From: Albert_Rich@msn.com   
      
   On Friday, August 19, 2016 at 8:42:33 PM UTC-10, clicl...@freenet.de wrote:   
      
   > Hah: There are still cube-root Goursat integrals FriCAS cannot do! And   
   > wow: a monic quadratic in ?^3 already too complicated! My Goursat   
   > integrability check, however, immediately certifies the whole integral   
   > as elementary:   
   >    
   >   goursat5a((a + b*x)/(3 + x^2), x, 1, 0, -1, 0)   
   >    
   >   [false, false, false, true, true]   
   >    
   > the final true signifying integrability. And its evaluation does not   
   > naturally decompose into pure a and b parts:   
   >    
   > INT((a + b*x)/((3 + x^2)*(1 - x^2)^(1/3)), x)   
   >  = 2^(1/3)*(a + 3*b)/24*(LN((1 - x)^3 + 2*(1 - x^2))   
   >  - 3*LN((1 - x) + 2^(1/3)*(1 - x^2)^(1/3))   
   >  + 2*SQRT(3)*ATAN(1/SQRT(3)*(1 - 2^(2/3)*((1 - x)/(1 - x^2)^(1/3)))))   
   >  - 2^(1/3)*(a - 3*b)/24*(LN((1 + x)^3 + 2*(1 - x^2))   
   >  - 3*LN((1 + x) + 2^(1/3)*(1 - x^2)^(1/3))   
   >  + 2*SQRT(3)*ATAN(1/SQRT(3)*(1 - 2^(2/3)*((1 + x)/(1 - x^2)^(1/3)))))   
   >    
   > Alternative formulations are possible, but none has one part pure a and   
   > the other pure b.   
   > [...]   
      
   Martin's antiderivative for (a + b*x)/((3 + x^2)*(1 - x^2)^(1/3)) may be   
   optimal.  But if either a or b is 0, there exists better ones.  Using the   
   substitution u=x^2, the following antiderivative of x/((3 + x^2)*(1 -   
   x^2)^(1/3)) is relatively trivial:   
      
       Sqrt[3]/(2*2^(2/3)) * ArcTan[(1 + 2^(1/3)*(1 - x^2)^(1/3))/Sqrt[3]] -    
       1//(4*2^(2/3)) * Log[3 + x^2] +    
       3/(4*2^(2/3)) * Log[2^(2/3) - (1 - x^2)^(1/3)]   
      
   The following antiderivative of 1/((3 + x^2)*(1 - x^2)^(1/3)) involves two   
   arctangents and two logs, instead of Martin's two arctangents and four logs:   
      
       1/(2*2^(2/3)*Sqrt[3]) *    
         ArcTan[(1 - 2^(2/3)*(1 - x)^(2/3)/(1 + x)^(1/3))/Sqrt[3]] -    
       1/(2*2^(2/3)*Sqrt[3]) *    
         ArcTan[(1 - 2^(2/3)*(1 + x)^(2/3)/(1 - x)^(1/3))/Sqrt[3]] -    
       1/(4*2^(2/3)) * Log[(1 - x)^(2/3) + 2^(1/3)*(1 + x)^(1/3)] +    
       1/(4*2^(2/3)) * Log[2^(1/3)*(1 - x)^(1/3) + (1 + x)^(2/3)]   
      
   I am trying to derive a general rule for integrating expressions of the form   
   1/((a + b*x^2)*(c + d*x^2)^(1/3)).  Can Martin's Goursat integrability checker   
   determine the relationship, if any, between the parameters a, b, c and d that   
   must hold in order    
   for an elementary antiderivative to exist?  Of course, help anyone can provide   
   with the rule itself would be much appreciated...   
      
   Albert   
      
   --- SoupGate-Win32 v1.05   
    * Origin: you cannot sedate... all the things you hate (1:229/2)