From: fateman@cs.berkeley.edu   
      
   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   
      
      
      
      
      
      
      
      
      
      
      
      
   On 4/15/2017 12:13 PM, Nasser M. Abbasi wrote:   
   > On 4/15/2017 11:41 AM, clicliclic@freenet.de wrote:   
   >   
   >> I am not surprised that Maxima too isn't perfect :). Sympy would   
   >> be another candidate to check.   
   >>   
   >> Martin.   
   >>   
   >   
   > Fyi;   
   >   
   > sympy 1.0 =========   
   >> python   
   > Python 3.6.0 |Anaconda 4.3.1 (64-bit)| (default, Dec 23 2016,   
   > 12:22:00) [GCC 4.4.7 20120313 (Red Hat 4.4.7-1)] on linux   
   >>>> from sympy import * x=symbols('x') integrand=log(x)*log(x**2)   
   >>>> sol=integrate(integrand,x)   
   >   
   > 2*x*log(x)**2 - 4*x*log(x) + 4*x   
   >   
   >>>> integrand_back=diff(sol,x)   
   >   
   > 2*log(x)**2   
   >   
   >>>> simplify( integrand_back-integrand)   
   >   
   > (2*log(x) - log(x**2))*log(x)   
   >   
   > Maple 2016.2 ============   
   >   
   > restart; integrand:=ln(x)*ln(x^2); sol:=int(integrand,x);   
   >   
   > ln(x)*ln(x^2)*x+4*x-2*x*ln(x)-x*ln(x^2)   
   >   
   > integrand_back:=diff(sol,x);   
   >   
   > ln(x)*ln(x^2)   
   >   
   > simplify( integrand_back-integrand);   
   >   
   > 0   
   >   
   > --Nasser   
      
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