Example 4. Find the integral $\int \operatorname{coth}^{2} x d x$.
Solution.

$$
\begin{aligned}
\int \operatorname{cth}^{2} x d x & =\int \frac{\operatorname{ch}^{2} x}{\operatorname{sh}^{2} x} d x=\int \frac{1+\operatorname{sh}^{2} x}{\operatorname{sh}^{2} x} d x=\int \frac{d x}{\operatorname{sh}^{2} x}+\int d x= \\
& =-\operatorname{cth} x+x+C=x-\operatorname{cth} x+C
\end{aligned}
$$