I'd like a closed form solution for $E(X_1 \mid X_2 > X_3)$ where $(X_1, X_2, X_3)$ is multivariate normal with possibly arbitrary mean vector and covariance matrix.

The conditional distribution $f(X_1 \mid X_2, X_3)$ 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 $X_2 > X_3$. Any help would be appreciated!