A definite integral: $\int_{0}^{\pi/2} \frac{\sin x~\mathrm dx}{\sin x+\cos x+ e^x}$ Mathematica can do this integral,
$$\int_{0}^{\pi/2} \frac{\sin x~ \mathrm dx}{\sin x+\cos x+ e^x}\,,$$
the question is: how to do it by hand?

Rewrite $$ \frac{\sin(x)}{e^x+\sin (x)+\cos (x)}=\frac{1}{2}-\frac{1}{2}\frac{e^x+\cos (x)-\sin (x)}{e^x+\sin (x)+\cos (x)}$$
$$\int \frac{\sin(x)}{e^x+\sin (x)+\cos (x)}\,dx=\frac{x}{2}-\frac{1}{2} \log \left(e^x+\sin (x)+\cos (x)\right)$$
