I would like to evaluate the following function analytically
$$\int_{-\infty}^{\infty} ds \text{Ai}(s)e^{2\pi i s x}e^{\frac i 3 (2\pi s)^3}$$
I know from evaluating this numerically that the answer is of the form
$$c_0\text{Ai}(x)+c_1\text{Ai}'(x)$$
but I do not understand why.
Note that
$$\int_{-\infty}^{\infty} ds e^{2\pi i s x}e^{\frac i 3 (2\pi s)^3}=\text{Ai}(x)$$
Another potentially useful feature of the Airy function is orthogonality.