Please help in
Integrate[ Sinh[x]/((Cosh[x] + Cosh[b])*(h^2 - x^2)), {x, 0, \[Infinity]}]
This integration is used by condensed matter communities. Do you think residues theorem may help
I doubt there's a closed-form for the Integral. Another maybe simple solution:
Regards M.I.
Your integral never converges, because of its behaviour as x->h:
x->h
Asymptotic[Sinh[x]/((Cosh[x] + Cosh[b])*(h^2 - x^2)), x -> h]
