$$I=\int \frac{e^{\cos x} \sin x-(\sin x+\cos x)e^{\sin x+\cos x}}{(e^{\sin x}-1)^2}dx$$
I'm supposing this is of the form $$\int (f(x)+f'(x))e^x dx=f(x)e^x$$ or $$\int (f(x)-f''(x))e^x dx=(f(x)-f'(x))e^x$$ but I'm unable to reduce it to that form. Or am I on the completely wrong track? I've plugged this into Wolfram Alpha and the closed form does exist. Can anybody set me in the right direction? Note: I don't want the full solution, but just a hint on how to proceed towards the answer.
If you write the integrand as $$ \frac{-\sin(x) \mathrm{e}^{\cos(x)} (\mathrm{e}^{\sin(x)}-1) - \cos(x) \mathrm{e}^{\sin(x)} \mathrm{e}^{\cos(x)}}{(\mathrm{e}^{\sin(x)} -1)^2} \, , $$ you may recognise one of the usual differentiation rules.