Answer
$$\frac{-1}{9}\csc^{9} x+\frac{2}{7} \csc ^{7} x-\frac{1}{5}\csc ^{5} x+c$$
Work Step by Step
\begin{align*}
\int \cot ^{5} x \csc ^{5} x d x&=\int\left(\csc ^{2} x-1\right)^{2} \csc ^{4} x(\cot x \csc x) d x\\
&=\int\left(\csc ^{8} x-2 \csc ^{6} x+\csc ^{4} x\right)(\cot x \csc x d x) \\
&=\frac{-1}{9}\csc^{9} x+\frac{2}{7} \csc ^{7} x-\frac{1}{5}\csc ^{5} x+c
\end{align*}
