A variation on: Another 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 $a>b>0$.
Attempt
I suppose, like before, I have to simplify the integrand to seperate the $a$ and $b$ into different integrals. So far I have managed to do this:
$1 - \frac{1}{\left(e^{bx}+1\right)} - \frac{(e^{bx})}{\left(e^{ax}+1\right)\left(e^{bx}+1\right)}$
But I am stuck. Not sure if the question is unsolvable this way or if I can't see the trick.
Additional Info
This is not an "official" question from a textbook, course or quiz. There might be no nice solutions.