First an example which I know how to solve. If we have the following integral
there is a very practical way to evaluate it by interpreting it as some particular parametrization of a closed contour over a complex function. It works since the relevant residues of that underlying complex function can be readily obtained. The whole procedure is very well explained here:
Now, let us make the integrand more complicated. Especially, I am interested in the following case:
Since now different powers of $t$ are in the exponential functions the substitution as described in the Wikipedia article does not directly give a complex function whose residues could be easily obtained. That spoils the whole procedure. Any suggestion on how to evaluate?