Evaluate $$\int_0^\pi \frac{x}{(\sin x)^{\sin (\cos x)}}\,\mathrm{d}x.$$

I tried using by parts and complex numbers along with series expansion but I was unable to find the answer.
Please Help!

  • 2
    $\begingroup$ Why would this have a closed form? The limits are not very natural for the integrand. $\endgroup$ – user111187 Nov 21 '14 at 11:40
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    $\begingroup$ Finding a closed form for the antiderivative seems (at least to me) to be a dream. $\endgroup$ – Claude Leibovici Nov 21 '14 at 11:46
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    $\begingroup$ Just for hee-haws, it turns out that $$\frac{\cos{x}}{(\sin{x})^{\sin{\cos{x}}}}$$ has an antiderivative in closed form. $\endgroup$ – Ron Gordon Nov 21 '14 at 11:47
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    $\begingroup$ @RonGordon. Please, tell more and show us. I stop my dream and I am awake. How did you get an answer ? $\endgroup$ – Claude Leibovici Nov 21 '14 at 11:52
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    $\begingroup$ The one equals $$3.9905179137710221001990755970868667785510732535206. $$ $\endgroup$ – user64494 Nov 21 '14 at 12:40

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