How to prove $\int_0^\pi \frac{dx}{2+2\sin x+\cos x}=\log3$? How can we prove that: $$\int_0^\pi \frac{dx}{2+2\sin x+\cos x}=\log3$$
I don't have any ideas, the $f(\pi-x)$ thing doesn't work as well. Please help :)

$$
\begin{aligned}
&\int_0^\pi\frac{1}{2+2\sin x+\cos x}dx  
\\=& \int_0^\pi \frac{dx}{2[\cos^2(x/2)+\sin^2(x/2)]+4\cos(x/2)\sin(x/2) + \cos^2(x/2)-\sin^2(x/2)}\\
=&\int_0^\pi\frac{dx}{3\cos^2(x/2)+4\cos(x/2)\sin(x/2)+\sin^2(x/2)}\\
=&\int_0^\pi \frac{\sec^2(x/2)dx}{\tan^2(x/2)+4\tan(x/2)+3}\\
=&\int_0^\infty \frac{2du}{u^2+4u+3}
\end{aligned}
$$
