Suppose you have a general $n$-th dimensional random Gaussian vector with probability distribution function $\mathcal{N}\left(\mathbf{x}|\boldsymbol{\mu},\boldsymbol{\Sigma}\right)$. What is the computational complexity (both space or time) of calculating the cumulative distribution function

$\int_{-\infty}^{y_1}dx_{1}\cdots\int_{-\infty}^{y_n}dx_{n}\mathcal{N}\left(\mathbf{x}|\boldsymbol{\mu},\boldsymbol{\Sigma}\right) $

?

This should be a function of $n$. If the suggested integration method is approximate (e.g., using monte-carlo methods), then the complexity may also depend on some required mean relative accuracy (std/mean) $\epsilon$.

Thanks in advance!