Albert Rich schrieb:   
   >   
   > On Friday, September 22, 2017 at 6:58:04 AM UTC-10, clicl...@freenet.de   
   wrote:   
   > >   
   > > Albert Rich schrieb:   
   > > >   
   > > > As Martin pointed out, when m=0 and n=1 or when m=2 and n=3 these   
   > > > integrals are pseudo-elliptic.  However for other even values of   
   > > > m, the optimal antiderivatives do seem to require a single, simple   
   > > > elliptic term.  Do you concur?   
   > >   
   > > I suppose this is most easily answered by means of a capable   
   > > algebraic Risch integrator.   
   >   
   > For the antiderivative of x^2/((2-x^2)*(x^2-1)^(1/4)) Rubi 4.13.3 gets   
   >   
   >   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]   
   >   
   > involving only a single EllipticE function.  If this was actually a   
   > seudo-elliptic integral, that would imply EllipticE[ArcSin[x]/2,2]   
   > could be expressed in terms of elementary functions.  But surely that   
   > is not the case(?).  Ergo the integral is elliptic.   
   >   
      
   I too believe that E(phi, k) := INT(SQRT(1 - k^2*SIN(p)^2), p, 0, phi)   
   cannot be expressed in terms of elementary function unless k^2 = 0 or   
   k^2 = 1, but I cannot name a source for this. The Digital Library of   
   Mathematical Functions at  may be a good point   
   to start digging. For specific values of k^2, however, an on-line proof   
   can be ordered at:   
        
   Scroll down to the bottom of the page, enter:   
      
   \begin{axiom}   
   setSimplifyDenomsFlag(true)   
   integrate(your_elliptic_integrand, your_integration_variable)   
   \end{axiom}   
      
   into the grey text box, and hit the Preview Button. After the screen   
   has been updated, inspect the result: if your integral is returned   
   unevaluated, FriCAS claims that it cannot be expressed in terms of   
   elementary functions.   
      
   Martin.   
      
   PS: Use lower-case sqrt(), sin(), etc.   
      
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