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 don't think the example on which your rule 482 is based was provided   
   by me (no such example was in my files until very recently), but perhaps   
   you transformed an example containing (1 - x^2)^(1/3) by making this   
   radical the new integration variable. Anyway, the following result   
   should be much simpler than the elementary antiderivative returned by   
   FriCAS:   
      
   INT(18*x/((x^3 + 8)*SQRT(x^3 - 1)), x)   
    = SQRT(3)*ATANH(SQRT(3)*(x - 1)/SQRT(x^3 - 1))   
    - ATAN(3/SQRT(x^3 - 1))   
    + ATAN((x - 1)^2/(3*SQRT(x^3 - 1)))   
      
   where (if you prefer):   
      
   - ATAN((x - 1)^2/(3*SQRT(x^3 - 1)))   
    = ATAN((x - 1)/SQRT(x^3 - 1)) + ATAN((2*x + 1)/SQRT(x^3 - 1))   
      
   Martin.   
      
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