oldk1331@gmail.com schrieb:   
   >   
   > First, a clarification, the result I got is computed by FriCAS,   
   > and I think Axiom can not compute (or take days), because Axiom   
   > does not implement 'setSimplifyDenomsFlag' and associated changes.   
   >   
   > Second, this integration clearly belongs to the Risch algorithm   
   > pure algebraic case, so the inability to give an elementary answer   
   > (or prove such elementary answer doesn't exist) is considered by me   
   > as a bug.  When there exists elementary answer, I think answers with   
   > non-elementary functions is "half-correct".   
   >   
   > Lots of tests from "CAS integration tests" have this problem,   
   > which I think it is unfair to systems like FriCAS which implements   
   > most of Risch algorithm (ALBI, Algorithm Based Integration) compared   
   > with RUBI systems.   
   >   
   > In "Specialized integral tests" section of "CAS integration tests",   
   > I analyzed the cases which Mathematica or RUBI fails and with   
   > elementary answer.  There are 33 cases, and FriCAS solves 31.   
   >   
   > Finally, FYI, I list all pure algebraic integration of the   
   > "Specialized integral tests" section:   
   >   
   > * A.F. Timofeev   
   > ** 319      mathematica fails   
   > [...]   
   > ** 324      mathematica fails   
   > [...]   
   >   
   > * Vladimir Bondarenko   
   > ** 15       mathematica fails   
   > [...]   
   > ** 16       mathematica fails   
   > [...]   
   > ** 18       mathematica fails   
   > [...]   
   >   
   > * Manuel Bronstein   
   > ** 6        both fail   
   > [...]   
   >   
   > * Tony Hearn   
   > ** 204      rubi fails   
   > [...]   
   > ** 279      both fail   
   > [...]   
   > ** 280      both fail   
   > [...]   
   >   
   > * Martin Welz   
   > ** 9        both fail   
   > [...]   
   > ** 10       both fail   
   > [...]   
   > ** 12       mathematica fails   
   > [...]   
   > ** 13       mathematica fails   
   > [...]   
   >   
   > Of these 13 pure algebraic integration, FriCAS only fails at   
   > Timofeev#319.   
      
   Sorry for writing Axiom instead of FriCAS: I knew I was using FriCAS,   
   but the site error threw me off, and having had to type in things like   
   \begin{axiom} and having been reminded by the error message that the   
   site URL starts on "axiom-wiki" obviously had some subconscious effect!   
      
   I am not so sure about counting purely algebraic Risch failures as bugs.   
   Even the purely algebraic case has never been implemented more fully   
   than in FriCAS - apparently because it proves too hard a task.   
   Considering that this situation has persisted for three decades, one may   
   suspect that the algorithm is not really practical.   
      
   I agree that non-elementary results returned when elementary ones exist   
   should not be counted as full successes; Albert Rich should perhaps   
   consider counting them as "nonoptimal" in his tables. And of course, a   
   Risch result correctly certifying an integral as undoable in elementary   
   terms must be considered as a full success.   
      
   Timofeev#319 is example 112 from Timofeev's Chapter 4:   
      
     INT(1/((3*x + 3*x^2 + x^3)*(3 + 3*x + 3*x^2 + x^3)^(1/3)), x)   
      
   However, on a considerable number of other algebraic Timofeev   
   integrands FriCAS succeeds only because they are first transformed by   
   pattern matching; some of these Rubi-type rules were added specifically   
   to increase the Timofeev score of FriCAS, I believe. Axiom (sic) should   
   thus presumably fail on integral 21 from Chapter 4, for example:   
      
     INT(((x - 1)^2*(x + 1))^(1/3)/x^2, x)   
      
   Martin.   
      
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