Waldek Hebisch schrieb:   
   >   
   > clicliclic@freenet.de wrote:   
   > >   
   > > Your use of the term optimization is futuristic in that it is   
   > > relative to a procedure that has been described but not been   
   > > implemented in FriCAS or elsewhere.   
   >   
   > FriCAS uses rationalizing substitution before trying Trager   
   > procedure.  In particular at this point FriCAS does not know   
   > if integral would hit unimplemented part or not.  And in cases   
   > handled by current version of Trager procedure execution time   
   > using substitution usually is much better.   
   >   
      
   Using the term like you do would allow to call Rubi the ultimate   
   optimization of Trager's procedure.   
      
   > >   
   > > And also about ten lines of Mathematica code in the Appendix - but   
   > > this line count does not include a separate interface package used   
   > > to access Singular on whose Groebner basis engine the code relies   
   > > (without real need, I suppose).   
   > >   
   >   
   > AFAICS this code does not handle any complications which are   
   > mentiond in the paper (that is caller must prepare data in   
   > a special way).   
   >   
   > BTW: Trager procedure builds divisors (that is colections of   
   > zeros and poles) corresponding to candidate integrals.  Then   
   > divisors have to be combined and checked if they correspond to   
   > real funtions.  If not, there is no elementary integral.   
   > If there is rationalizing substitution, then no combining   
   > is needed: any divisor gives you a function.   
   >   
   > Compared to that Kauers uses incomplete divisors (without   
   > part at infinity) and skipped combining.  His Groebner   
   > heuristic attemps to find a function which has arbitrary   
   > behaviour at infinity while in finite places has zeros   
   > and poles given by the divisor.  AFAICS his procedure is   
   > incomplete in at least two places: for some functions   
   > combining divisors is needed and he uses fixed bound   
   > for order of divisors while it is know that there are   
   > divisors of order higher than any fixed bound   
   > (and Trager procedure can compute bound on the order).   
   > It is not clear for me if Groebner heuristic is complete   
   > for the problem of checking if divisor corresponds to   
   > a function.   
   >   
   > How this compares to what FriCAS is doing: at first   
   > step of Trager procedure FriCAS obtains incomplete   
   > divisors (without part at infinity).  They have to   
   > be combined to get a complete divisors.  Normaly   
   > divisors come in pairs, one for zeros and other   
   > for poles.  In such case FriCAS typically can   
   > recognize that parts of divisor should be paired and   
   > checks combined divisor.  In your examples there   
   > is no such pattern, one gets three parts.  ATM I do not   
   > know if one needs to combine all parts or (more likely)   
   > if one can handle them separately.  If parts can be   
   > handled separately then Kauers heuristic has chance   
   > to succeed.   
      
   Goursat pseudo-elliptics involving cube roots of cubics are very similar   
   to those involving square roots of quartics: either there is a single   
   rationalizing substitution, or the rational part of the integrand needs   
   to be split in two such that each component integral succumbs to a   
   single substitution; three components are never needed. The examples   
      
   INT((2 - (k + 1)*x)/((1 - (k + 1)*x)*(x*(1 - x)*(1 - k*x))^(1/3)), x)   
      
   INT((1 - k*x)/((1 + (k - 2)*x)*(x*(1 - x)*(1 - k*x))^(2/3)), x)   
      
   require no splitting, the example   
      
   INT((a + b*x + c*x^2)/((1 - x + x^2)*(1 - x^3)^(1/3)), x) =   
   1/2*INT(((c*x^2 + b*x + a)/(x^2 - x + 1) +   
   #i*(c*x^4 - x^3*(2*a + b + c) + x*(a + 2*(b + c)) - 2*c)/   
   (SQRT(3)*x*(x^3 + 1)))*(1 - x^3)^(-1/3), x) +   
   1/2*INT(((c*x^2 + b*x + a)/(x^2 - x + 1) -   
   #i*(c*x^4 - x^3*(2*a + b + c) + x*(a + 2*(b + c)) - 2*c)   
   /(SQRT(3)*x*(x^3 + 1)))*(1 - x^3)^(-1/3), x)   
      
   must be split, one possibility being shown. Goursat's 1887 prescriptions   
   for square root cases (and their equivalents for cube roots) permit to   
   detect integrals that need to be split: for square roots these satisfy   
   the four-term equation (5) on p. 111 but none of the two term equations   
   F + F(Si) = 0 (i = 1,2,3); the prescriptions then also provide a   
   suitable splitting. The substituted integrals are of the type   
   INT(R(t^2), t)) in the square-root case, and of the types INT(R(t^3),   
   t)) or INT(R(t^3)*t, t)) for exponents -1/3 and -2/3.   
      
   Martin.   
      
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