Chapter 8 - Techniques of Integration - 8.4 Integrals Involving Hyperbolic and Inverse Hyperbolic Functions - Exercises - Page 415: 16

$$\ln|\cosh x|-\frac{1}{2}\tanh ^2x+C$$

\begin{aligned} \int \tanh ^{3} x d x &=\int \tanh ^{2} x \tanh x d x \\ &=\int\left(1-\operatorname{sech}^{2} x\right) \tanh x d x \\ &=\int \tanh x-\operatorname{sech}^{2} x \tanh x d x \\ &=\int \tanh x d x-\int \operatorname{sech}^{2} x \tanh x d x \\ &=\ln|\cosh x|-\frac{1}{2}\tanh ^2x+C \end{aligned}