Several threads (e.g. Integration of the product of pdf & cdf of normal distribution ) have shown that

$E[\Phi(x)]=\Phi(\mu/\sqrt{\sigma^2+1})$ when $x\sim N(\mu,\sigma^2)$.

I'd like to compute $Var(\Phi(x))$ for such an $x$, ideally without numerically integrating. Does anyone have any ideas?