I should solve the following tricky integral:
$$\int_{0}^{+\infty}~\text{e}^{-ak^2}\left(\frac{b+ck^2}{\sqrt{(k^2-\alpha)(k^2-\beta)}}\sinh(d\sqrt{(k^2-\alpha)(k^2-\beta)})+\cosh(d\sqrt{(k^2-\alpha)(k^2-\beta)})\right)dk $$
with $a,\,b,\,c,\,d>0$ and $\alpha,\,\beta\in\mathbb{C}$.
Thanks
Since $a,d$ depends on time, if I compute the Laplace transform of the integrand, I obtain (making all the constants more explicit):
$$\mathcal{L^{-1}}\int_0^{+\infty} \exp(-4\pi^2\omega^2k^2)\,\frac{s+\lambda+\gamma+k^2(D_1+D_2)}{(s+\lambda+k^2D_1)(s+\gamma+k^2D_2)-\lambda\gamma}dk$$
where $D_1,\,D_2,\,\lambda,\,\gamma,\,\omega>0$.
That i'm not quite sure if it might help or not!