Answer
$$-\frac{1}{2} \cot x \csc x+\frac{1}{2} \ln |\csc x-\cot x|+c$$
Work Step by Step
Given $$\int \csc ^{3} x d x $$
Use
$$\int \csc ^{m} x d x=-\frac{\cot x \csc ^{m-2} x}{m-1}+\frac{m-2}{m-1} \int \csc ^{m-2} x d x $$
Then
\begin{aligned} \int \csc ^{3} x d x &=-\frac{\cot x \csc x}{2}+\frac{1}{2} \int \csc x d x \\ &=-\frac{1}{2} \cot x \csc x+\frac{1}{2} \int \csc x d x \\ &=-\frac{1}{2} \cot x \csc x+\frac{1}{2} \ln |\csc x-\cot x|+c \end{aligned}
