recently, i need to compute this kind of integral: $$ \int ^\infty _c \Phi(ax+b) \phi(x) dx$$ where a, b and c are all constants and $\Phi(x)$ denotes the CDF of standard normal distribution and $\phi(x)$ denotes the PDF of standard normal

I have looked up similar questions both in "math.stackexchange.com" and here, but i don't find any satisfactory answer. if "c" is negative infinity here, it would be relatively easy. but here it's not.

later i find if i can find a solution to: $$ \int ^\infty _c x^2\Phi(ax+b) \phi(x) dx$$ then the former integral could be calculated.

can someone help me? if no closed solution exists, is there any practical approximation to that integral?

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