I am trying to compute the following integral $$I \triangleq \int_0^{+\infty}\exp{\left(-x-\frac{1}{cx}\right)}x^{-(b+1)}dx,$$
for $b, c$ strictly positive reals. From Wolfram, I obtain that $$I = 2 c^{b/2}K_b\left(\frac{2}{\sqrt{c}}\right),$$
where $K_b$ is the modified Bessel function of the second kind (as defined in e.g. Wikipedia).
I have tried to obtain this formula by applying Schläfli's integral $$J_b(z) = \frac{1}{2 \pi i}\left(\frac{z}{2} \right)^b\int_{-\infty}^0\exp{\left(x-\frac{z}{4x}\right)}x^{-(b+1)}dx,$$
where $z$ is a complex number in a domain such that $z^b$ is well-defined (e.g. $\mathbb{C}\setminus i\mathbb{R}_-$) and $J$ is the Bessel function of the first kind (see also the link above). Schläfli's integral is stated in DLMF, Bessel Functions.
Using the change of variables $x = -t$ and Schläfli's integral, we obtain $$I = (-1)^{b+1}\int_{-\infty}^0\exp{\left(t-\frac{-1}{ct}\right)}t^{-(b+1)}dt = (-1)^{b+1} 2 \pi i c^{b/2}i^{-b}J_b\left(\frac{2}{\sqrt{2}}i\right).$$
I think it is possible to conclude from this using relations between Bessel functions, but I haven't been able to do so. Does anyone have any ideas ? Thanks a lot !