Answer¶
The given transform is of no standard function. However, if we consider the derivative of this transform, we are greeted by a few familiar functions
\[
\begin{aligned}
F(s) &= \ln\roundbr{1 + \frac{\omega^{2}}{s^{2}}} = \ln\roundbr{s^{2} + \omega^{2}} - \ln s^{2}\newline
\diffone{F}(s) &= \frac{2s}{s^{2} + \omega^{2}} - \frac{2}{s}\newline
&= L\roundbr{2\cos \omega t} - L(2) = L\roundbr{2\cos \omega t - 2}
\end{aligned}
\]
We know
\[
\begin{aligned}
L^{-1}\roundbr{\diffone{F}(s)} &= -tf(t)\newline
\implies 2\cos \omega t - 2 &= -tf(t)\newline
f(t) &= 2\frac{1 - \cos \omega t}{t}
\end{aligned}
\]