Find : \[\int \frac{\cos x d x}{1+\cos x+\sin x} \]

\[\begin{aligned} I & =\int \frac{\cos x}{1+\cos x+\sin x} d x \\ & =\int \frac{\cos ^{2} \dfrac{x}{2}-\sin ^{2} \dfrac{x}{2}}{2 \cos ^{2} \dfrac{x}{2}+2 \sin \dfrac{x}{2} \cos \dfrac{x}{2}} d x \\ & =\int \frac{\cos \dfrac{x}{2}-\sin \dfrac{x}{2}}{2 \cos \dfrac{x}{2}} d x \\ & =\frac{1}{2} \int\left(1-\tan \frac{x}{2}\right) d x \\ & =\frac{1}{2}\left[x+2 \log \cos \frac{x}{2}\right]+C \end{aligned} \]