From: nma@12000.org   
      
   On 3/12/2020 12:47 AM, Albert Rich wrote:   
   > On Wednesday, March 11, 2020 at 6:22:34 AM UTC-10, anti...@math.uni.wroc.pl   
   wrote:   
   >> [...]   
   >>   
   >> And of course, having no better method one could use Rubi way:   
   >> add lookup table and retrive precomputed answer from the table.   
   >>   
   >> --   
   >>                                Waldek Hebisch   
   >   
   > Rubi does NOT use a lookup table of specific precomputed integrals.  Rather   
   it uses generic reduction and terminal rules to iteratively integrate large   
   classes of expressions.  For example, when Martin presents a specific example   
   Rubi cannot integrate,    
   it usually leads to a generic rule able to produce optimal antiderivatives for   
   the whole class of integrands for which the example is a special case.   
   >   
   > Instead of using rule-based integration when there is “no better   
   method”, I contend it should be used BEFORE resorting to advanced methods   
   like Risch.  There are numerous advantages of a properly implemented   
   rule-based integrator:   
   >   
   > 1. If a rule does apply, the optimal antiderivative will quickly and   
   reliably be found.   
   > 2. Rule-based systems can show the rules applied and the resulting   
   intermediate steps.   
   > 3. The rules are self-contained and easily verified by differentiation.   
   > 4. The individual rules are elementary in nature and thus comprehensible to   
   mere humans, like first year calculus students.   
   > 5. This makes rule-based systems great pedagogical tools in the classroom.   
   > 6. Rules can be developed, debugged and tested in a modular fashion rather   
   than as a monolith.   
   > 7. Holes in the rule-based decision tree point to where new mathematical   
   knowledge (in this case, integration formulas) is crying out to be discovered.   
   > 8. It’s quickly determined when no rule applies, so the delay in resorting   
   to advanced methods is negligible, especially compared to the amount time such   
   methods often require.   
   >   
   > Albert   
   >   
      
   Good summary.   
      
   But I really like this idea of using Rule-based (Like Rubi) as   
   a first phase to the integrate command, and if that fails, then   
   a second phase is called which uses the traditional methods currently   
   implemented by CAS systems (i.e. Risch, etc...).   
      
   May be someone from Wolfram could look into this (given that Rubi is   
   already implemented in Mathematica (i.e. Wolfram Language), this   
   will be easier to integrate it into the Mathematica Integrate command than   
   with other CAS systems.   
      
   --Nasser   
      
   --- SoupGate-Win32 v1.05   
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