This problem comes from here, I wonder whether it has an elementary closed form or not.
Maybe, we can make a transformation first as follows \begin{align*} \int \frac{e^{-\sin x}\sin 2x}{(\sin x-\cos x)^4}{\rm d}x=\int \frac{e^{-\sin x}\sin 2x}{(\sin 2x-1)^2}{\rm d}x=\int \frac{e^{-\sin x}}{\sin 2x-1}{\rm d}x+\int \frac{e^{-\sin x}}{(\sin 2x-1)^2}{\rm d}x, \end{align*} but this seems helpless, and WA gives no results.