Solving or estimating $\int_{-\infty}^\infty \frac{\exp(-a/(x^2+b))}{x^2+c}dx$

Question

Can the integral $$\int_{-\infty}^\infty \frac{\exp(-a/(x^2+b))}{x^2+c}dx$$ be made explicit ($a,b,c>0$)? I'm also asking those of you who have access to CAS which can solve it. In case it can't, is there a very good upper bound? I need a better one than the obvious bound $$\int_{-\infty}^\infty \frac{1}{x^2+c}dx = \frac{\pi}{\sqrt{c}}$$ Thanks a lot!

Answer 1

Hint:

$$\begin{align}\int_{-\infty}^\infty\dfrac{e^{-\frac{a}{x^2+b}}}{x^2+c}~dx&=2\int_0^\infty\dfrac{e^{-\frac{a}{x^2+b}}}{x^2+c}~dx \\&=2\int_0^\frac{\pi}{2}\dfrac{e^{-\frac{a}{b\tan^2x+b}}}{b\tan^2x+c}~d(\sqrt b\tan x) \\&=2\sqrt b\int_0^\frac{\pi}{2}\dfrac{e^{-\frac{a}{b\sec^2x}}\sec^2x}{b\sec^2x+c-b}~dx \\&=2\sqrt b\int_0^\frac{\pi}{2}\dfrac{e^{-\frac{a\cos^2x}{b}}}{b+(c-b)\cos^2x}~dx \\&=2\sqrt b\int_0^\frac{\pi}{2}\sum\limits_{n=0}^\infty\dfrac{(-1)^na^n\cos^{2n}x}{b^n(b+(c-b)\cos^2x)}~dx\end{align}$$