Let us fix an integer $n\in2\mathbb{Z}+1$ and consider the function $f(x)=\exp(\pi i n \{x\})$ that is a step function attaining $1$ on intervals of the form $[2k,2k+1]$ and $-1$ on intervals of the form $[2k+1,k]$ (with $k\in\mathbb{Z}$). The integral $$\int_{\mathbb R}\exp(-ax^2+iJx)dx,\qquad a\in\mathbb R_+,J\in\mathbb R$$ is quite standard and can be computed using completion to a sqaure. However I wonder about the "modified" integral $$\int_{\mathbb R}f(x)\exp(-ax^2+iJx)dx,\qquad a\in\mathbb R_+,J\in\mathbb R.$$ In this case completion to square allegedly does not work and as long as $J\neq 0$ we lose a lot if we try to bound this integral naively by an absolute value.
Does the second integral has a closed form and can be computed explicitly?