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the definite integral has a closed-form expression for $\mu=0$,

$$\int_a^\infty x \Phi(cx+d) \phi\left(\frac{x}{\sigma}\right) dx=$$ $$\frac{c \sigma^3 e^{-\frac{d^2}{2 c^2 \sigma^2+2}}}{2 \sqrt{2 \pi } \sqrt{c^2 \sigma^2+1}}\left[1-\text{erf}\left(\frac{c \sigma^2 (c a+d)+a}{\sigma \sqrt{2 c^2 \sigma^2+2}}\right)\right]$$ $$\qquad-\sigma^2(2 \sqrt{2 \pi })^{-1}e^{-\frac{a^2}{2 \sigma^2}} \left[\text{erfc}\left(\frac{c a+d}{\sqrt{2}}\right)-2\right]$$

the definite integral has a closed-form expression for $\mu=0$,

\begin{align} & \int_a^\infty x \Phi(cx+d) \phi\left(\frac{x}{\sigma}\right) dx \\[8pt] = {} & \frac{c \sigma^3 e^{-\frac{d^2}{2 c^2 \sigma^2+2}}}{2 \sqrt{2 \pi } \sqrt{c^2 \sigma^2+1}}\left[1-\operatorname{erf}\left(\frac{c \sigma^2 (c a+d)+a}{\sigma \sqrt{2 c^2 \sigma^2+2}}\right)\right] \\[8pt] & -\sigma^2(2 \sqrt{2 \pi })^{-1}e^{-\frac{a^2}{2 \sigma^2}} \left[\operatorname{erfc}\left(\frac{c a+d}{\sqrt{2}}\right)-2\right] \end{align}

the definite integral has a closed-form expression for $\mu=0$,

$$\int_a^\infty x \Phi(cx+d) \phi\left(\frac{x}{\sigma}\right) dx=$$ $$\frac{c \sigma^3 e^{-\frac{d^2}{2 c^2 \sigma^2+2}}}{2 \sqrt{2 \pi } \sqrt{c^2 \sigma^2+1}}\left(1-\text{erf}\left(\frac{c \sigma^2 (c a+d)+a}{\sigma \sqrt{2 c^2 \sigma^2+2}}\right)\right)-\sigma^2(2 \sqrt{2 \pi })^{-1}e^{-\frac{a^2}{2 \sigma^2}} \left(\text{erfc}\left(\frac{c a+d}{\sqrt{2}}\right)-2\right)$$

the definite integral has a closed-form expression for $\mu=0$,

$$\int_a^\infty x \Phi(cx+d) \phi\left(\frac{x}{\sigma}\right) dx=$$ $$\frac{c \sigma^3 e^{-\frac{d^2}{2 c^2 \sigma^2+2}}}{2 \sqrt{2 \pi } \sqrt{c^2 \sigma^2+1}}\left[1-\text{erf}\left(\frac{c \sigma^2 (c a+d)+a}{\sigma \sqrt{2 c^2 \sigma^2+2}}\right)\right]$$ $$\qquad-\sigma^2(2 \sqrt{2 \pi })^{-1}e^{-\frac{a^2}{2 \sigma^2}} \left[\text{erfc}\left(\frac{c a+d}{\sqrt{2}}\right)-2\right]$$

the definite integral has a closed-form expression for $\mu=0$,

$$\int_a^\infty x \Phi(cx+d) \phi\left(\frac{x}{\sigma}\right) dx=$$ $$\frac{c \sigma^3 e^{-\frac{d^2}{2 c^2 \sigma^2+2}}}{2 \sqrt{2 \pi } \sqrt{c^2 \sigma^2+1}}-\sigma^2(2 \sqrt{2 \pi })^{-1} \left(\frac{c \sigma e^{-\frac{d^2}{2 c^2 \sigma^2+2}} \text{erf}\left(\frac{c \sigma^2 (c a+d)+a}{\sigma \sqrt{2 c^2 \sigma^2+2}}\right)}{\sqrt{c^2 \sigma^2+1}}+e^{-\frac{a^2}{2 \sigma^2}} \left(\text{erfc}\left(\frac{c a+d}{\sqrt{2}}\right)-2\right)\right)$$

the definite integral has a closed-form expression for $\mu=0$,

$$\int_a^\infty x \Phi(cx+d) \phi\left(\frac{x}{\sigma}\right) dx=$$ $$\frac{c \sigma^3 e^{-\frac{d^2}{2 c^2 \sigma^2+2}}}{2 \sqrt{2 \pi } \sqrt{c^2 \sigma^2+1}}\left(1-\text{erf}\left(\frac{c \sigma^2 (c a+d)+a}{\sigma \sqrt{2 c^2 \sigma^2+2}}\right)\right)-\sigma^2(2 \sqrt{2 \pi })^{-1}e^{-\frac{a^2}{2 \sigma^2}} \left(\text{erfc}\left(\frac{c a+d}{\sqrt{2}}\right)-2\right)$$