On Friday, April 10, 2020 at 2:46:28 PM UTC+10, clicl...@freenet.de wrote:   
   > antispam@math.uni.wroc.pl schrieb:   
   > >   
   > > samuel.thomas.blake@gmail.com wrote:   
   > > >   
   > > > Here's a couple of other error messages I found in 1.2.6 (on OSX).   
   > > >   
   > > > [...]   
   > > >   
   > > > integrate(((-1+x^4)*(1+x^2+x^4)*(-1+x^2-x^4)^(1/2))/(1+x^4)^3,x);   
   > > >   
   > > >    >> Error detected within library code:   
   > > >    catdef: division by zero   
   > > >   
   > > >   
   > > > integrate(((4+x^6)*(-4+x^4+2*x^6)*(32-14*x^4-32*x^6-4*x^8+   
   *x^10+8*x^12)^(1/2))/(x^9*(-2+x^6)),x);   
   > > >   
   > > >    >> Error detected within library code:   
   > > >    integrate: implementation incomplete (residue poly has multiple   
   non-linear factors)   
   > >   
   > > The two above are known problems, each will take more work to fix.   
   > > Actually, the first one requires reorganization of integration   
   > > routines.  The second one is king of open problem: known methods   
   > > of solving may require prohibitivly long time and nobody knows   
   > > of really efficient method.   
   > >   
   >   
   > By the way, the radicand sqrt(-1+x^2-x^4) is negative everywhere on the   
   > real axis, and FriCAS 1.3.6 succeeds when its sign is inverted:   
   >   
   >   integrate(((-1+x^4)*(1+x^2+x^4)*(1-x^2+x^4)^(1/2))/(1+x^4)^3,x)   
   >   
   > (((-5)*x^8+(-10)*x^4+(-5))*atan((2*x*(x^4+(-1)*x^2+1)^(1/2))   
   > /(x^4+(-2)*x^2+1))+((-6)*x^5+(-4)*x^3+(-6)*x)*(x^4+(-1)*x^2+1)^(1/2))   
   > /(16*x^8+32*x^4+16)   
   >   
   > But this may be just an accident.   
   >   
   > Martin.   
      
   Below is a link to an experimental heuristic for computing some    
   seudo-elliptic integrals. For example, the previous example uses the   
   substitution u = (-1 - x^2)/x   
      
   In[5026]:= int[((-1 + x^4) (1 + x^2 + x^4) Sqrt[1 - x^2 + x^4])/(1 + x^4)^3, x]   
   AlgebraicIntegrateHeuristic`Private`RationalSubstitution   
   D[%% // Last, x] - ((-1 + x^4) (1 + x^2 + x^4) Sqrt[1 - x^2 + x^4])/(1 +   
   x^4)^3 // Simplify   
      
   Out[5026]= {0, 0, (Sqrt[1 - x^2 + x^4] (-((3 x)/8) - x^3/4 - (3 x^5)/8))/(1 +   
   x^4)^2 + 5/8 ArcTan[Sqrt[1 - x^2 + x^4]/x]}   
      
   Out[5027]= (-1 - x^2)/x   
      
   Out[5028]= 0   
      
   https://github.com/stblake/algebraic_integration   
      
   Hopefully this method isn't too silly.   
      
   Sam   
      
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