From: Albert_Rich@msn.com   
      
   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.  But   
   when Derive IS able to determine that n is not -1, it returns the simpler   
   antiderivative x^(n+1)/(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.   
      
   As far as testing antiderivatives for optimality, leaf count is only one of   
   many factors that should be taken into account.  Others include continuity,   
   imaginary unit free, elementary vs. advanced functions, finite limiting   
   values, and most importantly    
   validity over the complex plane.  Programs that compare the results of various   
   systems on large test suites should take into account all these factors, and   
   ignore quibbles over minor differences in leafcounts that result from putting   
   answers over common    
   denominators, etc.   
      
   Albert   
      
   --- SoupGate-Win32 v1.05   
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