Let $s > 0, t \in \mathbf{R}$, and define
\begin{align} I (s, t) = \int_{x \in \mathbf{R}} \exp \left( -\frac{sx^2}{2} \right) \cdot \frac{\cosh (tx) }{ \cosh (x) } dx. \end{align}
I am seeking a moderately-concrete analytic form for $I$, e.g. if a pure analytic form is not available (which I consider fairly likely), then a rapidly-convergent infinite sum of analytic terms would do just fine.
Some things which I have tried (without success):
I have had a go at this integral with a pretty standard rectangular contour. Going upwards in the complex plane causes the Gaussian term to grow quickly, and so I am not sure how productive this approach is. Perhaps a different contour would be more forgiving.
I also tried some formal calculations where I took partial derivatives of $I$ with respect to $(s, t)$, which led me to the PDE $\partial_t^2 I + 2 \partial_s I = 0$, which looks to me like a backwards heat equation. I can compute that $I(s, 0) = \sqrt{2 \pi / s}$, but I suspect that this is not enough to determine a solution (noting that $s$ is the `time' variable of the PDE).
I represented $1/\cosh(x)$ as a scale mixture of Gaussians (i.e. $1/\cosh(x) \propto \int_{t > 0} p(t) \cdot \mathcal{N} (x | 0, t) dt$) and then exchanged the order of integration. This gives rise to an integral with respect to $t$ which is at least as challenging as the original form, as the particular $p(t)$ is not so nice.
Any suggestions for other approaches I could try? (or, indeed, a full solution)