Non-Elementary Integration · Hard problem

Glasser's master theorem gives $\displaystyle\int_{-\infty}^{\infty}f\!\left(x-\dfrac1x\right)dx=\int_{-\infty}^{\infty}f(x)\,dx$ for integrable $f$. Evaluate $\displaystyle\int_{-\infty}^{\infty}e^{-\left(x^2+1/x^2\right)}\,dx$.