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 $$\begin{array}{rl}F(s)& =\ln (1+\frac{{\omega }^2}{s^2})=\ln (s^2+{\omega }^2)-\ln s^2\\ F^{\prime }(s)& =\frac{2s}{s^2+{\omega }^2}-\frac{2}{s}\\ & =L(2\cos \omega t)-L(2)=L(2\cos \omega t-2)\end{array}$$

We know $$\begin{array}{rl}{L}^{-1}(F^{\prime }(s))& =-tf(t)\\ ⟹2\cos \omega t-2& =-tf(t)\\ f(t)& =2\frac{1-\cos \omega t}{t}\end{array}$$