@vv I'm not at all an expert in CAS implementations, 

but I do understand that dealing with branch points is tricky. 

This is often the cause of misbehaviors with CAS.

This is not the case here, where we are dealing with an entire

function. 

I'm surprised that Maple first compute an (non-elementary)

anti-derivative to evaluate this definite integral. Naively, I

thought that the integration meta algorithm would first try 

the simplest weapons before relying on the H bomb.   

@vv exp(a*X) = sum a^n*X^n/n!  n=0..infinity) 

Now, replaxe X by exp(I*x) and you get the Fourier series 

of exp(a*exp(I*x)). Isn't it trivial?

Perfect CAS do not exist but at least they should be 

trustable for trivial computations. 

Thanks all for the replies.

My problem is that I have to deal  with complicated expressions 

involving zillions of such elementary integrals with plenty of different

parameters. I cannot check all the integrals individually, nor I can set

the constants to be real or whatever  just to check.

@vv Exp is an entire function. The Fourier series of the integrand 

has coefficients a^n/n! so the integral is 2*Pi for all a. I don't 

need Maple to check this trivial result.