Chi-squared transform of an inverse Gaussian variable ($\lambda (X-\mu {)}^2/({\mu }^{2}X)\sim {\chi }_1^2$)

For $X\sim IG(\mu ,\lambda )$, $h(X)=\lambda (X-\mu {)}^2/({\mu }^{2}X)$ has the chi-squared-one law. Its two inverse branches contribute weights $\mu /(\mu +x_{-})$ and $\mu /(\mu +x_{+})$ to the transformed density, and reciprocal roots make those weights sum to one.