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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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Maple doesn't do anything to this as an antiderivative. So its "likely" there's no simple closed form antiderivative. I also put it into maple including the limits, and again no answer; to me this means that it might be doable using contour integration or the like, but would be difficult (maple seems to know a lot of contour type definite integrals). – coffeemath Oct 30 '12 at 17:04
Thank you. I, too, ran it through Maple and it gave a numerical solution of $-.2708685659...$ I read a solution was supposed to be $\frac{-\pi \sqrt{2}}{48}\left({\pi}^{2}+12\pi \ln(2)+24\ln^{2}(2)\right)$. But this does not check out numerically with what Maple gives, so I assume it is incorrect. – Cody Oct 30 '12 at 17:35
The Answer is $$I=-\frac{\pi\sqrt{2}}{8}(3ln^{2}2+2{\pi}ln{2}-4G-\frac{\pi^2}{6})$$ – user178256 Feb 23 '15 at 15:52
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. – Cody Mar 3 '15 at 13:54

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