Chapter 8 - Techniques of Integration - Chapter Review Exercises - Page 460: 30

$$-\frac{1}{3}\csc ^3\left(x\right)+\csc \left(x\right)+C$$

\begin{aligned} \int \cot ^{3} x \csc x d x &=\int \cot ^{2} x \cot x \csc x d x \\ &=\int\left(\csc ^{2} x-1\right) \cot x \csc x d x \\ &=\int \csc ^{3} x \cot x-\cot x \csc x d x \\ &=\int \csc ^{3} x \cot x d x-\int \cot x \csc x d x\\ &=-\frac{1}{3}\csc ^3\left(x\right)+\csc \left(x\right)+C \end{aligned}