Integral of log of inverse erf from 0 to 1

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Theorem

The following formula holds: $$\displaystyle\int_0^1 \log(\mathrm{erf}^{-1}(x)) \mathrm{d}x = \dfrac{\gamma}{2} + \log(2),$$ where $\mathrm{erf}^{-1}$ denotes the inverse error function, $\log$ denotes the logarithm, and $\gamma$ denotes the Euler-Mascheroni constant.

Proof

References