Chapter 8 - Techniques of Integration - 8.4 Integrals Involving Hyperbolic and Inverse Hyperbolic Functions - Exercises - Page 415: 15

$$\frac{1}{32} \sinh (4 x)-\frac{1}{8} x+C $$

\begin{aligned} \int \sinh ^{2} x \cosh^{2} x d x &=\frac{1}{4} \int \sinh ^{2}(2 x) d x \\ &=\frac{1}{8} \int \cosh (4 x)-1 d x \\ &=\frac{1}{32} \sinh (4 x)-\frac{1}{8} x+C \end{aligned}