I've been experimenting/playing with the Rubi 4.14.1 package for MMA, see here. Unfortunately I have not been able to find too much documentation.
For the following function of
c I was surprised MMA returned an answer, while Rubi returned... well I don't quite know how to interpret what was produced. I'd appreciate any light you can shed
fc = -(1/2) Erf[((2 m - k^2) - 2 Log[c + b] + 2 Log[a])/(2 Sqrt k)] - 1/2 fcresmma = Assuming[m \[Element] Reals && k > 0 && c >= 0 && a > 0 && b > 0, Integrate[fc, c]] fcres = Assuming[ m \[Element] Reals && k > 0 && c >= 0 && a > 0 && b > 0, Int[fc, c]]
The MMA (11.2) results is
1/2 (-c - (b + c) Erf[(-k^2 + 2 m + 2 Log[a] - 2 Log[b + c])/( 2 Sqrt k)] + a E^m Erf[(k^2 + 2 m + 2 Log[a] - 2 Log[b + c])/(2 Sqrt k)])
The Rubi result is:
-(c/2) - Dist[1/2, Subst[ Int[ Erf[ (-k^2 (1 - (2 (m + Log[a]))/k^2) - 2 Log[c])/(2 Sqrt k) ], c], c, b + c], c]
Update: Confirmed and Resolved in Rubi 4.14.3
... Rubi 4.14.3 is able to integrate any expression of this form including when m is 0, numeric or symbolic. In fact, it can integrate any expression of the form
(e x)^m F[d (a+b log(c x^n))]
where F is Erf, Erfc, Erfi, FresnelS, FresnelC, ExpIntegralEi, SinIntegral, CosIntegral, SinhIntegral, CoshIntegral or Gamma (incomplete).