Integral related to $\sum\limits_{n=1}^\infty\sin^n(x)\cos^n(x)$ Playing around in Mathematica, I found the following:
$$\int_0^\pi\sum_{n=1}^\infty\sin^n(x)\cos^n(x)\ dx=0.48600607\ldots =\Gamma(1/3)\Gamma(2/3)-\pi.$$
I'm curious... how could one derive this?

I think you are making this much more difficult than it has to be. Since $|\sin(x)\cos(x)|< 1$ you have that 
$$\sum_{n=1}^{\infty}(\sin(x)\cos(x))^n=\frac{\sin(x)\cos(x)}{1-\sin(x)\cos(x)}=\frac{\sin(2x)}{2-\sin(2x)}$$
And you can just find through normal calc that 
$$\int \frac{\sin(2x)}{2-\sin(2x)}\mathrm dx=-\left(x+\frac{2}{\sqrt{3}}\arctan\left(\frac{1-2\tan(x)}{\sqrt{3}}\right)\right)$$
