Evaluate the integral $$ \int{\frac{e^{-\sin x}\sin 2x}{\left( \sin x-\cos x \right) ^4}\text{d}x} $$
I tried a lot of ways but none of them worked. Also I tried to use Mathematica to solve it, but there's no result either. So my question is, does this integral have elementary closed forms? If not, how to prove it? (We may use Liouville's theorem)