antispam@math.uni.wroc.pl schrieb:   
   >   
   > clicliclic@freenet.de wrote:   
   > >   
   > > antispam@math.uni.wroc.pl schrieb:   
   > > >   
   > > > clicliclic@freenet.de wrote:   
   > > > >   
   > > > > Thanks to the (still unpublished?) Masser-Zanier counterexample   
   > > > > Waldek informed us about, we know that there are infinitely many   
   > > > > (though increasingly complicated algebraic) u^2 for which   
   > > > >   
   > > > >   INT(x/((x^2 - u^2)*SQRT(x^3 - x)), x)   
   > > > >   
   > > > > has a solution (also of increasing complexity) in terms of   
   > > > > elementary functions. Consequently, via the usual Moebius   
   > > > > transformation of the integration variable, there are also   
   > > > > infinitely many v = (u^2 - 1)/(u^2 + 1) for which   
   > > > >   
   > > > >   INT((y^2 - 1)/((y^2 + 2*v*y + 1)*SQRT(y^4 - 1)), y)   
   > > > >   
   > > > > has such an elementary solution. I too expect these patterns to be   
   > > > > exceptions rather than the rule, the rule being that the number of   
   > > > > pseudo-elliptic cases is finite and small. But does this pair   
   > > > > already exhaust the store of patterns involving simple square-root   
   > > > > radicands and giving rise to infinitely many pseudo-elliptic   
   > > > > integrals?   
   > > >   
   > > > Well, any torsion point leads to a pseudo-elliptic integral.  Over   
   > > > algebraically closed field there are infintely many torsion points   
   > > > on any curve.  So example with _single_ parameter may be exceptional,   
   > > > put once you allow algebraic extentions there is see of strange   
   > > > examples -- just two parameters family:   
   > > >   
   > > > integrate((1/(x - a)  - b)/sqrt(P), x)   
   > > >   
   > > > has no elementary integral valid for continuous family of a and b   
   > > > and infintely many integrable cases when P is a polynomial of   
   > > > degree 3 without multiple factors.   
   > > >   
   > >   
   > > Yes, my comment was about the dependence on a single parameter, with any   
   > > additional parameters kept fixed.   
   >   
   > To be clear: above P is fixed.  And b is a function of a (there is   
   > at most one b for which the integral is elementary).  I think that   
   > b can be given as a simple expression, but it would take extra   
   > effort to find it.  If you are really bothered by b you can pass   
   > to Jacobi form, that is take P = (1 - x^2)*(1 - m*x^2).  Then   
   > b = 0.   
   >   
      
   There is no need to fix P = (x-u)*(x-v)*(x-w) in INT((1/(x-a) - b)/   
   SQRT(P), x). But the search for pseudo-elliptic instances can be   
   narrowed down to u = 0, v = 1 without loss of generality: any linear   
   transformation x' = p*x + q just takes the integrand into some   
   (1/(x'-a') - b')/SQRT(P') of the same form with P' = (x'-u')*(x'-v')*   
   (x'-w').   
      
   For u = 0, v = 1, the system of equations to be satisfied by the   
   pseudo-elliptic instances considered by Goursat is easily solved:   
      
     [w = a^2, a*b = -1/2]   
      
     [w = a*(2 - a), b*(a - 1) = -1/2]   
      
     [w*(a - 1/2) = 1/2*a^2, b*a*(1 - a) = a - 1/2]   
      
   In terms of a, the three solutions correspond to the integrals:   
      
     INT((x + a)/((x - a)*SQRT(x^3 - x^2*(a^2 + 1) + a^2*x)), x)   
      
     INT((x + a - 2)   
         /((x - a)*SQRT(x^3 + x^2*(a^2 - 2*a - 1) + a*x*(2 - a))), x)   
      
     INT((x*(2*a - 1) - a)   
         /((x - a)*SQRT(x^3*(2*a - 1) - x^2*(a^2 + 2*a - 1) + a^2*x)), x)   
      
   I am confident that FriCAS can solve all three, and suspect that there   
   are no other (i.e. non-Goursat) pseudo-elliptics of this general form.   
   By the way, a rule-based integrator would not need to solve polynomial   
   equations, it only needs to check if a given instance of the integrand   
   satisfies them.   
      
   Martin.   
      
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