Let $\phi(\cdot)$ and $\Phi(\cdot)$ be the probability and cumulative density functions, respectively, of a random variable with distribution $\text{N}(0,\,1)$. That is:
$$\forall x\in\mathbb{R}:\,\phi(x)=\frac{1}{\sqrt{2\cdot\pi}}\cdot\text{e}^{-x^2/2}$$ and
$$\forall x\in\mathbb{R}:\,\Phi(x)=\int_{-\infty}^{x}\phi(u)\,\text{d}u.$$ I was wondering if you could help me to compute
$$\int_{-\infty}^{w}\phi(x)\cdot\Phi(a+b\cdot x)\,\text{d}x\mbox{,}$$ where $(a,\,b,\,w)\in\mathbb{R}^3$ and $b\neq 0$, please.
Thanks a lot for your help.