From: Albert_Rich@msn.com   
      
   On Tuesday, October 3, 2017 at 5:58:21 PM UTC-10, Albert Rich wrote:   
      
   > However, there may still remain a problem.  For the pseudo-elliptic   
   integrand 1/((2-x^2)*(x^2-1)^(1/4)), Rubi returns   
   >    
   >     ArcTan[x/(Sqrt[2]*(x^2-1)^(1/4))]/(2*Sqrt[2]) +    
   >     ArcTanh[x/(Sqrt[2]*(x^2-1)^(1/4))]/(2*Sqrt[2])   
   >    
   > which is never real on the real line, although the integrand is when x^2>1.    
   This is analogous to the way log(x) is not real when x<0, although 1/x is real   
   on the entire real line.  So is the above antiderivative for 1/(   
   2-x^2)*(x^2-1)^(1/4)) optimal    
   in terms of reality?   
   >    
   > Albert   
      
   Converting the above arctanh term to a log you get    
      
       ArcTan[x/(Sqrt[2]*(x^2-1)^(1/4))] / (2*Sqrt[2]) +    
       Log[(Sqrt[2]*x+2*(x^2-1)^(1/4))/   
           (Sqrt[2]*x-2*(x^2-1)^(1/4))] / (4*Sqrt[2])   
      
   for the antiderivative of 1/((2-x^2)*(x^2-1)^(1/4)), which magically is real   
   in all the right places and hence optimal.   
      
   Albert   
      
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