Is there a solution to this integral?


$$\int\_{0}^{y} x^{-a} \exp \left[- \frac{(b - cx^{-d})^2}{2} \right] dx,$$
where $a > 0$ and $d > 0$.



Solution to $\int_{0}^{y} x^{-a} \exp \left[- \frac{(b - cx^{-d})^2}{2} \right] dx$
By using series expansion, change of variable and Eq. (3.381.9) of the book: "I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, 8th ed. Burlington, MA, USA: Academic Press, 2015", I was able to find this solution:


$$\int\_{0}^{y} x^{-a} \exp \left[- \frac{(b - cx^{-d})^2}{2} \right] dx 
\\= \exp(-b^2/2) \sum\_{k=0}^{\infty} \frac{(bc)^k}{k!} \Gamma\left(\frac{dk+a-1}{2d}, \frac{c^2 y^{-2d}}{2} \right)\frac{1}{2d(c^2/2)^{\frac{dk+a-1}{2d}}}.$$


I've run some simulations and it works.However, there might be some convergence problem for some values, I think so.


