$\int^{+\infty}_{-\infty}\ln\left({1\over2}+{1\over2}\text{erf}\left\{{x\over\sqrt{2}}\right\}\right)H_n(x)e^{-\frac{x^2}{2}}~dx$
Where $H_n$ are "probabilists' Hermite polynomials".
2026-05-15 18:26:51.1778869611
Analytical evaluation of the following integral
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I get (with the help of Maple) $$ \int _{-\infty }^{\infty }\!\ln \left( 1/2+1/2\, {{\rm erf}\left(1/2\,x\sqrt {2}\right)} \right) {{\rm e}^{-1/2\,{x}^{2 }}}{dx}=-\sqrt {2\pi } $$ but not the case $n=1$.