Finding $\int e^{-\sin^2x}{(\cos x -3 x \sin(x)+2 x \sin^3(x))}dx$

Question

I need to find $$\int e^{-\sin^2x}{(\cos x -3 x \sin(x)+2 x \sin^3(x))}dx$$ .

I know that $$\int e^{g(x)}{(f(x)g'(x)+f'(x))}dx = e^{g(x)}f(x)$$.

But I cannot find cannot $f(x)$ in the above expression.It seems really difficult to guess what $f(x)$ could be.Any other faster methods?How do I get $f(x)$ ?