Inverse laplace transform of $\frac{\tanh\sqrt{j\omega}}{\sqrt{j\omega}-\tanh\sqrt{j\omega}}$

Question

Good morning, I am struggling in finding the inverse Laplace transform of the following function

$\mathcal{L}_s^{-1}\biggl[\frac{\tanh\sqrt{j\omega}}{\sqrt{j\omega}-\tanh \sqrt{j\omega}}\biggl]$

For help I know the antitrasform of

$\mathcal{L}_s^{-1}\biggl[\frac{k_2 }{\sqrt{j\omega}}\tanh\left(\frac{k_1}{k_2}\sqrt{j\omega}\right)\biggl] = 2\frac{k_2^2}{k_1} \sum_{n=1}^\infty \exp\left(-\frac{(2n-1)^2\pi^2k_2^2}{4k_1^2}\,s\right) $

Thanks for any support.

Answer 1

Assuming that you mean $s=j\omega$, one can divide numerator and denominator by $\tanh\sqrt{s}$ and arrive at the problem posed here.

Edit: I have updated my answer to that question and now provide semi-analytical and numerical results to it.