Computing the integral $\int_{-\pi/2}^{\pi/2}\cos^2(x) \exp(\cos^2(x))dx$
Mathematica says:
$\frac{1}{2}\sqrt{e}\pi\left(I_0\left(\frac{1}{2}\right)+I_1\left(\frac{1}{2}\right)\right)$
For the original integral.
Since the definition of the Bessel function $I$ is close to your integral, I don't think there is much to add.