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A closed form expression in terms of a special function exists for $b=0$, when $$\int_0^y x^{-a} \exp \left(-\tfrac{1}{2} c^2 x^{-2d} \right) \,dx=\frac{1}{2 d}y^{1-a} E_{1-\frac{a-1}{2 d}}\left(\tfrac{1}{2} c^2 y^{-2 d}\right),\;\;c,d>0,$$ with $E_n(x)$ the exponential integral function.