For a positive integer $n$ the terms given by \begin{align} -\int_0^1 x^n \sin(\pi x) x^x (1-x)^{1-x} dx \\ = \int_0^1\int_0^1 (xy)^n \sin(\pi xy) (xy)^{xy} \frac{(1-xy)^{1-xy}}{\ln (xy)} dx dy \end{align} are rationals mutiple of $e\pi$. The left side integral, when $n=0$, can be found here: Prove that $\int_{0}^{1}\sin{(\pi x)}x^x(1-x)^{1-x}\,dx =\frac{\pi e}{24} $
I'm interessed in knowing about \begin{align} \int_0^1\int_0^1 x^py^q \sin(\pi xy) (xy)^{xy} \frac{(1-xy)^{1-xy}}{\ln (xy)} dx dy \end{align} where $p,q$ are positive integers. Maybe someone can show a way to evaluate a specific case and I will try to conjecture a general exepression.
