Waldek Hebisch schrieb:   
   >   
   > clicliclic@freenet.de wrote:   
   > >   
   > > ... and another quick FriCAS 1.2.3 on-line experiment:   
   > >   
   > > integrate((a + b*x + c*x^2)/((1 - x + x^2)*(1 - x^3)^(1/3)), x)   
   > >   
   > > >> Error detected within library code:   
   > >    integrate: implementation incomplete   
   > >    (residue poly has multiple non-linear factors)   
   > >   
   > > Is there no workable treatment of "multiple non-linear factors" other   
   > > than Kauer's Groebner-basis heuristics?   
   > >   
   >   
   > This one can be handled by simple method: since the integrand   
   > is linear in a, b, c each term can be integrated separately.   
   > FriCAS can integrate each part, however this is slow   
   > (I used 'setSimplifyDenomsFlag(true}' before integration to   
   > get resonable time) and result is ugly (sum over roots of   
   > sextic of large function).   
   >   
      
   Hmm, an integrator that can (incompletely) do individual terms but not   
   their sum. But this trick doesn't help with Example 21 from Timofeev   
   Chapter 4:   
      
   INT(((x - 1)^2*(x + 1))^(1/3)/x^2, x) = ?   
      
   Another way would be to split the rational factor into its complex   
   partial fractions:   
      
   (a + b*x + c*x^2)/(1 - x + x^2) = c   
    + (3*(b + c) + SQRT(3)*#i*(2*a + b - c))/(3*(2*x + SQRT(3)*#i - 1))   
    + (3*(b + c) - SQRT(3)*#i*(2*a + b - c))/(3*(2*x - SQRT(3)*#i - 1))   
      
   But this once again produces ">> Error detected within library code:   
   impossible".   
      
   However, the full result then doesn't involve any unresolvable roots:   
      
   INT(1/(1-x^3)^(1/3),x)=1/2*LN((1-x^3)^(1/3)+x)-1/SQRT(3)*ATAN(((~   
   1-x^3)^(1/3)-2*x)/(SQRT(3)*(1-x^3)^(1/3)))   
      
   INT(1/((2*x+SQRT(3)*#i-1)*(1-x^3)^(1/3)),x)=2^(2/3)*(1+SQRT(3)*#~   
   i)/32*(LN(4+((2*x+1-SQRT(3)*#i)/(2*(1-x^3)^(1/3)))^3)-3*LN(2^(2/~   
   3)+(2*x+1-SQRT(3)*#i)/(2*(1-x^3)^(1/3)))+2*SQRT(3)*ATAN(1/SQRT(3~   
   )*(1-(2*x+1-SQRT(3)*#i)/(2^(2/3)*(1-x^3)^(1/3)))))   
      
   For the third integral replace #i by -#i.   
      
   Martin.   
      
   --- SoupGate-Win32 v1.05   
    * Origin: you cannot sedate... all the things you hate (1:229/2)