I am trying to solve an improper integral that can be reduced to the following form
$$\int_{-\infty}^{\infty} \frac{1}{|x|}\exp\left(-\frac{a}{x^2} - \frac{(x + b)^2}{c|x|} \right)dx$$
for $a,b,c \in \mathbb{R}$, known scalar constants.
It's similar in spirit to this integral and this other one which is just the second term within the exponential. Both of those integrals have closed-form solutions in terms of the modified Bessel function of the second kind.
I've tried giving a similar treatment to this specific integral but to no avail. I've evaluated it numerically and I know it converges. Its plot resembles modified Bessel functions, so my intuition was that I should be able to obtain a closed-form solution for it, likely also in terms of the modified Bessel function family.
Any ideas on how to tackle it? Is there a known integral representation that has this shape?