\(\frac{(1-\cos x)^{\frac{2}{7}}}{(1+\cos x)^{\frac{9}{7}}} dx\) is

\(\frac{7}{11}\left(\tan \frac{x}{2}\right)^{\frac{11}{7}}+c\)

\(\begin{aligned}\int \frac{(1-cos x)^{\frac{2}{7}}}{(1+\cos x)^{\frac{9}{7}}}dx&=\frac{1}{2}\int \frac{\left(\sin \frac{x}{2}\right)^{\frac{4}{7}}}{\left(\cos \frac{x}{2}\right)^{\frac{18}{7}}}dx\\&=\frac{1}{2}\int \left(\tan \frac{x}{2}\right)^{\frac{4}{7}}\sec^{2} \frac{x}{2} dx\\ &=\frac{1}{2} \int 2t^{\frac{4}{7}}dt\\ &=\frac{7}{11}\left(\tan \frac{x}{2}\right)^{\frac{11}{7}}+c\end{aligned}\)