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I have been trying to solve the following integral $$\int_{0}^{\frac {\pi}{2}} \ln\left (\frac {\ln^2 (\sin x)}{\pi^2+\ln^2 (\sin x)}\right) \frac {\ln \cos x}{\tan x} dx$$ I tried substituting for $\frac {\ln^2 (\sin x)}{\pi^2+\ln^2 (\sin x)}$ and using the properties of definite integrals, but I am not able to proceed with the integral as the $\ln \cos x$ term doesn't get substituted. Is there any other trick that I can employ?

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marked as duplicate by David H, Anastasiya-Romanova 秀, Micah, user147263, anomaly Oct 31 '14 at 4:34

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  • $\begingroup$ from where did you got this integral? $\endgroup$ – Dr. Sonnhard Graubner Oct 25 '14 at 13:52
  • $\begingroup$ Oh I saw it on a forum a couple of days ago and wanted to work on it. $\endgroup$ – Artemisia Oct 25 '14 at 13:59
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    $\begingroup$ @Artemisia, math.stackexchange.com/questions/985686/integral-contest $\endgroup$ – Galc127 Oct 25 '14 at 14:12
  • $\begingroup$ Oh I found it on quora haha. I guess someone was trying to solve it on there :/ $\endgroup$ – Artemisia Oct 25 '14 at 14:19
  • $\begingroup$ This problem has been stolen, the Prosecutor General has started an investigation into this grave matter. $\endgroup$ – Count Iblis Oct 25 '14 at 19:12