I ran across an integral with $\ln(\sin(x))$, and have not been able to make much headway.
Perhaps there is no closed form, but I thought I would post it to see if anyone has some clever ideas.
$\displaystyle \int_{0}^{\frac{\pi}{2}}x\sqrt{\tan(x)}\cdot \ln(\sin(x))dx$
I tried various subs, but they did not yield anything manageable. At first glance, it looks like the classic Beta integral is in there somewhere. How to deal with that $\sqrt{\tan(x)}$ poses the challenge.
Does anyone have ideas on a way to tackle this one?.