I am trying to compute the expected value of the ratio of two normal CDFs. Specifically, I like to compute the expected value of $\Phi(X+Y)/\Phi(X)$ where $X$ and $Y$ are independent normally distributed variables with means $\mu_X$ and $\mu_Y$ and variances $\sigma_X^2$ and $\sigma_Y^2$, and where $\Phi(.)$ is the standard normal CDF function.
I am not sure whether there is a closed form solution. In case there isn't any approximation to the derivative of the expected value in $\mu_X$ would be very helpful for my problem, too.
I would appreciate any help or suggestions where to look further.