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 essentialy diofantine phenomana depending on arithmetic   
   of elliptic curves.  ATM Rubi contains rules for handful   
   of curves.  But there are infintely many essentially different   
   elliptic curves and tables giving "simple" ones contain   
   thousends 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 numberes, only entries have much   
   more complicated nature.   
      
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