Consider the two independent random variables $X$ and $Y$ where each random variable is 3-dimensionally normal distributed with $X \sim \mathcal{N}(\mathbf{0},\Sigma_X)$ and $Y \sim \mathcal{N}(\mathbf{\mathbf{\mu}},\Sigma_Y)$, with covariance matrices $\Sigma_X$ and $\Sigma_Y$ and mean $\mathbf{0} $ and $\mathbf{\mu},$ respectively.
My Question is: What is the probability that $X$ and $Y$ have a distance that is smaller than the number $r$, i.e. $P(\lVert X - Y \rVert^2 < r^2) $?
I am also particularly interested in the special cases $\Sigma_X = \sigma_X^2 \mathbf{I} $ and $\Sigma_Y = \sigma_Y^2 \mathbf{I} $.