for $b=0$ one has $$\int x^{-a} \exp \left[- \frac{( cx^{-d})^2}{2} \right] dx=\frac{1}{2 d}x^{1-a} E_{1-\frac{a-1}{2 d}}\left(\tfrac{1}{2} c^2 x^{-2 d}\right),$$ with $E_n(x)$ the exponential integral function.
Carlo Beenakker
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