$\int \sqrt{\frac{{\cos }^{3}x}{{\sin }^{11}x}}dx=$

$I=\int \sqrt{\frac{{\cos }^{3}x}{{\sin }^{11}x}}dx$

$=\int \sqrt{\frac{{\cot }^3\mathrm{x}}{{\sin }^8\mathrm{x}}}\mathrm{dx}=\int \left({\cot }^{3/2}\right)\cdot {\mathrm{cosec}}^4\mathrm{x}\mathrm{dx}$

$$\begin{gathered} =\int \left({\cot }^{3/2}x\right)\left(1+{\cot }^{2}x\right){\mathrm{cosec}}^{2}xdx \\ put cot x=t\Rightarrow -\cos e{c}^{2}x dx=dt \end{gathered}$$

$I=\int t^{3/2}\left(1+t^2\right)(-dt)$

$=-\frac{t^{5/2}}{5/2}-\frac{t^{9/2}}{9/2}+c$

$=-2\left[\frac{1}{5}\cot x+\frac{1}{9}{\cot }^{9/2}\right]+c$

$=-2\left[\frac{1}{5}{\tan }^{-5/2}x+\frac{1}{9}{\tan }^{-9/2}\right]+c$