I am trying to solve the integral of a gaussian cumulative distribution function and a gaussian probability function. On this site I have seen solutions of similar, less general integrals (e.g. Integral of the product of Normal density and cdf) . Does a closed form solution for this specific case exist? $$\int_{s_2=-\infty }^{s_2=y} \phi(s_2;X_{22}\beta;B_{2i})\times \Phi(X_{21i}\beta; cs_2;B_{1i}) \text{d}\boldsymbol{s}_2 $$ where Φ is the cdf of a gaussian distribution, and ϕ its density.

I am trying to solve the integral of a gaussian cumulative distribution function and a gaussian probability function. On this site I have seen solutions of similar, less general integrals (e.g. Integral of the product of Normal density and cdf) . Does a closed form solution for this specific case exist? $$\int_{s_2=-\infty }^{s_2=y} \phi(s;\mu;\sigma)\times \Phi(r; cs_2;\tau) \text{d}{s} $$ where Φ is the cdf of a gaussian distribution, and ϕ its density.