From: fateman@cs.berkeley.edu   
      
   On 9/18/2017 10:53 PM, Albert Rich wrote:   
   > As Martin makes clear, these exercises are pseudo-elliptic integrals,   
   meaning they can be expressed in terms of elementary, non-elliptic functions.    
   And as the antiderivatives given above show, they are relatively simple   
   expressions easy to evaluate    
   numerically.   
      
   There are two atans, an atanh, one log, some fourth roots and square   
   roots, but   
   there also seem to be common subexpressions.   
   This formula would  be fairly easy to evaluate   
   numerically.  However, the time to produce that formula symbolically   
   must also be considered.  Possibly add in the time to convert it to   
   a numerical program  (optimized to reduce the redundant computation).   
   This cost could be amortized across the number of times the   
   formula is evaluated. If it is evaluated many times, it might   
   be a winner.  That's certainly one argument I have made.   
      
     If the process is followed to get one or a few points,   
   it is less clear.  Also sometimes there is an issue, as   
   Albert points out, that the integration result may be   
   poorly behaved.   I don't know if this correlates 100%   
   to misbehavior in the integrand.   
      
   Here's an example illustrating a hazard where the formula   
   is a loser.  I don't know if Rubi gets this, but Mathematica gets   
   a formula with 64 distinct terms, each with atan, atanh, some with logs,   
   and 32 different sines and cosines ...   
   vs. the simple integrand below..   
      
   integrate(1/(x^64+1),x)   
      
      
      
   details   
   http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.25.7855   
      
   I think we have to keep an open mind on what is best, not knowing   
   in advance what people will do next with our results. :)   
   RJF   
      
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