How to verify integral with hypergeometric function

Question

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?

Answer 1

Expand rhs into series and differentiate. Or to make a change of variable in lhs $t=e^{-\beta \omega}$, expand into series and integrate.

Answer 2

Differentiate the RHS to get the integrand on the LHS? Note: $$ \frac{d}{dz} {}_2F_1(a,b;c;z) = \frac{ab}{c}\;{}_2F_1(a+1,b+1;c+1;z) $$