Integral of a product of a cosine function with argument x and a confluent hypergeometric function with argument $x^2$

Question

I'm trying to integrate the following

\begin{equation} \int_0^1{_1 F_1 (a;2a;i \alpha x^2)}\cos{\beta x}dx \end{equation}

where $\alpha$ and $\beta$ are real numbers.

Answer 1

You integral equals:

$$ \int_{0}^{1}\cos(\beta x)\sum_{n\geq 0}\frac{\Gamma(a+n)\Gamma(2a)}{n!\Gamma(a)\Gamma(2a+n)}(i\alpha x^2)^n \,dx\\ =\frac{1}{B(a,a)}\int_{0}^{1}\sum_{n\geq 0}\frac{B(a,a+n)}{n!}(i\alpha x^2)^n\cos(\beta x)\,dx\\=\frac{1}{B(a,a)}\int_{0}^{1}\int_{0}^{1}z^{a-1}(1-z)^{a-1}e^{i\alpha zx^2}\cos(\beta x)\,dz\,dx \\=\text{Re}\left[\frac{1}{B(a,a)}\int_{0}^{1}\int_{0}^{1}z^{a-1}(1-z)^{a-1}e^{i(\alpha zx^2+\beta x)}\,dz\,dx\right].$$