my efforts.