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this is not a homework and my problem is much bigger for me to give full details here. I came across this integral

[itex]\mathcal{I}(\xi)=\int^{\xi_c}_{\xi}{\rm d}\xi^\prime\exp\left[\sqrt{2}\sigma\,{\rm Erf}^{-1}\left(1-\frac{8\pi}{3}{\xi^\prime}^3\right)\right][/itex]

where Erf[itex]^{-1}[/itex] is the inverse error function and

[itex]\xi_c=\left[\frac{3}{8\pi}\left(1-{\rm Erf}\left(\frac{\sigma^2-\sqrt{2}\sigma\,{\rm Erf^{-1}(2\beta-1)}}{\sqrt{2}\sigma}\right)\right)\right]^{1/3}[/itex]

with [itex]0\le\beta\le1[/itex].

I would like get an analytical approximation but I can't figure out a way to do that, even with software like Mathematica. I tried solving the integral numerically and I find a reliable solution, however, I'm mostly interested in points where [itex]\xi\to\xi_c^-[/itex], and here the inverse error function diverges.

Do you have any ideas on how to approximate this integral?

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# How to solve an integral with the Inverse error function

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