I'm afraid that method doesn't work, as the integral is non convergent for -1 < Re(e) < 1 or Im(e) != 0 so there is obviously something fishy with that substitution
I have now figured it out, to do it (semi) rigorously, you taylor expand the (1+e*cos(f))^(-gamma-2) term which is a convergent series for -1<e<1 and integrate term by term, all of which evaluate to zero.
Thanks for the help though