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 $$\begin{equation*} \pi^3\int\prod_{i=1}^6\left( \frac{1}{\sqrt{\pi}}\exp{(-x_i^2)}\right) (\mathbf{x}-\mathbf{y})^2dx_1.\dots dx_6. \end{equation*}$$