I've come across the following integral:
I know how to approximate this integral using the Laplace method, just wondering if:
a) Does this integral have an exact answer?
b) Is there a better approximation than Laplace method for this integral? If so, under what conditions will it be better?
My thinking is that it will be a hypergeometric function (mainly because every hard integral I've come across turns out to be one of these). Conditions (if needed) are $A>R>0$, and $N$ is an integer.