Albert Rich schrieb:   
   >   
   > On Tuesday, January 23, 2018 at 4:40:12 AM UTC-10, anti...@math.uni.wroc.pl   
   wrote:   
   > >   
   > > I do not see Albert's message, only this reply so here I am   
   > > answering to Albert.   
   > >   
   > > AFAICS x/((x^3+9)*sqrt(x^3-1)) has no elementary antiderivative.   
   > > I am surprised that you go that way.  Examples that Martin gave   
   > > are essentially diofantine phenomana depending on arithmetic   
   > > of elliptic curves.  ATM Rubi contains rules for handful   
   > > of curves.  But there are infinitely many essentially different   
   > > elliptic curves and tables giving "simple" ones contain   
   > > thousands of entries (google Cremona).  Each such curve leads   
   > > to its own family of pseudoelliptics.  You may be able to   
   > > find some patterns, but apparently number theorists believe   
   > > that core behaviour is essentially random.  So this task   
   > > is like table of prime numbers, only entries have much   
   > > more complicated nature.   
   > >   
   >   
   > Sorry, I made a typo.  I meant to type x/((x^3+8)*sqrt(x^3-1)), the   
   > integrand of Martin's second pseudo-elliptic integral example.   
   >   
   > As far as the density of pseudo-elliptic integrals, Martin is better   
   > equipped to address that question than I.  However, I would bet that   
   > only a finite number of rules are required to find elementary   
   > antiderivatives for pseudo-elliptic integrands of the form   
   >   
   >     x^m * (a+b*x^3)^n * (c+d*x^3)^(p/2)   
   >   
   > where m, n and p are integers.   
   >   
   > Note that rather than finding optimal antiderivatives for all   
   > possible expressions, the more modest goal of Rubi is to find such   
   > antiderivatives for all elements of well-defined classes, like the   
   > form given above.   
   >   
      
   Thanks to the (still unpublished?) Masser-Zanier counterexample Waldek   
   informed us about, we know that there are infinitely many (though   
   increasingly complicated algebraic) u^2 for which   
      
     INT(x/((x^2 - u^2)*SQRT(x^3 - x)), x)   
      
   has a solution (also of increasing complexity) in terms of elementary   
   functions. Consequently, via the usual Moebius transformation of the   
   integration variable, there are also infinitely many v = (u^2 - 1)/   
   (u^2 + 1) for which   
      
     INT((y^2 - 1)/((y^2 + 2*v*y + 1)*SQRT(y^4 - 1)), y)   
      
   has such an elementary solution. I too expect these patterns to be   
   exceptions rather than the rule, the rule being that the number of   
   pseudo-elliptic cases is finite and small. But does this pair already   
   exhaust the store of patterns involving simple square-root radicands   
   and giving rise to infinitely many pseudo-elliptic integrals?   
      
   As regards pseudo-elliptics involving square roots, FriCAS appears to   
   be reliable: earlier failures for many of the Masser-Zanier integrals   
   could be attributed to silly coding bugs rather than design oversights.   
   Is there any moderately-sized square-root pseudo-elliptic integral on   
   which FriCAS is still known to fail?   
      
   Pseudo-elliptics involving higher roots appear to be a different story.   
      
   Martin.   
      
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