From: fateman@cs.berkeley.edu   
      
   On 1/4/2016 10:51 AM, clicliclic@freenet.de wrote:   
   > Mathematica's designers may have considered explicit results in terms of   
   > the minimal algebraic extension of limited relevance, and thus prefer   
   > antiderivatives in terms of symbolic denominator root objects instead.   
      
   If so, I think this is a bad decision, and that using the minimal   
   algebraic extension is an important (and solved!) part of the routine.   
      
   >   
   > All rational integrands can be handled in this way, whereas FriCAS will   
   > never be able to handle arbitrary algebraic integrands by adding   
   > specific rewrite rules.   
      
   What I think is interesting here is the commentary on what it is that   
   Axiom/Fricas can or cannot do relative to implementation of Risch   
   methods.  Though the purely algebraic case is,I think algorithmic,   
   the transcendental simplification issues make the whole deal not   
   computable in general...  But if Axiom/Fricas doesn't do the algebraic   
   case and Mathematica sort-of does, then people who believe that   
   there is a complete implementation and it is Axiom, are misinformed.   
      
   The need to add arbitrary numbers of rules to Rubi is maybe not   
   such an issue if there is a way of identifying where it fails, e.g.   
   "Rubi cannot handle algebraic extensions of degree XXX" or some   
   other recognizable characterization.   
      
   By the way, I don't see this as a big deal.  The general world   
   views definite integration by numerical methods as a thoroughly   
   examined and mostly-solved problem.  Indefinite integration   
   (anti-differentiation) is a kind of boutique theory problem with   
   a very few applications in multiple-integral problems.  Some of   
   these are important to some people doing Feynman diagram calcs.   
   It was the motivation for Hearn's REDUCE system.  Should   
   these then be a benchmark for systems not being used for   
   such things?   
      
   Independent of any applications, there is an interesting   
   theoretical history of the problem of Integration in Finite Terms   
   from Liouville (with many many contributors, only a scattering   
   of whom committed their work to computer implementations.)   
   RJF   
      
      
      
      
   RJF   
      
   --- SoupGate-Win32 v1.05   
    * Origin: you cannot sedate... all the things you hate (1:229/2)