Waldek Hebisch schrieb: > > email@example.com wrote: > > > > I bumped into this and am surprised by 70% failures (and not just > > timeouts) of the Axiom integrator on algebraic integrals possessing > > logarithmic antiderivatives that involve the single simple radical > > SQRT(x^3+1) or SQRT(x^4+1): > > > > <http://www.risc.jku.at/publications/download/risc_3427/Ka01.pdf> > > > > This must be where the missing computation of the "splitting field > > of the Trager resultant" comes in. > > If one wants to describe the reason in one sentence, then yes. > [...] > > FYI, FriCAS contans code which in many cases avoids both > problems. Namely, FriCAS tries to split integral and > integrate parts separately. This handles the sqrt(1 + x^4) > cases. Currently in sqrt(1 + x^3) case FriCAS is unable > to find working split and gives up. >
Thanks for the explanation.
If the failure to suitably split elementary integrals involving the radical sqrt(1 + x^3) = sqrt((1 + x)*(1 - x + x^2)) is somehow caused by the presence of an odd power in the radicand, the substitution t^2 = 1 + x might help, which introduces the radical sqrt(3 - 3*t^2 + t^4) where all powers of t are even. A better substitution would be t^2 = (1 - x + x^2)/(1 + x), which introduces sqrt(-3 + 6*t^2 + t^4), because here t remains real whenever the original radical is real.