= Gaussian integral with an inverse-square term
{c}
{title2=$\displaystyle\int_{-\infty}^{\infty}e^{-(ax^2+b/x^2)/2}dx=\sqrt{2\pi/a}\,e^{-\sqrt{ab}}$}
For $a,b>0$, the integrand is even and integrable at zero and infinity. On the positive half-axis the <change of variables> $u=\sqrt a\,x-\sqrt b/x$ maps $(0,\infty)$ bijectively onto $\mathbb R$ and satisfies $ax^2+b/x^2=u^2+2\sqrt{ab}$. Solving the quadratic for the positive root gives
$$
\frac{dx}{du}=\frac1{2\sqrt a}\left(1+\frac u{\sqrt{u^2+4\sqrt{ab}}}\right).
$$
After doubling this half-axis integral, the odd part of the transformed integrand integrates to zero and the remaining <Gaussian integral> yields the formula. The reciprocal substitution $x=1/t$ also gives
$$
\int_{-\infty}^{\infty}x^{-2}e^{-(ax^2+b/x^2)/2}dx=\sqrt{2\pi/b}\,e^{-\sqrt{ab}}.
$$
The positive parameters ensure that the substitutions and improper integrals are valid; the second identity can also be recovered by differentiating the first in $b$ under the integral sign.
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