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\begin{equation}  I(\mu, \sigma) = \frac{\mu}{2} + \frac{\sigma}{\sqrt{2\pi}} + \frac{\sigma}{\sqrt{2\pi}}\left[ \exp\left(-\frac{\mu^2}{2\sigma^2}\right) -1\right] + \frac{\mu}{2} \,\,{\rm erf} \left( \frac{\mu}{\sqrt{2}\sigma} \right)  \end{equation}  And this implies that   \begin{equation}  F' = I(\mu, \sigma) + I(-\mu, \sigma)  \end{equation}  \begin{equation}  F' = \sqrt{\frac{2}{\pi}}\sigma + \frac{\sigma}{\sqrt{2\pi}}\left[ \exp\left(-\frac{\mu^2}{2\sigma^2}\right) -1\right] + \frac{\mu}{2} \,\,{\rm erf} \left( \frac{\mu}{\sqrt{2}\sigma} \right)  \end{equation}  ===