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

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.

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