$\int \sqrt{\frac{{\cot }^{2}x.\log \left(\sin x\right)}{1-{\left(\log \left(\sin x\right)\right)}^3}}dx=---$

$\int \sqrt{\frac{{\cot }^{2}x.\log \left(\sin x\right)}{1-{\left(\log \right(\sin x\left)\right)}^3}}dx$

$\begin{array}{l}=\int \cot x\frac{\sqrt{\log (\sin x)}}{\sqrt{1-{\left([\log (\sin x){]}^{\frac{3}{2}}\right)}^2}}dx\\ \text{ Let }(\log (\sin x){)}^{\frac{3}{2}}=t\\ \frac{3}{2}\cot x\sqrt{\log (\sin x)}dx=dt\\ =\frac{2}{3}\int \frac{1}{\sqrt{1-t^2}}dt=\frac{2}{3}{\sin }^{-1}\left(t\right)+c\end{array}$

$=\frac{2}{3}{\sin }^{-1}\left({\left(\log \right(\sin x\left)\right)}^{\frac{3}{2}}\right)+c$