Chapter 8 - Techniques of Integration - 8.2 Trigonometric Integrals - Exercises - Page 403: 42

$$ -\frac{1}{3}\cot ^{3} x-\frac{1}{5}\cot ^{5} x+c$$

\begin{align*} \int \cot ^{2} x \csc ^{4} x d x&=\int \cot ^{2} x \csc ^{2} x \csc ^{2} xd x\\ &=\int \cot ^{2} x(1+\cot ^{2} x ) \csc ^{2} xd x\\ &=\int( \cot ^{2} x+\cot ^{4} x ) \csc ^{2} xd x\\ &= -\frac{1}{3}\cot ^{3} x-\frac{1}{5}\cot ^{5} x+c \end{align*}