With Maple 18.02 and 2017.2 on OSX 10.13.6: 

restart: 
int( exp(a*exp(I*x)) ,x=-Pi..Pi); 
                               0 

but we get the correct answer for numerical a, e.g. 

restart: 
int( exp(1*exp(I*x)) ,x=-Pi..Pi); 
                              2 Pi 

Is this fixed in version 2018.1?