\begin{align} L^{-1}\roundbr{\frac{s}{\roundbr{s^{2} - 9}^{2}}} &= \frac{1}{12}L^{-1}\roundbr{\frac{12s}{\roundbr{(s - 3)^{2}{(s + 3)^{2})}}}}\newline &= L^{-1}\roundbr{\frac{1}{12}\roundbr{\frac{1}{(s-3)^{2}} - \frac{1}{(s+3)^{2}}}}\newline &= L^{-1}\roundbr{\frac{1}{12}\roundbr{L\roundbr{e^{3t}t} - L\roundbr{e^{-3t}t}}}\newline \implies f(t) &= \frac{t}{6} \frac{e^{3t} - e^{-3t}}{2} = \frac{t}{6}\sinh 3t \end{align}