Integral involving Airy functions and exponential function

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Has anyone any idea about the following integral:

$$\int_ {-\infty}^\infty \operatorname{Ai}(x)~\operatorname{Ai}(x-2it) e^{2(x-it-0.5t^2)} dx $$ Where x is a complex number. I'm studying "Airy Functions and Applications to Physics" (Soares and Valee) but I cannot figure it out.