Chapter 6 - Applications of Integration - 6.10 Hyperbolic Functions - 6.10 Exercises - Page 503: 35

$$\tanh x + x + C$$

$$\eqalign{ & \int {{{\tanh }^2}xdx} \cr & {\text{hyperbolic identity 1}} - {\tanh ^2}x = {\operatorname{sech} ^2}x \cr & = \int {\left( {{{\operatorname{sech} }^2}x + 1} \right)} dx \cr & {\text{split the integrand}} \cr & = \int {{{\operatorname{sech} }^2}x} dx + \int {dx} \cr & {\text{find the antiderivative}} \cr & = \tanh x + x + C \cr} $$