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I would like to show directly,

[tex]\int \frac{e^{at}}{e^{it}+e^{-it}}dt=\frac{e^{(i+a) t} \text{Hypergeometric2F1}\left[1,\frac{1}{2}-\frac{i a}{2},\frac{3}{2}-\frac{i a}{2},-e^{2 i t}\right]}{i+a}[/tex]

I realize I can differentiate the antiderivative to show the relation but was wondering how to integrate the integral directly to obtain the hypergeometric function or are these type integrals evaluated using a different technique? I can express it as:

[tex]\int \frac{e^{(i-a)t}}{1+e^{2it}}dt[/tex]

and at least it has the [itex]e^{2it}[/itex] in there.

How are any integrals expressed in terms of hypergeometric functions?