Evaluate $\int \frac{dx}{1+\sin x+\cos x}$ Evaluate $$\int \frac{1}{1+\sin x+\cos x}\:dx$$
I tried several ways but all of them didn't work
I tried to use Integration-By-Parts method but it's going to give me a more complicated integral
I also tried u-substitution but all of my choices of u didn't work
Any suggestions?

Write $\sin(x)$ in the denominator as $2\sin(x/2)\cos(x/2)$ and $\cos(x)$ as $2(\cos(x/2))^2-1$. Now the 1s get cancelled. Take $(\cos(x/2))^2$ out from the denominator and send it to numerator as $(\sec(x/2))^2$. Now you arrived at $\frac{\sec(x/2)^2}{2 \tan(x/2) + 2}$. Proceed.
