Related to: An Integral involving $e^{ax} +1$ and $e^{bx} + 1$
Evaluate the integral $I(a,b)=\int_{0}^{1}\frac{e^{ax}-e^{bx}}{\left(e^{ax}+1\right)\left(e^{bx}+1\right)}dx$ for $a>b>0$.
Just like the related question, this is a "putnam practice" type of question that is meant to be solved in less than 5 minutes using "simple" mathematics.
Additional Info
I am not sure if the question is actually solvable. There might be crucial details missing (this was found scribbled on old notes).