integrate cosx/(1+sinx+cosx)

The substitution $x = \tfrac12\pi-y$ shows that $I \; = \; \int_0^{\frac12\pi} \frac{\cos x}{1 + \sin x + \cos x}\,dx \; = \; \int_0^{\frac12\pi} \frac{\sin y}{1 + \sin y + \cos y}\,dy$ The substitution $t=\tan\tfrac12x$ shows that $J \; = \; \int_0^{\frac12\pi} \frac{1}{1 + \sin x + \cos x}\,dx \; = \; \int_0^1 \frac{dt}{1+t} \; = \; \ln2$ Thus $2I + J \; = \; \int_0^{\frac12\pi}\,dx \; = \; \tfrac12\pi$ and hence $I \; = \; \tfrac14\pi - \tfrac12J \; = \; \tfrac14\pi - \tfrac12\ln2 \; = \; \boxed{0.4388245731}$