I am trying to compute the following integral
$$ \int_{-\infty}^\infty e^{i \omega x} e^{i k^2 \sinh(x)} dx $$
with $k^2$ a positive, real constant.
For $k^2 = 1$, one can compute this integral numerically on Mathematica with seeming great precision. The imaginary part of the integral vanishes due to a parity argument, and the plot of the real part of the integral as a function of $\omega$ is produced here: Image of the Fourier transform. Because the plot looks quite nice, I hope that one might express this Fourier transform in terms of some known functions. I have been able to prove via saddle point methods that the Fourier transform decays exponentially in $\omega$ as $\omega \rightarrow \pm \infty$ and have shown rigorously that the Fourier transform is bounded for any choice of $k^2$. Nevertheless, the actual answer (if there is, indeed, a nice expression) continues to evade me. I have tried to compute the Fourier transform of $\exp(ik^2 e^{\pm \theta})$ in effort to use the convolution theorem to derive the final result. Unfortunately, this method ultimately leads to an integral of a product of $\Gamma$ functions, which seems even more challenging. I have also tried a few more standard tricks with no success.
Any help would be greatly appreciated!
P.S. In fact, I am ultimately hoping to compute the more general integral
$$ \int_{-\infty}^\infty e^{i \omega x} e^{i k^2 \sinh(x)} \frac{\cosh(x + i \pi \alpha)}{\cosh(x)}dx $$
where $\alpha \in \mathbb{R},$ but the $\alpha = 0$ case of the first integral may be an easier place to start. Thoughts in this direction would be particularly helpful.