Example 3. Find the integral $\int \operatorname{sh}^{2} x \operatorname{ch}^{2} x d x$.
Solution.

$$
\begin{gathered}
\int \sinh^{2} x \cosh^{2} x d x=\int(\sinh x \cosh x)^{2} d x=\int\left(\frac{1}{2} \sinh 2 x\right)^{2} d x= \\
=\frac{1}{4} \int \sinh^{2} 2 x d x=\frac{1}{4} \int \frac{\cosh 4 x-1}{2} d x=\frac{1}{8} \int \cosh 4 x d x-\frac{1}{8} \int d x= \\
=\frac{1}{32} \int \cosh 4 x d(4 x)-\frac{1}{8} \int d x=\frac{1}{32} \sinh 4 x-\frac{1}{8} x+C
\end{gathered}
$$