Distribution of $r.(\ln(X)-X)$ when $X$ is beta distributed. I need some help and this is certainly not a home work question, as it appears in my research. 
If $X$ is a beta distributed random variable, is there a known distribution for 
\begin{equation} 
\begin{split}
Y & = r.(ln(X)-X)
\end{split}
\end{equation}
where $r$ is a constant? If not, is there at least an analytic form for the distribution of Y? 
I tried to derive one, but got stuck when I had to solve for $e^{y/r} = xe^{-x}$. Any help will be appreciated.
Thanks! 

From user3658307 insightful reply above, we have $x = -W(-e^{y/r})$, where $W$ is the Lambert $W$ function. Since the pdf of beta distribution is described by
$$f(x;\alpha,\beta) = \frac{\Gamma(\alpha+\beta)}{\Gamma(\alpha)\Gamma(\beta)} x^{\alpha-1}(1-x)^{\beta-1}$$ the pdf of $Y$ should technically follow:
$$f(y;\alpha,\beta) = \frac{\Gamma(\alpha+\beta)}{\Gamma(\alpha)\Gamma(\beta)} \left(-W(-e^{y/r})\right)^{\alpha-1}\left(1+W(-e^{y/r})\right)^{\beta-1}$$
