The question is related to the Gaussian copula.
Let $\Phi(x)$ denote the cdf of standard Normal distribution. Let $(X_1, X_2) \sim \mathcal{N}(0,\Sigma)$ be joint Normal with covariance matrix $$ \Sigma = \begin{pmatrix} 1 & \rho \\ \rho & 1 \end{pmatrix}. $$ I wonder, does $\mathbb{E}\Big[\Phi(X_1)\Phi(X_2)\Big]$ has an analytical expression? I tried by myself but the integration looks very messy.