I am a PhD student (not in maths). I have recently encountered a problem in my research which requires a solution to the following integral question:

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Let:

$\displaystyle f(x)=\frac{1}{\sigma _{1}\sqrt{2\pi}} e^{-\frac{1}{2} (\frac{x-\mu _{1}}{\sigma _{1}}\, )^{2}}$

$\displaystyle \phi (x)=\int_{-\infty }^{x} f(x)\, dx$

$\displaystyle g(x)=\frac{1}{\sigma _{2}\sqrt{2\pi}} e^{-\frac{1}{2} (\frac{x-\mu _{2}}{\sigma _{2}}\, )^{2}}$

Find an expression or approximate expression of several terms for the following integral:

$\displaystyle Pr(\mu _{1}\, , \sigma _{1}\, ,\mu _{2}\, , \sigma _{2}\)=\int_{-\infty}^{\infty} \phi (x) g(x) \, dx $

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It looks like a classical question, and must have been solved by mathematicians long time ago. However I am not a maths student, so I don't know.

If someone in this forum knows, please tell me. Exact solution or approximate solution with several terms are all OK.

Any solution must be in general form involving those 4 parameters.

Or please tell me about any journal article or maths textbook that discuss this question or similar question. Thank you!