Wow, that I have been looking over this so many times... Thanks!

Anyhow, I will have to move away from the analytical solution as finding the inverse of a function is not that straightforward anymore as soon as the function to be inverted gets quite complicated (which is the case as I am trying to fit functions to measured BSDF data).

Seems I'm back where I started by performing my monte carlo simulations using distribution functions that are being sampled instead of using an analytical solution.