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?

Please help,


  • $\begingroup$ Closed-form: A closed form for a primitive of that beast seems unlikely. Approximate expression: are you sure you need an approximate expression for an indefinite integral? $\endgroup$ – Pierpaolo Vivo Jan 7 '16 at 18:07
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    $\begingroup$ A 1969 paper by JPL scientists provides a table of integrals of the error functions, might that be helpful? There is also an addendum here $\endgroup$ – njuffa Jan 7 '16 at 22:18
  • $\begingroup$ 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 $$\lambda\simeq-0.2427834044630287264768290\dots$$ $\endgroup$ – Lucian Jan 8 '16 at 5:53
  • $\begingroup$ @PierpaoloVivo Thanks for the reply. I think the main culprit here is the 'Log of that beast (erf)' :( However, in my purpose an approximation is sufficient. In fact I want to compare the solution of a numerical model with the analytical expression that contains the above integral. Just comparison purpose, an approximation applying any special condition may be enough. By the way, for practical applications, range of x can also be considered as [0, infinite] or even [0, to any finite number] if that helps with anything. $\endgroup$ – Muhammad Muniruzzaman Jan 8 '16 at 11:40
  • $\begingroup$ @njuffa Thanks for the link! Interesting stuff! But somehow, anything involving log(erf) seems to be missing :( $\endgroup$ – Muhammad Muniruzzaman Jan 8 '16 at 11:44

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