Let $\varepsilon$ be a Gaussian distributed random variable with mean $\mu_0$ and standard deviation $\sigma_0$. Is it possible to compute/approximate the expected value
$$ \begin{eqnarray} & &\mathbb{E}\left[\Phi\left(\frac{\varepsilon-\mu_1}{\sigma_1}\right)\mid\varepsilon<c\right]=\cdots\\ & &\cdots =\int_{-\infty}^{c}d\varepsilon \frac{1}{\sqrt{2\,\pi\,\sigma_0^2}}\,\exp\left(-\frac{\left(\varepsilon-\mu_0\right)^2}{2\,\sigma_0^2}\right)\,\int_{-\infty}^{\frac{\varepsilon-\mu_1}{\sigma_1}}dt\,\frac{1}{\sqrt{2\,\pi}}\,\exp\left(-\frac{t^2}{2}\right) ? \end{eqnarray} $$
Note that while the integral
$$\int \exp\left(-x^2\right)\,\textrm{erf}\left(x\right)\,dx$$ exists (check on http://integrals.wolfram.com/index.jsp), any integral of the kind
$$ \int \exp\left(-(x-a)^2/b\right)\,\textrm{erf}\left((x-c)/d\right)\,dx $$ does not exist (at least according to Wolfram integrals).