Any solution for the following definite integaral? Here $\Phi(x)$ represents the cumulative distributive function of standard normal distribution $$\int_{\frac{-d}{\sqrt2\sigma}}^{\frac{d}{\sqrt2\sigma}} \frac{1}{2\sigma^2\sqrt{2\pi}} \left( \Phi\left(\alpha+\sqrt{d^2-\frac{x^2}{2\sigma^2}}\right)-\Phi\left(\alpha-\sqrt{d^2-\frac{x^2}{2\sigma^2}}\right)\right) \exp\left(\frac{-x^2}{2}\right) \; dx$$
The above function can be represented in terms of error function as $$\frac{1}{4\sigma^2\sqrt{2\pi}}\int_{\frac{-d}{\sqrt2\sigma}}^{\frac{d}{\sqrt2\sigma}}\exp\left(\frac{-x^2}{2}\right) \left(\text{erf}\left(\frac{\alpha+\sqrt{d^2-\frac{x^2}{2\sigma^2}}}{\sqrt{2}}\right)-\text{erf}\left(\frac{\alpha-\sqrt{d^2-\frac{x^2}{2\sigma^2}}}{\sqrt{2}}\right)\right)\; dx$$
any more help???