Chapter 8 - Techniques of Integration - 8.2 Trigonometric Integrals - Exercises - Page 403: 21

$$-\frac{1}{6} \cot ^{6} x+c$$

Let $$ u= \cot x\ \ \ \Rightarrow \ \ \ du =-\csc^2 xdx$$ Then \begin{aligned} \int \cot ^{5} x \csc ^{2} x d x &=-\int u^{5} d u \\ &=-\frac{1}{6} u^{6}+c \\ &=-\frac{1}{6} \cot ^{6} x+c \end{aligned}