I'm trying to evaluate the following integral
$\int_{0}^{\infty}\left( 1-\frac{\gamma(a,bx)}{\Gamma(a)}\right) \, \ln\left( 1-\frac{\gamma(a,bx)}{\Gamma(a)}\right) \, dx$ for $b>0$ and $\gamma(a,bx)$, the lower, incomplete Gamma function.
It seems like a complete mess. I'm guessing this doesn't possess a closed form solution but what kind of special functions can be used to express it ?