Trying to evaluate the following integral, Mathematica returns this result:
$$ \int \frac{e^{-\tau \omega}}{1+e^{-\beta \omega}} d \omega = \frac{e^{(\beta - \tau) \omega} \cdot {}_2F_1(1, 1-\frac{\tau}{\beta}, 2 - \frac{\tau}{\beta}, -e^{\beta \omega})}{\beta - \tau} $$
$\beta$ and $\tau$ can be treated as constants at this point. Unfortunately, I do not have any clue how I could achieve the same result with pen and paper. Does anyone have an idea?