From: Albert_Rich@msn.com   
      
   On Saturday, April 15, 2017 at 11:46:56 AM UTC-10, Richard Fateman wrote:   
   > I think there are others more qualified to   
   > explain this, but my guess is that   
   > basically the Risch "algorithm"   
   > doesn't do what you want done.   
   >    
   > You want an antiderivative that is somehow analytically   
   > valid in the complex plane  (or some multi-dimensional   
   > extension). That's not what it does.   
   >    
   > If you have a differential field F  with an element x in it,   
   > then field G is a logarithmic extension of F if it   
   > includes s=log(x), where s has the property that diff(s,x) is   
   > diff(x)/x,  which is in F.   
   >    
   > That's all.   
   >    
   > It doesn't say that there is just one such logarithm,   
   >    
   > nor does it say anything about the value of log(0) or log(-1)   
   > or the relationship of log(x^2) to log(x), which could, I suppose,   
   > be considered uncorrelated, as you seem to prefer.   
   >    
   >   But then by the same kind of reasoning,   
   > log(-1)-log(-1) might not be zero, because they are different logs..   
   > each log(-1) is (2*n+1)*i*pi, but for different n.   
   >    
   > The Risch "algorithm" is interesting and fun, but it   
   > is solving a problem that is related, but not identical to   
   > what, I suppose, is the "integrate" command, which   
   > attempts to do Riemann integration under the assumption that   
   > continuity is magically guaranteed to be irrelevant to   
   > the task.   
   >    
   > I have in the past supervised an MS thesis involved in   
   > identifying singularities and branch cuts. In your example,   
   > we would have to do some analysis to show that   
   > log(x)*log(x^2) and 2*log(x)^2 are different functions   
   > in some places, and identical in others.   
   >    
   > This multivaluedness is not a problem uniquely associated   
   > with integrate(). It should be solved  more generally, and then it could   
   > be used by integrate(), simplification,   
   > comparisons, evaluations, etc.   
   >    
   > RJF   
      
   log(z^2) is not equal to 2*log(z) over half the complex plane.     
      
   It seems to me if indeed the Risch Algorithm cannot be trusted to produce   
   analytically valid antiderivatives over the complex plane, a computer algebra   
   system's Integrate command should first try using elementary methods that do.   
   Then iff that fails,    
   resort to the Risch Algorithm AND warn users the result may not be valid over   
   the complex plane.   
      
   As Rubi demonstrates, rule-based elementary methods can produce optimal,   
   analytically valid antiderivatives for a large class of integrands.   
      
   Albert   
      
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