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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?.

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    Thanks for the closed form. Sorry for not being around. I recently had surgery. Anyway, I had forgotten all about this, but it looks like a fun and challenging integral. How did you start this one?. I would love to see some of your ideas.2015-03-03

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