Answer
$$ - \operatorname{sech} x + C$$
Work Step by Step
$$\eqalign{
& \int {{{\operatorname{sech} }^2}x\sinh xdx} \cr
& {\text{use }}\operatorname{sech} x = \frac{1}{{\cosh x}} \cr
& = \int {\left( {\frac{1}{{{{\cosh }^2}x}}} \right)} \sinh xdx \cr
& = \int {\left( {\frac{1}{{\cosh x}}} \right)\left( {\frac{{\sinh x}}{{\cosh x}}} \right)} dx \cr
& {\text{hyperbolic identities}} \cr
& = \int {\operatorname{sech} x\tanh x} dx \cr
& {\text{find the antiderivative}} \cr
& = - \operatorname{sech} x + C \cr} $$
