In my research I came across the following integral: \begin{equation} \int_{-\infty}^{+\infty}\frac{\partial{p(t)}}{\partial{t}}\frac{1}{4}\Big(1-\operatorname{erf}\Big(\frac{t-a}{\sigma\sqrt{2}}\Big)\Big)\Big(1+\operatorname{erf}\Big(\frac{t-c}{\sigma\sqrt{2}}\Big)\Big)\,dt \end{equation}
where
\begin{equation} p(t)=\frac{1}{\sigma\sqrt{2\pi}}e^{-\frac{(t-b)^2}{2{\sigma}^2}} \end{equation}
\begin{equation} \operatorname{erf}(x)=\frac{2}{\sqrt{\pi}}\int_{0}^{x}e^{-s^2}ds \end{equation}
\begin{equation} a>b>c>0 \end{equation}
Does anyone know if this is solvable? I can tolerate a solution in terms of erf functions of $a,b,c,\sigma$. Thank you!