I have come across a very difficult integral that I do not know how to go forward with:
$$\int_{\mathbb{R}^3}\frac{1}{2\pi \hbar}\exp\left(\frac{-(x-x')^2-(p-p')^2+2i(p-p')(x+x')+4iyp-2(x-y)^2}{4\hbar}\right)dxdpdy$$
I only want to integrate over $x$ and $p$, and Mathematica tells me it should end up as:
$$\int_{\mathbb{R}}\exp\left(\frac{4iyp'-(x'-y)^2}{4\hbar}\right)dy$$
but I am unsure of how to realize this. I tried to substitute in new variables for $(x-x')$ and $(p-p')$ but the imaginary part of $2i(p-p')(x+x')$ gives me trouble. I tried switching to some polar coordinates in $x$ and $p$ but once again that term gives me trouble. I tried completing the square, and made no headway. I went to wikipedia, and tried to find some gaussian form that this looked like, and again made no headway as that aforementioned term causes trouble. If anyone could provide some insight, I would be most appreciative.