$\int \frac{(1-\cos \mathrm{\theta }{)}^{2/7}}{(1+\cos \mathrm{\theta }{)}^{9/7}}\mathrm{d\theta }=$

$$\begin{gathered} \int \frac{(1-\cos \mathrm{\theta }{)}^{2/7}}{(1+\cos \mathrm{\theta }{)}^{9/7}}\mathrm{d\theta }\Rightarrow \int \frac{(1-\cos \mathrm{\theta }{)}^{2/7+1}}{(1+\cos \mathrm{\theta }{)}^{9/7}}\cdot \frac{1}{(1-\cos \mathrm{\theta })}\mathrm{d\theta } \\ \Rightarrow \int {\left({\tan }^2\frac{\mathrm{\theta }}{2}\right)}^{9/7}\cdot \frac{1}{2{\sin }^2\frac{\mathrm{\theta }}{2}}\mathrm{d\theta } \\ \text{ put }\cot \mathrm{\theta }/2=\mathrm{t}\Rightarrow -\frac{1}{2}{\mathrm{cosec}}^2\frac{\mathrm{\theta }}{2}\mathrm{d\theta }=\mathrm{dt} \\ =\int {\left(\frac{1}{{\mathrm{t}}^2}\right)}^{9/7}\cdot \left(-\mathrm{dt}\right) \\ =\frac{7}{11}{\mathrm{t}}^{\frac{-11}{7}}+\mathrm{c} \\ =\frac{7}{11}{\left(\tan \frac{\mathrm{\theta }}{2}\right)}^{11/7}+\mathrm{c} \end{gathered}$$