Is there any solution (numeric or closed) to integration of ${\sin x}^{\cos x}$? I've tried so many ways to evaluate $\int{\sin x}^{\cos x}dx$ and even searched and used programs like matlab, maple and scipy library and got no answer! my question is clear, is there any numerical or closed form solution available for this integration?
 A: As said in comments, there is no closed form.
If you want to compute for small values of $a$
$$I_a=\int_0^a {[\sin (x)]}^{\cos (x)}\,dx$$ you could compose Taylor series for the integrand and get
$${[\sin (x)]}^{\cos (x)}=x- \left(\frac{\log (x)}{2}+\frac{1}{6}\right)x^3+\frac{1}{120}  \left(15 \log
   ^2(x)+15 \log (x)+11\right)x^5+O\left(x^7\right)$$ and get
$$I_a=\frac{a^2}{2}- \frac{12\log (a)+1}{96}a^4+\frac{ 45
   \log ^2(a)+30 \log (a)+28}{2160}a^6+\cdots$$
For example, using $a=\frac \pi 4$, this would give $0.318492$ while numerical integrate would lead to $0.318453$ Which is not too bad (I hope).
Edit
Interesting (at least to me), is the expansion around $x=\frac \pi 2$
$${[\sin (x)]}^{\cos (x)}=1+\frac{1}{2} \left(x-\frac{\pi }{2}\right)^3+\frac{1}{8} \left(x-\frac{\pi
   }{2}\right)^6+\frac{1}{80} \left(x-\frac{\pi
   }{2}\right)^7+O\left(\left(x-\frac{\pi }{2}\right)^9\right)$$ which is very accurate
$$J_a=\int_{\frac \pi2}^{\frac \pi2+a} {[\sin (x)]}^{\cos (x)}\,dx=a+\frac{a^4}{8}+\frac{a^7}{56}+\frac{a^8}{640}+\cdots$$
