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I am looking for approximated or exact solution of

\begin{align}

I = \int_R \exp(cx^3-ax^2+bx)dx

\end{align}

where $a,b,c$ are complex numbers defined as:

\begin{align}

c &= \frac{1}{3}i\pi\phi'''(t) \notag\\

a &= \dfrac{1}{2\sigma^2}-i\pi \phi''(t) = re^{i\varphi}~~\text{with}~~~ r = \dfrac{1}{2\sigma^2}\sqrt{1+4\pi^2\sigma^4\phi''(t)^2} ~~\text{and}~~\varphi = arctan(-2\pi\sigma^2\phi''(t))\notag\\

b &= -i2\pi\eta

\end{align}

The fact that I computed the following :

\begin{align}

\int_{\mathbb{R}} \exp(i\alpha x^3)dx = \frac{2}{3} \frac{\alpha^{-1/3}\pi}{\Gamma(\frac{2}{3})}

\end{align}

Any help is greatly appreciated!

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# I Integral of third order polynomial exponential

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