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This integral comes up in a problem in Statistics involving power laws. Here are some notes if anyone is interested. The integral in question would be related to equation (7) therein.

I would like to compute $$ \int_0^\infty x^a\exp(-bx)\left(\frac{1}{\text{erfc}(c\sqrt{x})}\right)^{2a} dx, $$ where $b > 0$, $a >0$ and $c \in \mathbb{R}$ and erfc is the complementary error function.

This looks hopeless. Anything I could try to squeeze out a closed-form solution?

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