Chapter 8 - Techniques of Integration - 8.4 Integrals Involving Hyperbolic and Inverse Hyperbolic Functions - Exercises - Page 415: 14

$$\frac{\cosh ^{9} x}{9}-\frac{\cosh ^{7} x}{7}+C$$

\begin{aligned} \int \sinh ^{3} x \cosh ^{6} x d x &=\int \sinh ^{2} x \cosh ^{6} x \sinh x d x \\ &=\int\left(\cosh ^{2} x-1\right) \cosh ^{6} x \sinh x d x \\ &=\int \cosh ^{8} x \sinh x-\cosh ^{6} x \sinh x d x \\ &=\int \cosh ^{8} x \sinh x d x-\int \cosh ^{6} x \sinh x d x \\ &=\frac{\cosh ^{9} x}{9}-\frac{\cosh ^{7} x}{7}+C \end{aligned}