Please help in
Integrate[ Sinh[x]/((Cosh[x] + Cosh[b])*(h^2 - x^2)), {x, 0, \[Infinity]}]
I doubt there's a closed-form for the Integral. Another maybe simple solution:
Regards M.I.
\int_0^{\infty } \frac{\sinh (x)}{\left(h^2-x^2\right) (\cosh (b)+\cosh (x))} \, dx=\\\int_0^1 \frac{e^b \left(-1+t^2\right)}{t \left(e^b+t+e^{2 b} t+e^b t^2\right) \left(-h^2+\log ^2(t)\right)} \, dt=\\\int_0^1 \left(\frac{1}{t (h-\log (t)) (h+\log (t))}-\frac{1}{\left(e^b+t\right) (h-\log (t)) (h+\log (t))}-\frac{e^b}{\left(1+e^b t\right) (h-\log (t)) (h+\log (t))}\right) \, dt=\\\int_0^1 \left(-\frac{1}{\left(e^b+t\right) \left(h^2-\log ^2(t)\right)}-\frac{e^b}{\left(1+e^b t\right) \left(h^2-\log ^2(t)\right)}\right) \, dt=\\\Re\left(\sum _{m=0}^{\infty } \frac{(-1)^m e^{-((b+h) (1+m))} \left(\Gamma (0,-h (1+m))-e^{2 h (1+m)} \Gamma (0,h+h m)\right)}{2 h}\right)+\Re\left(\sum _{m=0}^{\infty } -\frac{(-1)^{2 m} h^{2 m} \left(-i \pi \text{Li}_{-1-2 m}\left(-e^b\right)+2 \log (h) \text{Li}_{-1-2 m}\left(-e^b\right)-2 \psi ^{(0)}(2+2 m) \text{Li}_{-1-2 m}\left(-e^b\right)-2 \text{PolyLog}^{(1,0)}\left(-1-2 m,-e^b\right)\right)}{2 (1+2 m)!}\right)
Maybe exist closed form I can't find them.
Numerically is easy to compute
This integration is used by condensed matter communities. Do you think residues theorem may help
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]
Setting values for b and h results in warnings that the integration doesn't converge on (0,\infty). Here's an example:
Integrate[Sinh[x]/((Cosh[x] + Cosh[b])*(h^2 - x^2)) /. {h -> 1, b -> 1}, {x, 0, \[Infinity]}]
Integrate::idiv: Integral of -(Sinh[x]/((-1+x^2) (Cosh[1]+Cosh[x]))) does not converge on {0,[Infinity]}.
Do you have some reason to think the integral exists for either specific values of h and b or on a different interval?
