From: fateman@cs.berkeley.edu   
      
   There are a few points that I think are worth mentioning.   
      
   1. The Risch/ etc/ methods determine algebraic anti-derivatives.   
   It is a bit of a stretch sometimes to identify them with functions   
   with certain properties. It is typically possible to multiply or add   
   or otherwise muck around with these functions by, for example, use   
   of delta or step functions. Or adding strange constants.   
      The results of a subsequent differentiation need not be affected.   
      
   2. Either we are dealing with functions of a single variable (e.g. x)   
   or multiple variables / parameterized or multidimensional integral.   
      
      a. If multiple variables, the notion of a singularity becomes much   
   more complicated, and insisting on a continuous function much more   
   hazardous.  Try to find useful theorems about what amounts to   
     functions of several complex variables. Not too many that I have found.   
      
      b. If a single variable, you are presumably being presented with a   
   function that is computable and continuous [yeah, well some version of   
   continuous for some version of integration]  within the limits of   
   integration (yes, I know you might not know the limits.  But you really   
   have a lot of nerve asking for the integral of a function you are not   
   willing to define adequately between the limits of interest.)   
   Continuing on this -- a continuous function of the kind being discussed   
   here can almost always be integrated by numerical methods.  That is   
   given f(x), excellent methods exist to produce a program F(a,b)   
   which returns the integral of f from a to b  (where a and b are   
   numerical constants).   
     Arbitrarily high precision methods are available, as are error   
   bounds.  Chance of being fooled very low.   
      
   These methods avoid the problems encountered with symbolic methods which   
   include  (i) the uselessness of an implicit proof that there is no   
   elementary expression for the antiderivative  or  (ii) there may be one   
   but we can't tell for sure because there is a bug in our complicated   
   program or (iii) because the procedure requires a sub-algorithm that is   
   not computable [zero equivalence] we can't say for sure about (i) or (ii).   
      
   3. You might argue  (certainly I have) that symbolic results display   
   more information than (say) a table of numbers for various a,b.  Yet   
   if the result is going to be run through a plotting program, not so   
   convincing.  A partial symbolic result can be obtained by doing a   
   numerical-partial-fraction decomposition of some expressions --   
   any denominator that looks like a polynomial in one variable with   
   coefficients that can be resolved to floating-point numbers can be   
   factored into a product of linear factors, thereby requiring only   
   integration of  / (x-r)^n.   
      
      
   4. You might argue that you need the results for a high-dimension   
   multiple integration (calculation of Feynman diagrams was a major   
   selling point of integration in the Reduce system). Eh, then you   
   have multiple variables, but given limits, I think.   
      
   Now if you want to posit a problem that is to find an antiderivative   
   that has the fewest singularities, that may be useful. Another challenge   
   is to find an antiderivative that is, by some data-structure complexity   
   measure, "the simplest". This seems to be the objective of Rubi.   
      
   These problems are interesting in a computer-sciency computer algebra   
   systems algebraic-geometry context.   
      
   I'm not aware of any current clamor from actual or potential users of   
   computer algebra systems for solution of these problems.  Are there   
   any Feynman diagram people still out there?   
      
   I'm not saying that Bronstein's work, or the programmers attempting to   
   solve the problems in this thread are "wrong", just that the context and   
   relevance to (say) scientific computation is easy to misconstrue.   
      
   Cheers.   
   RJF   
      
      
      
   On 1/31/2017 8:39 AM, clicliclic@freenet.de wrote:   
   >   
   > clicliclic@freenet.de schrieb:   
   >>   
   >> antispam@math.uni.wroc.pl schrieb:   
   >>>   
   >>> My calculations indicate that points of order 3 correspond to   
   >>> u^2 = 1 +- 2*sqrt(3)/3.  FriCAS says that integal is nonelementary,   
   >>> so something is wrong...   
      
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