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.