$$\int_{-\infty}^\infty \exp (a t^3 + b t^2 + c t) \mathrm{d}t,\;\;(a,c)\in\mathbb{I}, \; \; \Re(b)\le0$$
i.e. an oscillation with frequency $3\Im(a)t^2 + 2\Im(b)t + \Im(c)$ and phase $0$, multiplied by a gaussian window $\exp(\Re(b)t^2)$
Does this even have an analytic form? What sort of methods might help evaluate it?
In the case $b=0,\;\Im(a)>0$, defining $p = \Im(a)$ and $q = \Im(c)$:
$$I = \int_{-\infty}^\infty \exp(i(p t^3 + q t)) \mathrm{d}t = 2\int_0^\infty \cos(p t^3 + q t) \mathrm{d}t$$
define $t^\prime = (3p)^{-\frac{1}{3}}t \implies \mathrm{d}t^\prime = (3p)^{-\frac{1}{3}} \mathrm{d t}$
$$\begin{align} I =& \;2 \int_0^\infty \cos(p ((3p)^{-\frac{1}{3}}t)^3 + q (3p)^{-\frac{1}{3}}t) \mathrm{d}t^\prime \\ =& \;2(3p)^{-\frac{1}{3}} \int_0^\infty \cos(\frac{1}{3}t^3 + q (3p)^{-\frac{1}{3}}t) \mathrm{d}t\\ =& \;2\pi(3p)^{-\frac{1}{3}} \mathrm{Ai}((3p)^{-\frac{1}{3}} q) \end{align}$$
where $\mathrm{Ai}$ is the Airy function.
I don't mind the $\Im(a)>0$ constraint, but I really need the whole $b$ domain.
given $A=\exp(\frac{2 b^3}{27 a^2} - \frac{bc}{3a})$, $T = t + \frac{b}{3a}$ and $\xi = \Im(\frac{b}{3a}) = \frac{-\Re(b)}{3 \Im(a)}$ $$ I = A \int_{-\infty + i \xi}^{\infty + i \xi} \exp(a T^3 + (c - \frac{b^2}{3a})\;T\:)\; \mathrm{d}T $$
but my complex analysis skills are too weak to get me past there... Also, $\Re(\frac{b^2}{3a}) \ne 0$ so I don't think it converges.
It is important to note that the depressed cubic approach is only suitable when $\dfrac{b}{3a}\in\mathbb{R}$ , i.e. $b\in\mathbb{I}$
For general $b$ , we can use the ODE approach:
Let $I=\int_{-\infty}^\infty e^{at^3+bt^2+ct}~dt$ ,
Then $\dfrac{dI}{dc}=\int_{-\infty}^\infty te^{at^3+bt^2+ct}~dt$
$\dfrac{d^2I}{dc^2}=\int_{-\infty}^\infty t^2e^{at^3+bt^2+ct}~dt$
$\therefore3a\dfrac{d^2I}{dc^2}+2b\dfrac{dI}{dc}+cI=\int_{-\infty}^\infty(3at^2+2bt+c)e^{at^3+bt^2+ct}~dt=\int_{-\infty}^\infty e^{at^3+bt^2+ct}~d(at^3+bt^2+ct)=[e^{at^3+bt^2+ct}]_{-\infty}^\infty=0$
This belongs to the ODE of the form http://eqworld.ipmnet.ru/en/solutions/ode/ode0204.pdf, which still can get the general solution in terms of the Airy function.