I need to find the inverse Laplace transform with respect to $s$ of the following function:
$$\tilde{F}\left(s,y,\omega\right)=\dfrac{s\cos\left(py\right)\cos\left(qd\right)}{4s^{2}pq\sin\left(pd\right)\cos\left(qd\right)+\left(q^{2}+s^{2}\right)^{2}\cos\left(pd\right)\sin\left(qd\right)}$$
where:
$$p=\sqrt{\frac{\omega^{2}}{c_{L}^{2}}+s^{2}}\qquad q=\sqrt{\frac{\omega^{2}}{c_{T}^{2}}+s^{2}}$$
Here $c$ is a positive constant and $\omega$, $y$ can take real values.
This is quite a monster and I don't know if there is a closed form for $\mathcal{L}^{-1}\left[\tilde{F}\right]$. So my question is twofold. Can you tell whether $\mathcal{L}^{-1}\left[\tilde{F}\right]$ will have a closed form? If there is a closed form, which is it? Moreover, any suggestions to simplify are welcome. Thanks.
Liouville's "Rational Integral Theorem" may be of some use for your first question, although I'm not that sure without looking into it further. See Fitt's paper, p.230.