Define for real $x>0$ and $\epsilon>0,$ the function $$ f(x,\epsilon):= \int_{\epsilon}^\infty \frac{\mathrm{d}s}{s} \frac{1}{\sinh^2 s/2} e^{-sx}. $$
Question: is it possible to compute explicitly $f$ by a residue calculation?
The idea would be to obtain something like $$f(x)=\sum_g F_g x^{2-2g} + O(e^{-x}).$$