"Nasser M. Abbasi" schrieb:   
   >   
   > On 1/24/2017 1:33 AM, clicliclic@freenet.de wrote:   
   > >   
   > >   "Never forget to check that the output is a continuous function!"   
   > >   
   > > Thus James H. Davenport reminded us in a conference presentation on the   
   > > "Complexity of Integration, Special Values, and Recent Developments" at   
   > > last year's International Congress on Mathematical Software.   
   > >   
   > > Now, in a posthumous publication in the Institutiones Calculi Integralis   
   > > Tomus 4, pages 36 - 48, Leonhard Euler showed that:   
   > >   
   > >   SQRT(a + b*x^2 + c*x^4)/(a - c*x^4)   
   > >   
   > > has an elementary antiderivative. In fact, the integrand is Goursat   
   > > pseudo-elliptic.   
   > >   
   > > Does your favorite symbolic integrator then succeed in returning a (i)   
   > > real, (ii) elementary, and (iii) continuous result for:   
   > >   
   > >   INT(SQRT(1 - x^4)/(1 + x^4), x)   
   > >   
   > > ? Yes, a simple such a solution does exist.   
   > >   
   > > Martin.   
   > >   
   > > PS: Derive 6.10 just returns the integral with its denominator factored   
   > > into real quadratics.   
   > >   
   >   
   > Maple 2016.2   
   > ===============   
   >   
   > int( sqrt(1 - x^4)/(1 + x^4), x);   
   >   
   > -(1/4)*arctan((-x^4+1)^(1/2)/x+1)+(1/4)*   
   > arctan(-(-x^4+1)^(1/2)/x+1)-(1/8)*   
   > ln(((1/2)*(-x^4+1)/x^2-(-x^4+1)^(1/2)/x+1)/((1/2)*(-x^4+1)/x^2   
   (-x^4+1)^(1/2)/x+1))   
      
   This is (i) real and (ii) elementary, but (iii) discontinuous at x=0.   
   There are also imaginary discontinuities at x^2 = 1 + SQRT(2).   
      
   >   
   > Mathematica 11.0.1   
   > ==================   
   > Integrate[ Sqrt[1 - x^4]/(1 + x^4), x]   
   > x AppellF1[1/4, -(1/2), 1, 5/4, x^4, -x^4]   
      
   This is not elementary. I am not surprised.   
      
   >   
   > Rubi 4.10   
   > ===========   
   > ShowSteps = False;   
   > Int[ Sqrt[1 - x^4]/(1 + x^4), x]   
   >   
   > -EllipticF[ArcSin[x], -1] + EllipticPi[-I, ArcSin[x], -1] +   
   >   EllipticPi[I, ArcSin[x], -1]   
      
   This is neither real nor elementary. I am surprised.   
      
   >   
   > FriCAS 1.3   
   > ==========   
   > integrate( sqrt(1 - x^4)/(1 + x^4), x)   
   >   
   > (log((2*x*((-1)*x^4+1)^(1/2)+((-1)*x^4+2*x^2+1))/(x^4+1))+((-2)*atan((((-1)*   
   >    x^4+1)^(1/2)+x)/x)+2*atan(x^2)))/4   
      
   This is (i) real and (ii) elementary, but (iii) discontinuous at x=0.   
   There are no other discontinuities.   
      
   -   
      
   Only Maple 2016.2 and FriCAS 1.3 return elementary answers, but both are   
   discontinuous at x=0.   
      
   More work is therefore needed to make the output of these integrators   
   Davenport compatible.   
      
   Martin.   
      
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