clicliclic@freenet.de wrote:   
   >   
   > 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 Thanks to the (still unpublished?)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').   
      
   My point is that you can find solutions for _any_ P of allowed   
   form.  Of course which 'a' lead to integrable case depend on P.   
   But for each fixed P you get coutable set of 'a'.  This is   
   answer to Albert question: can finite number of rules cover   
   integrals of form:   
      
   integrate(1/((x - a)*sqrt(P)), x)   
      
   When P is of form (1 - x^2)*(1 - m*x^2) the answer is: no   
   finite system of reasonable rules will cover cases integrable   
   using elementary functions.  Namely, assuming polynomial type   
   conditions on 'a' each rule can catch only finite number of   
   'a': there are infintely many nonitegrable cases so polynomial   
   can be zero only for finite number of 'a'.  Also, there   
   are infintely many different forms of integral.  More preciely   
   you can write integral as   
      
   log((p + q*sqrt(P))/(p + q*sqrt(P)))   
      
   where p and q are polynomials, but there is no bound on   
   degree.  In fact degrees must grow to infinity.   
      
   For P of degree 3 with no multiple factors no finite system   
   of resonable rules will cover all case integrable using   
   elementary functions for family:   
      
   integrate((1/(x - a) - b)*(1/sqrt(P)), x)   
      
   If you insist to handle only 'b = 0' than in general   
   the integral   
      
   integrate(1/((x - a)*sqrt(P)), x)   
      
   need elliptic integral F.  However, if you insist on "optimal"   
   integral than presumably you will be not satisfied with   
   generic answer which is sum of elliptic integral F and   
   elliptic integral Pi and you will want logarithm in cases   
   when elliptic integral Pi is elementary.  Again, no finite   
   system of rules is enough to cover cases when integral F   
   + logarith is elementary.   
      
   And of course special role of 'b' somewhat shakes Rubi   
   belief that it can split integrals at will: splitting   
      
   integrate((1/(x - a) - b)*(1/sqrt(P)), x)   
      
   will give   
      
   integrate((1/((x - a)*sqrt(P)), x)   
      
   and   
      
   integrate(b/sqrt(P), x)   
      
   The second one always leads to elliptic integral F.   
   If the first one gives elliptic integral F + elementary   
   part than F parts can cancel.  But if you take easy   
   way of generating elliptic integral Pi, then it will   
   remain in final result, even for cases when sum is   
   elementary.   
      
   --   
                                 Waldek Hebisch   
      
   --- SoupGate-Win32 v1.05   
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