From: Albert_Rich@msn.com   
      
   On Tuesday, October 3, 2017 at 6:40:04 AM UTC-10, clicl...@freenet.de wrote:   
   > The appearance of the imaginary unit #i should render this result   
   > meaningless to many students of analysis.   
   >    
   > The Rubi antiderivative above does not have this defect, but another   
   > recent result (from Albert's post of Fri, 22 Sep 2017 in the thread   
   > "can your system handle 4th-root pseudo-elliptics?") does:   
   >    
   >   ArcTan[x/(Sqrt[2]*(x^2-1)^(1/4))]/Sqrt[2] +    
   >   ArcTanh[x/(Sqrt[2]*(x^2-1)^(1/4))]/Sqrt[2] -    
   >   2*(1-x^2)^(1/4)/(x^2-1)^(1/4)*EllipticE[ArcSin[x]/2,2]   
   >    
   > Can you spot the problem?   
   >    
   > Martin.   
      
   The problem is the (1-x^2)^(1/4) factor in the EllipticE term which is   
   imaginary when x^2>1; whereas x^2/((2-x^2)*(x^2-1)^(1/4)) is real when x^2>1.    
   The forthcoming Rubi 4.14 resolves this problem as follows:   
      
   For the antiderivative of 1/(x^2-1)^(3/4), Rubi 4.13 returns   
      
       2*(1-x^2)^(3/4) / (x^2-1)^(3/4) * EllipticF[ArcSin[x]/2, 2]   
      
   Although nice and compact, it is imaginary on the entire real line; whereas   
   the integrand is real when x^2>1.  Using the nonobvious piecewise-constant   
   extraction sqrt(x^2)/x, followed by the substitution u=(x^2-1)^(1/4), Rubi   
   4.14 will return   
      
       Sqrt[x^2/(1+Sqrt[x^2-1])^2] * (1+Sqrt[x^2-1])/x *    
         EllipticF[2*ArcTan[(x^2-1)^(1/4)],1/2]   
      
   which is real and continuous on the real line when the integrand is real.   
      
   The same piecewise-constant extraction and substitution will be used to find a   
   similarly optimal antiderivative of 1/(x^2-1)^(1/4).  Since   
      
       x^2/((2-x^2)*(x^2-1)^(1/4)) =    
         2/((2-x^2)*(x^2-1)^(1/4)) - 1/(x^2-1)^(1/4),   
      
   the integral of 1/(x^2-1)^(1/4) is the elliptic term of the integral of   
   x^2/((2-x^2)*(x^2-1)^(1/4)) which will now be real when x^2>1.   
      
   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/(S   
   rt[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   
      
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