From: Albert_Rich@msn.com   
      
   On Tuesday, January 23, 2018 at 4:40:12 AM UTC-10, anti...@math.uni.wroc.pl   
   wrote:   
   > clicliclic@freenet.de wrote:   
   > >    
   > > Albert Rich schrieb:   
   > > >    
   > > > On Sunday, January 21, 2018 at 10:05:01 PM UTC-10, clicl...@freenet.de   
   wrote:   
   > > >    
   > > > > Thank you for analyzing this. To me it makes no sense to have Rubi   
   > > > > equipped with a rule for INT(x/((4 - x^3)*SQRT(1 - x^3)), x) but no   
   > > > > rule for the very similar INT(x/((x^3 + 8)*SQRT(x^3 - 1)), x). This   
   > > > > should be remedied, while Rubi's handling of the third and fourth   
   > > > > integrals need not be changed in my view.   
   > > > >   
   > > >    
   > > > As I recall, I derived Rubi 4.14.4 rule 482 by generalizing a specific    
   > > > example of a Goursat pseudo-elliptic integral for which you had   
   > > > provided the elementary antiderivative.  If you provide me the   
   > > > elementary antiderivative of x/((x^3+9)*sqrt(x^3-1)) wrt x, I will try   
   > > > to generalize it to make a rule analogous to 482.   
   > > >    
   >    
   > 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.   
   >     
   > --    
   >                               Waldek Hebisch   
      
   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.   
      
   Albert   
      
   --- SoupGate-Win32 v1.05   
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