Answer
$$\frac{1}{4}\sinh 2x - \frac{1}{2}x + C$$
Work Step by Step
$$\eqalign{
& \int {{{\sinh }^2}xdx} \cr
& {\text{hyperbolic identity }}{\sinh ^2}x = \frac{{\cosh 2x - 1}}{2} \cr
& = \int {\frac{{\cosh 2x - 1}}{2}} dx \cr
& {\text{split the integrand}} \cr
& = \int {\frac{{\cosh 2x}}{2}} dx - \int {\frac{1}{2}} dx \cr
& = \frac{1}{4}\int {\cosh 2x\left( 2 \right)} dx - \int {\frac{1}{2}} dx \cr
& {\text{find the antiderivative}} \cr
& = \frac{1}{4}\sinh 2x - \frac{1}{2}x + C \cr} $$
