Albert Rich schrieb:   
   >   
   > [...]   
   >   
   > However, I am curious as to why the arguments of the arctan and   
   > arctanh are systematically, and I think unnecessarily, the reciprocals   
   > of each other in Martin's expression of the optimal antiderivatives.   
   > Could it be due to an artifact of Derive's simplifer?   
   >   
      
   Oh, I just took reciprocal ATAN or ATANH arguments as needed to avoid   
   discontinuities in plots of the antiderivatives on Derive. Since   
   Derive's definition of ATANH in particular is unusual, the arguments   
   should be reconsidered for Rubi on Mathematica. ACOT and ACOTH may also   
   be used.   
      
   As Rubi is referring to Euler for this, you may also want to cite his   
   E695, "Integratio succincta formulae integralis maxime memorabilis ..."   
   and Legendre's "Traité des Fonctions Elliptiques", Tome I, Chapitre XXVI   
   (De l'intégrale ...) if dedicated rules are supplied for those integrals   
   too.   
      
   Martin.   
      
   PS: Might Rubi be interested in:   
      
   INT(x^2/((2 - x^2)*(x^2 - 1)^(3/4)), x) = SQRT(2)/2   
   *(ATANH(SQRT(2)*(x^2 - 1)^(1/4)/x) - ATAN(x/(SQRT(2)*(x^2 - 1)^(1/4))))   
      
   and friends as well?   
      
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