From: nma@12000.org   
      
   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.   
   >   
      
   hi   
      
   fyi;   
      
   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))   
      
   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]   
      
   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]   
      
   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   
      
   --Nasser   
      
   --- SoupGate-Win32 v1.05   
    * Origin: you cannot sedate... all the things you hate (1:229/2)