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Without loss of generality, the problem is equivalent to computing the probability $P(X_1 \geq \max(X_2, \ldots, X_n))$. We can transform the coordinates as $X_2-X_1, X_3-X_1, \ldots, X_n-X_1$ which is a fully general multivariate normal distribution of degree $n-1$. The problem is thus equivalent to finding the orthant probability for which there is no known closed-form solution for $n-1>3$