Albert Rich schrieb:   
   >   
   > 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.   
   >   
      
   That's just why I juggled around with reciprocal arguments for the   
   evaluation:   
      
     INT(1/((2 - x^2)*(x^2 - 1)^(1/4)), x)   
      = SQRT(2)/4*(ATANH(SQRT(2)*(x^2 - 1)^(1/4)/x)   
                  + ATAN(x/(SQRT(2)*(x^2 - 1)^(1/4))))   
      
   posted on Wed, 20 Sep 2017 20:37:57 +0200. Plotted on Derive, this   
   antiderivative exhibits an imaginary jump at x = 0, but is continuous   
   otherwise, and real for all x^2 > 1, reproducing the behavior of your   
   LOG variant. By contrast, the version:   
      
      = SQRT(2)/4*(ACOTH(x/(SQRT(2)*(x^2 - 1)^(1/4)))   
                  + ATAN(x/(SQRT(2)*(x^2 - 1)^(1/4))))   
      
   on Derive shows imaginary jumps at x^2 = 1, and is real for x > 1 only.   
   And the version:   
      
      = SQRT(2)/4*(ACOTH((SQRT(2)*(x^2 - 1)^(1/4))/x)   
                  + ATAN(x/(SQRT(2)*(x^2 - 1)^(1/4))))   
      
   jumps only at x = 0, but shows an imaginary offset for all x^2 > 0.   
      
   Martin.   
      
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