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My work has arrived at needing to solve the integral below for $a,b,c,\sigma>0$

$$I=\int_{-\infty}^{\infty}\phi\left(x\right)\Phi\left(\frac{a}{\sqrt{b+c\mathrm{e}^{(x-\mu)/\sigma}}}\right)dx$$

I have tried substitution: $u=\frac{a}{\sqrt{b+c\mathrm{e}^{\frac{x-\mu}{\sigma}}}}$ and then a couple of rounds of integration by parts. However it does not seem to be getting closer to finding an integration that can be done directly (i.e. without needing a further by parts integration).

Is there another route to solving this, or does it not have a closed-form solution?

Thanks

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    $\begingroup$ i have strong doubts that there is a closed for solution even in the special cases like $b=0$ $\endgroup$ – tired Jul 29 '15 at 17:01
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    $\begingroup$ How are $\phi$ and $\Phi$ related, if at all? $\endgroup$ – Mark Viola Jul 29 '15 at 17:39
  • $\begingroup$ Sorry, I should have clarified. They are the normal distribution pdf and cdf respectively. $\endgroup$ – AndrewLong Jul 29 '15 at 18:27
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    $\begingroup$ Can someone explain to me why almost every parametric integral proposed on MSE has a completely useless extra parameter? The integral above just depends on $\frac{b}{a^2}$ and $\frac{c}{a^2}$ (together with $\mu,\sigma$), so why to introduce syntactic garbage? $\endgroup$ – Jack D'Aurizio Jul 29 '15 at 20:42
  • $\begingroup$ Thank you for your insightful comments Jack. In fact, the parameters do have meaning in the wider context of how the integral arose, but I take the point that I could have done more to simplify before posting on MSE. Please accept my apologies! $\endgroup$ – AndrewLong Jul 30 '15 at 7:30
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This question is from the past, but I was curious about the answer. The closest thing that I could come up with is approximating the normal CDF.

Depending on the desired accuracy, I think we can substitute the CDF with one of the approximation methods that exist for normal CDF. A few examples are here: http://web2.uwindsor.ca/math/hlynka/zogheibhlynka.pdf

Then, we should be able to calculate the integral in close-form.

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