integral involving error function (erf)

Does anybody know if a closed form of this integral exist?

$\int \mbox{erf}(x) \ln(\mbox{erf}(x)) \Bbb dx$

where erf is so called error function.

In case there is no closed form solution. Is it possible to get an approximate one?

• Integrating by parts and simplifying, we have $$-I=-\int_0^\infty\text{erf}(x)~\ln\text{erf}(x)~dx ~=~ \lambda+\frac1{\sqrt\pi}~,$$ where $$\lambda~=~\lim_{t\to0}~\frac{\ln\text{erf}(t)}{\sqrt\pi} ~+~ \frac2\pi\int_t^\infty\frac{e^{-2x^2}}{\text{erf}(x)}~dx,$$ whose numerical value is about $$\lambda\simeq-0.2427834044630287264768290\dots$$ – Lucian Jan 8 '16 at 5:53