clicliclic@freenet.de schrieb:   
   >   
   > Here's a much more deadly failure:   
   >   
   > setSimplifyDenomsFlag(true)   
   >   
   > integrate((7*x + 6*sqrt(17*sqrt(2) + 23) + 9*sqrt(2) + 34)   
   > /((x^2 + 2*x*(3*sqrt(sqrt(2) + 1) + 1) + 6*sqrt(58*sqrt(2) + 82)   
   >  - 18*sqrt(2) - 26)*sqrt(x^3 - 30*x - 56)), x)   
   >   
   > is returned unevaluated on the web interface. But my oracle says that   
   > the integand is pseudo-elliptic according to Goursat:   
   >   
   > goursat3((7*x + 6*SQRT(17*SQRT(2) + 23) + 9*SQRT(2) + 34)   
   > /(x^2 + 2*x*(3*SQRT(SQRT(2) + 1) + 1) + 6*SQRT(58*SQRT(2) + 82)   
   >  - 18*SQRT(2) - 26), x, -56, -30, 0, 1)   
   >   
   > [false, false, false, true]   
   >   
   > The last answer counts.   
   >   
      
   Since the oracle was characteristically cryptic, I should supply some   
   details. Putting x = (6 - 3*SQRT(2))*t - 4 one finds:   
      
   SQRT(x^3 - 30*x - 56) =   
   3*SQRT(12 - 6*SQRT(2))*SQRT(t*(1 - t)*(1 - (SQRT(2) - 1)^2*t))   
      
   So Legendre's modulus k of the elliptic radical takes the nice special   
   value SQRT(2) - 1. The quadratic in the denominator of the integrand   
   divides the division polynomial of order four for the elliptic curve   
   y^2 = x^3 - 30*x - 56. The antiderivative consists of two terms:   
      
   6*(2*SQRT(SQRT(2) + 1) - SQRT(2))/7   
    * INT((7*x + 6*SQRT(17*SQRT(2) + 23) + 9*SQRT(2) + 34)   
    / ((x^2 + 2*x*(3*SQRT(SQRT(2) + 1) + 1) + 6*SQRT(58*SQRT(2) + 82)   
    - 18*SQRT(2) - 26)*SQRT(x^3 - 30*x - 56)), x)   
    = 2/(SQRT(3*SQRT(2) + 6) - SQRT(3)*2^(3/4))   
    * ATANH((SQRT(3*SQRT(2) + 6) - SQRT(3)*2^(3/4))*(x - 3*SQRT(2) - 2)   
    / SQRT(x^3 - 30*x - 56)) - SQRT(6 - 3*SQRT(2))/3   
    * ATAN(SQRT(6*SQRT(2) + 12)*(x + 4)/SQRT(x^3 - 30*x - 56))   
      
   Having this integral declared non-elementary constitutes a bug that   
   seriously undermines trust in the FriCAS integrator.   
      
   Martin.   
      
   --- SoupGate-Win32 v1.05   
    * Origin: you cannot sedate... all the things you hate (1:229/2)