From: Albert_Rich@msn.com   
      
   On Monday, January 22, 2018 at 3:29:26 AM UTC-10, clicl...@freenet.de wrote:   
   >    
   > 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.   
      
   Thanks! I generalized the above result to integrate expressions of the form   
   x/((a+b*x^3)*sqrt(c+d*x^3)) when 8*b*c+a*d=0 and c is positive and when c is   
   negative.  The forthcoming Rubi 4.14.5 will incorporate these two new rules,   
   as well as two    
   corresponding rules for expressions of the form 1/((a+b*x^2)^(1/3)*(c+d*x^2))   
   when b*c-9*a*d=0.   
      
   However, for your first example integrand x/((4-x^3)*sqrt(1-x^3)), I have   
   tried in vain to discover a simple antiderivative involving just 3 or 4   
   arctan(h) and/or log terms. If you or someone derives an antiderivative   
   simpler than the one Rubi currently    
   returns, please let me know. Rubi will appreciate it.   
      
   Albert   
      
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