antispam@math.uni.wroc.pl schrieb:   
   >   
   > clicliclic@freenet.de wrote:   
   > >   
   > > I suppose "more algebraic functions" include those cube-root integrands   
   > > whose Trager resultants are polynomials in t^3. When some change in the   
   > > integrand coefficients makes such integrals non-elementary, does FriCAS   
   > > 1.3.0 certify them as non-elementary by returning them unevaluated, or   
   > > does it merely exit with an error message like "implementation   
   > > incomplete (trace 0)"?   
   >   
   > Trager resultant is of form t^3 - a (that us residues are third roots   
   > ) and integral is nonelementary current implementation will detect it.   
   > But as you showed adding extra part can change Trager resultant...   
   >   
      
   ... which doesn't sound promising. But I see that I should have been   
   specific. So, how does FriCAS 1.3.0 react to:   
      
     INT(1/(x^3 - 3*x^2 + 7*x - 4)^(1/3), x) = ?   
      
     INT(1/(x*(3*x^2 - 6*x + 5)^(1/3)), x) = ?   
      
     INT((1 - x^3)^(1/3)/(2 + x), x) = ?   
      
     INT((2 + x)/((1 + x + x^2)*(2 + x^3)^(1/3)), x) = ?   
      
   All four are 'detuned' versions of pseudo-elliptics considered earlier.   
      
      
   > > Analogy suggests that Trager's resultants for classical Goursat pseudo-   
   > > elliptics involving square roots of quartics or cubics are limited to   
   > > polynomials in t^2. Perhaps the problem integrand (x - SQRT(a^2+1) - a)   
   > > / ((x + SQRT(a^2+1) - a)*SQRT((x - a)*(x^2 + 1))) could thus also be   
   > > handled in finite time by hard-coding of the splitting field.   
   >   
   > The problem is quite different here.   
   >   
      
   Taking Goursat's elementary viewpoint, integrability detection and   
   antiderivative computation are closely related for the square-root and   
   cube-root problems. That's why I would be surprised if the Trager   
   resultant of this problem integrand were not just a polynomial in t^2.   
      
   Martin.   
      
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