Several threads (e.g. http://mathoverflow.net/questions/101469/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?