Gaussian integral with an inverse-square term (${\int }_{-\infty }^{\infty }{e}^{-(a{x}^2+b/x^2)/2}dx=\sqrt{2\pi /a}{e}^{-\sqrt{ab}}$)

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 $a{x}^2+b/x^2=u^2+2\sqrt{ab}$. Solving the quadratic for the positive root givesAfter 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 givesThe 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.