We can solve this integral by employing our brute force scheme, or using importance sampling and random variables distributed according to a gaussian PDF. For the latter, if we set the mean value \( \mu=0 \) and the standard deviation \( \sigma=1/\sqrt{2} \), we have $$ \begin{equation*} \frac{1}{\sqrt{\pi}}\exp{(-x^2)}, \end{equation*} $$ and using this normal distribution we rewrite our integral as $$ \begin{equation*} \pi^3\int\prod_{i=1}^6\left( \frac{1}{\sqrt{\pi}}\exp{(-x_i^2)}\right) (\mathbf{x}-\mathbf{y})^2dx_1.\dots dx_6. \end{equation*} $$