$\int_{-\infty}^{\infty}\frac{1}{3 t^2+1} {\rm exp}\left(\frac{at^2+ibt}{3 t^2+1}+itx\right){\rm d}t$ How to solve the integral?
$$
f(x)=\int_{-\infty}^{\infty}\frac{1}{3 t^2+1} {\rm exp}\left(\frac{a t^2+i b t}{3 t^2+1}+itx\right){\rm d}t\tag{1}
\\{\rm with}\,\, x,b\in \mathbb{R},a\in\mathbb{R}_{<0},i^2=-1$$
This question was previously posted in Mathematica SE, however algorithms of Mathematica couldn't solve it. A deeper mathematical analysis is needed. In the solution a potpourri of Error function, Bessel function, hypergeometric function or infinite series of them can be expected (see these solved similar integrals: integral1, integral2, integral3).
A plot of $f(x)$ for $a=-200,b=-100$:

 A: In a hurry, I pulled out of my head the equality of equations (1) & (2)  but had no time to write down all the steps or reduce to simpler functions that seems to be possible with low effort. I leave that for other users.
$$\begin{align}
f(x)&=\int_{-\infty}^{\infty}\frac{1}{3 t^2+1} {\rm exp}\left(\frac{a t^2+i b t}{3 t^2+1}+itx\right){\rm d}t\tag{1}\\
g(x)&=\frac{\pi e^{a/3}}{{\sqrt{3}}} \left(
\begin{array}{l}
 \begin{array}{l}
 \sum _{k,v=0}^\infty  \frac{c_-^k c_+^v}{k! (v!)^2}\left(\frac{-2x}{\sqrt{3}}\right)^{(k+v)/2}W_{(v-k)/2,-(k+v+1)/2}\left(\frac{-2x}{\sqrt{3}}\right) & x<0 \\
 \sum _{k,v=0}^\infty  \frac{c_-^k c_+^v}{(k!)^2 v!}\left(\frac{2x}{\sqrt{3}}\right)^{(k+v)/2}W_{(k-v)/2,-(k+v+1)/2}\left(\frac{2x}{\sqrt{3}}\right) & x>0 \\
 \sum _{k,v=0}^\infty  \frac{c_-^k c_+^v}{k! (v!)^2}(k+1)^{\overline{v}} & x=0 \\
\end{array}
\end{array}
\right)\tag{2}
\end{align}$$
with
$c_{\pm}=-\frac{a}{12}\pm\frac{b}{4 \sqrt{3}}$, rising factorial $(\cdot)^{\overline{v}}$, and Whittaker's $W$ function
$$W_{(k-v)/2,-(k+v+1)/2}(x)=\sqrt{\dfrac{x^{k-v}}{e^x}}\sum _{s=0}^k \binom{k}{s} \frac{(v+1)^{\overline{s}}}{x^s}\tag{3}$$
A comparison of $f(x),g(x),|f(x)-g(x)|$ for $a=-20,b=-10$ reveals numerical insufficiencies. They are higher for the example given in the question $(a=-200,b=-100)$.

