Albert Rich schrieb:   
   >   
   > On Tuesday, January 31, 2017 at 6:40:08 AM UTC-10, clicl...@freenet.de wrote:   
   > >   
   > > This can still be simplified considerably:   
   > >   
   > >   1/2*ATANH(x*(1 - x^2)/SQRT(1 - x^4))   
   > >   + 1/2*ATAN(x*(1 + x^2)/SQRT(1 - x^4))   
   > >   
   >   
   > Nice!  Is it possible to Riobooize the arctan term so the result is   
   > continuous at x^2=1?   
   >   
      
   As oldk1331 has pointed out, this is just a pointlike defect, which can   
   be automatically repaired by taking the usual limits x -> +-1.   
      
   Plain heresy: No general-purpose integrator needs to handle algebraic   
   integrands beyond Goursat pseudo-elliptics, and Goursat antiderivatives   
   do not exhibit discontinuities if done properly. This applies to both   
   square-root and cube-root integrands.   
      
   In general, Riobooizing a term c*ATAN(g(x)) of an antiderivative amounts   
   to rewriting g(x) as R(h(x)) where R is a rational function, and then   
   applying Rioboo's recursion to ATAN(R(h)). Here, c must be constant and   
   h(x) must remain finite in the x range of interest, else the entire   
   exercise would be pointless, or even disastrous.   
      
   Not seeing how the present ATAN argument x*(1 + x^2)/SQRT(1 - x^4) could   
   be expressed as some rational function R(h(x)), I am led to believe that   
   this ATAN term cannot be Riobooized.   
      
   Martin.   
      
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