I'd like a closed form solution for $E(X1 | X2 > X3)$ where $(X1, X2, X3)$ is multivariate normal with possibly arbitrary mean vector and covariance matrix.

The conditional distribution $f(X1 | X2, X3)$ is easy to get in closed form, but I am not sure if there is a closed-form solution to handling conditioning on the event $X2 > X3$. Any help would be appreciated!