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 Thanks to the (still unpublished?)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.   
      
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