Let $a\in [0,1]$, $\alpha\in [0,2\pi)$, $\beta\gg 1$. Viewing $a,\beta$ as fixed, consider the integral $$I_\alpha :=\int_{-\infty}^\infty e^{-|x-\alpha|^2 + \beta a\cos(x)}dx.$$ I want to show that this integral is maximized for $\alpha=0$.
In the case where $a=0$, this is straightforward by using the Fourier transform of the Gaussian. For $a\in (0,1]$, it seems intuitive that if $\alpha$ moves closer to zero, the maximum of $e^{-|x-\alpha|^2}$ moves closer to the maximum of $e^{\beta a \cos(x)}$, and therefore $I_\alpha$ should increase. But I am struggling to formalize this intuition. Computing the first derivative with respect to $\alpha$ and showing that it vanishes if and only if $\alpha=0$ doesn't seem so straightforward, as now I have a more complicated integral to deal with.