Graduate Difficult cosh integral using Leibniz rule?

[Kavorka]

I was wondering if I could get some pointers on how to at least start on this. In quantum mechanics we are using the WKB approximation, and we end up with a definite integral that looks like this:

∫(1 - a(cosh(x))^-2)^1/2 dx = ∫(1/cosh(x)) (1 - a(cosh(x))²)^1/2 dx

where a is a positive constant. I've tried everything I can think of to no avail, the answer on wolfram isn't pretty but it seems like if I can figure out what process to use I could reach it eventually. I asked the professor and he suggested Leibnitz rule, but not sure how differentiation under the integral sign would help here.

 

[Buffu]

definite integral

But it does not have limits on it. Do you mean indefinite ? I am asking because some indefinite integrals are easy to evalute with limits.

Are you just interested in final result ?

 

[Kavorka]

But it does not have limits on it. Do you mean indefinite ? I am asking because some indefinite integrals are easy to evalute with limits.

Are you just interested in final result ?

From how my professor was describing it it seemed like the limits wouldn't be too helpful, but yes it is a definite integral from -cosh^-1(a^0.5 to cosh^-1(a^0.5 (inverse hyperbolic cosh, not cosh^-1 )

 

[Buffu]

From how my professor was describing it it seemed like the limits wouldn't be too helpful, but yes it is a definite integral from -cosh^-1(a^0.5 to cosh^-1(a^0.5 (inverse hyperbolic cosh, not cosh^-1 )

I tried to integrate it and I was unable to do so except for $a = 1$ (which you can do easily). I think it is not integratable but I am no expert, I don't have any clue about gamma function, airy functions ...