Integration by parts, hyperbolic functions

Question

I tried solving this hyperbolic integration using integration by parts, and it doesn’t seem to work. $$ \begin{gather} \int \sinh ^ { 2 } u d u \\ = \sinh u \cosh u - \cosh ^ { 2 } u \end{gather} $$ But this is the actual solution: $$ \frac { 1 } { 2 } ( - x + \frac { 1 } { 2 } \sinh ( 2 x ) ) + C$$ I know I could do this using identities or write it in terms of $e$, but why does this not work? Thank you so much!