From: fateman@cs.berkeley.edu   
      
   On 6/19/2015 7:20 PM, Albert Rich wrote:   
   > On Wednesday, June 17, 2015 at 9:43:10 AM UTC-10, Richard Fateman   
   > wrote:   
   >   
   > [...}   
   >> HOWEVER There is another integral of x^n,  namely  z=   
   >> (x^(n+1)-1)/(n+1). It differs from the usual answer by 1/(n+1),   
   >> which is just another value for "plus a constant":)   
   >>   
   >> why is z preferable?   the limit of z as n -> -1  is -- magically   
   >> -- log(x).   
   >>   
   >> so One Might Argue that integrate(x^n,x)  is better expressed by   
   >> (x^(n+1)-1)/(n+1) VALID FOR ALL values of n including n=-1 if you   
   >> just take the limit.   
   >>   
   >> I know of no CAS that return this value, and presumably RUBI   
   >> testers would reject it as not being optimal from the perspective   
   >> of leafcount. Even though it might be "more correct".  It could   
   >> easily be changed in Maxima, unless the benchmark gives some guff.   
   >   
   > I guess you do not know of Derive which has long returned   
   > (x^(n+1)-1)/(n+1) for the antiderivative of x^n, precisely for the   
   > reason you give above.   
      
   Apologies!  I had forgotten it, and unfortunately I have not had a   
   working copy for decades.   
      
     I suspect that the precise reason is partly historical --   
     that my colleague, W. Kahan was a fan of Derive, and often derided   
   other systems by showing that Derive got the right answer contrary to   
   others. He also made suggestions which were often adopted by Derive but   
   not by other systems...  Like the integral for x^n, I think.   
      
     But when Derive IS able to determine that n   
   > is not -1, it returns the simpler antiderivative x^(n+1)/(n+1).   
      
   This is true of Maxima as well.   A simple "fix" is to say assume(n>-1).   
      
      BTW,   
   > Rubi always returns the simpler antiderivative, but users are free to   
   > change its behavior to mimic that of Derive by making a simple edit   
   > to just one of Rubi's 6000+ rules.   
      
   Ah yes.  Like the chiropractor who makes a simple adjustment.   
      
   Why not make that change in the standard system?  Is it   
   that the leafcount would be bigger and therefore the answer -- while   
   technically superior -- would not get as good a grade?   I think   
   there is more substance to an argument that says Rubi gets a more valid   
   answer, but it's your choice.   
   RJF   
      
   >   
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