Question: $\ $Prove or disprove that $$\int_0^1\sin(\pi x)\,x^x\,(1-x)^{1-x}\ dx\ =\ \frac{e\pi}{4!}$$
The numerical integration by Matlab showed that $$\left|\ \int_0^1\sin(\pi x)\,x^x\,(1-x)^{1-x}\ dx\ - \ \frac{e\pi}{4!}\ \right|\ <\ 10^{-15}$$ which made me kind of believe that the equality holds.
Therefore, I tried to prove it by contour integration, integration by parts and all other methods I could come up with. However, it seems that none of them would work.
Could anyone help me with this question? Any hint will be much appreciated.