I'm trying to find $\int_{\frac{a-b}{B}}^\infty\Phi\left(tA+ABx\right)\phi(x)\,dx$ where
$A = \frac{\sqrt{\gamma_{3}+\sigma_3^2}}{\gamma_{3}},\ B = \frac{\gamma_{2}}{\sqrt{\gamma_{2}+\sigma_{2}^2}},\ t=b-c$.
This integral arrives from trying the find $E[\max\{X,Y,a\}]$, where $X\sim N(b,B^2) $, $Y\sim N(c,\frac{1}{A^2})$, independent. I've tried differentiating w.r.t. $A$ but after solving the integral, I can't integrate back w.r.t. $A$. This is trying to follow the strategy from How can I calculate $\int^{\infty}_{-\infty}\Phi\left(\frac{w-a}{b}\right)\phi(w)\,\mathrm dw$.
Is there a closed form solution for this?
I realise that it is similar to rainbow options where they calculate $\max\{X,Y,k\}$ but $X$ and $Y$ follow lognormal distribution. The closed formed solution for that involves the bivariate normal distribution.