Closed form solution to $\int_0^{\pi/4}\frac{e^{ic\sqrt{1+ br^2(\theta)}}}{\sqrt{1+ br^2(\theta)}}\,d\theta$

Question

Is there closed form solution to this integral $$\int_0^{\pi/4}\frac{e^{ic\sqrt{1+ br^2(\theta)}}}{\sqrt{1+ br^2(\theta)}}\,d\theta$$

$r(\theta)=\frac{a}{\cos(\theta)}$ is radius vector from the origin tracing square with side 2a (0,45 deg).

$c=$ constant, $b=$ constant.