I am reading Flajolet et al. Airy Phenomena and Analytic Combinatorics of Connected Graphs. 2004.
We are presented with the following integral
\begin{equation} \frac{1}{\sqrt{2\pi}}\int^\infty_{-\infty}\exp\left(-\frac 12 z^2+xe^{iz\sqrt{\log q}}\right)dz \end{equation}
The integral is not carried out in closed form, but does have asymptotic results. I am wondering if a closed form solution exists in the near 20 years since this paper was published?