# Definite integral of hyperbolic function with hyperbolic arguments

I need to calculate the following integral $$\int^{\infty}_{0} {\sinh[a \cosh(t)] \over \cosh[a \cosh(t)] - b} dt\,.$$ I doesn't look too complicated but I have not been able to get a useful change of variables to make it look like something which could be in some integral tables.

Thank you for any help!

• "Doesn't look too complicated"? Geez, what kind of integrals are you used to? Do you know for sure if this has a closed-form value? Where did you find this problem? – Franklin Pezzuti Dyer Dec 21 '17 at 0:30
• Perhaps it is relevant that $$\frac{d}{dt}\ln(\cosh(a\cosh(t))-b)=\frac{a\sinh(t)\sinh(a\cosh(t))}{cosh(a\cosh(t))-b}$$ – Franklin Pezzuti Dyer Dec 21 '17 at 0:35
• Also, it doesn't seem to converge. The integrand approaches $1$ as $t$ approaches infinity. – Franklin Pezzuti Dyer Dec 21 '17 at 0:36
• So the question is easy. Answer $+\infty$ – GEdgar Dec 21 '17 at 11:37
• Thank you all the answers! The problem came up in a renormalization context. It does appear to give $\infty$ but I was going to derive it with respect to $a$ afterwards so I was looking for an expression in terms of $a$ and $b$, but I guess it isn't possible. – ABarr Dec 22 '17 at 18:19