Forgive me for asking this question. I am deriving the Wigner function, $$ W\left(x_{1},p_{1},x_{2},p_{2}\right)=\frac{1}{4\pi^{2}}\int dx'_{1}dx'_{2}e^{-ip_{1}x'_{1}-ip_{2}x'_{2}}\Psi\left(x_{1}+\frac{x'_{1}}{2},x_{2}+\frac{x'_{2}}{2}\right)\Psi^{\ast}\left(x_{1}-\frac{x'_{1}}{2},x_{2}-\frac{x'_{2}}{2}\right), $$ for $\Psi\left(x_{1},x_{2}\right)=\frac{\left(x_{1}+x_{2}\right)}{\sqrt{\pi\sigma_{-}\sigma_{+}^{3}}}e^{-\frac{\left(x_{1}+x_{2}\right)^{2}}{4\sigma_{+}^{2}}}e^{-\frac{\left(x_{1}-x_{2}\right)^{2}}{4\sigma_{-}^{2}}}$.
While calculating this, I encounter a problem solving this part, $$ \int dx'_{1}dx'_{2}\left(x'^{2}_{1}+2x'_{1}x'_{2}+x'^{2}_{2}\right)\exp\left\{-\frac{1}{4}\left(\frac{1}{\sigma^{2}_{+}}+\frac{1}{\sigma^{2}_{-}}\right)x'^{2}_{1}-\frac{1}{2}\left(\frac{1}{\sigma_{+}^{2}}-\frac{1}{\sigma_{-}^{2}}\right)x'_{1}x'_{2}-\frac{1}{4}\left(\frac{1}{\sigma_{+}^{2}}+\frac{1}{\sigma_{-}^{2}}\right)x'^{2}_{2}-ip_{1}x'_{1}-ip_{2}x'_{2}\right\} $$ Could you help me solve this particular integral? It would really helped me a lots.
Thank you!
Reference: S.P. Walborn, B.G. Taketani, A. Salles, F. Toscano, and R.L. de Matos Filho, Entropic Entanglement Criteria for Continuous Variables, Phys. Rev. Lett. 103, 160505 (2009)