$S=\int \frac{[\cot (\log x){]}^5{\mathrm{cosec}}^2(\log x)}{x}dx(x>0)\text{ is equal to }$

$S=\int \frac{{\left[\cot \left(\log x\right)\right]}^5\cos e{c}^2\left(\log x\right)}{x}dx$

$\log x=y$

$\begin{array}{l}\frac{1}{x}dx=dy\\ S=\int (\cot y{)}^5\cos e{c}^2(y)dy\end{array}$

$\text{ Put }\cot y=z$

$\begin{array}{l}\cos e{c}^{2}ydy=-dz\\ S=\int z^5(-dz)\\ =\frac{-z^6}{6}+C\\ =-\frac{1}{6}(\cot y{)}^6+C\\ S=-\frac{1}{6}(\cot (\log x){)}^6+C\end{array}$