Probability of multivariant gaussian random variables in different areas $\newcommand{\sgn}{\operatorname{sgn}}$Let $X_i$ is a gaussian random variable correlated with others. we want to find the probability of each possible case to find the expectation of following expression
$\mathbb{E} [\sgn(X_1) \sgn(X_2) \sgn(X_3) \sgn(X_4)] = \mathbb P(X_1>0, X_2>0, X_3>0, X_4>0) - \mathbb P(X_1<0, X_2>0, X_3>0, X_4>0) + \ldots$
here we have the closed form for $\mathbb P(X_1>0, X_2>0, X_3>0, X_4>0)$ based on this https://math.stackexchange.com/questions/869502/multivariate-gaussian-integral-over-positive-reals/3148280#3148280. but unfortunately not for other terms.
Please guide if you can. Thanks for your help in advance.
 A: If you know $P(X_1>0,X_2>0,X_3>0,X_3>0)$ for any multivariate Gaussian random vector $(X_1,X_2,X_3,X_4)$, then you know all such probabilities. E.g.,, $P(X_1<0,X_2<0,X_3>0,X_3>0)=P(Y_1>0,Y_2>0,Y_3>0,Y_4>0)$, where $(Y_1,Y_2,Y_3,Y_4):=(-X_1,-X_2,X_3,X_4)$ is multivariate Gaussian as well.
Similarly, if you know $P(X_1>0,X_2>0,X_3>0,X_3>0)$ for any zero-mean multivariate Gaussian random vector $(X_1,X_2,X_3,X_4)$, then you know all such probabilities. E.g.,, $P(X_1<0,X_2<0,X_3>0,X_3>0)=P(Y_1>0,Y_2>0,Y_3>0,Y_4>0)$, where $(Y_1,Y_2,Y_3,Y_4):=(-X_1,-X_2,X_3,X_4)$ is zero-mean multivariate Gaussian as well.

Added: According to a comment by the OP, the $X_i$'s are, in fact, zero-mean. In that case, one can express each of the probabilities in questions as the sum of $\binom42=6$ ordinary integrals each with an integrand that is the product of an algebraic expression and the $\arctan$ of an algebraic expression. This is done by formulas (7) and (6) of Plackett. Here the signs of the correlation coefficients do not matter, and the integrals are much, much simpler than the ones in the answer you linked in your post.
