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. However, the actual Axiom limitation is more severe: it can only handle integrals where Trager resultant has at most one nonlinear factor. Additionally, Axiom can handle directly only integrands with no poles at infinity. Kauers examples have pole at infinity, which Axiom handles by change of variables. After change of variables Trager resultant changes and has two nonlinear factors and Axiom gives up.
What is surprising is 12% sucess rate: Kauers examples of given series are variation of a single example -- only numerical coefficients differ but structure remains the same unless there is degeneracy. Apparently distribution of coefficients (which Kauers did not gave) is such that degeneracies are relatively frequent.
Axiom restriction on single nonlinear factor of Trager resultant is quite severe. Namely, given logand with no zero or pole at infinity it will lead to at least _two_ residues (one corresponding to zero another one correspondig to a pole). So typically we will get a pair of nonlinear factors.
Main difficulty of Kauers examples were already observerd by Davenport. Namely, given derivative of
r1*log(f1) + r2*log(f2)
where r1 and r2 are rationals Trager method will consider both terms together. If r1 and r2 are integer this leads to
f1^r2*f2^r1 may be quite large even if f1 and f2 are small.
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.