Truncated trivariate normal - conditional expectation

I am working on a paper in which I'd need to use the two following conditional expectations:

1. $E(X_{1}|a \leq X_{2} \leq b)$

2. $E(X_{1}|a \leq X_{2} \leq b, a \leq X_{3} \leq b)$

where $X_{1}, X_{2}, X_{3}$ come from a trivariate normal distribution and $a$ and $b$ are some real numbers ($X_{2}$ and $X_{3}$ have the same bounds).

I have checked several textbooks and references on the web, including those mentioned in previous related posts (e.g., Tallis 1961), but did not find anything relevant. Could anyone help me derive these conditional expectations, or else point me to any good reference(s) I may have missed?