clicliclic@freenet.de wrote:   
   >   
   > Albert Rich schrieb:   
   > >   
   > > On Tuesday, January 23, 2018 at 4:40:12 AM UTC-10, anti...@m   
   th.uni.wroc.pl wrote:   
   > > >   
   > > > I do not see Albert's message, only this reply so here I am   
   > > > answering to Albert.   
   > > >   
   > > > AFAICS x/((x^3+9)*sqrt(x^3-1)) has no elementary antiderivative.   
   > > > I am surprised that you go that way.  Examples that Martin gave   
   > > > are essentially diofantine phenomana depending on arithmetic   
   > > > of elliptic curves.  ATM Rubi contains rules for handful   
   > > > of curves.  But there are infinitely many essentially different   
   > > > elliptic curves and tables giving "simple" ones contain   
   > > > thousands of entries (google Cremona).  Each such curve leads   
   > > > to its own family of pseudoelliptics.  You may be able to   
   > > > find some patterns, but apparently number theorists believe   
   > > > that core behaviour is essentially random.  So this task   
   > > > is like table of prime numbers, only entries have much   
   > > > more complicated nature.   
   > > >   
   > >   
   > > Sorry, I made a typo.  I meant to type x/((x^3+8)*sqrt(x^3-1)), the   
   > > integrand of Martin's second pseudo-elliptic integral example.   
   > >   
   > > As far as the density of pseudo-elliptic integrals, Martin is better   
   > > equipped to address that question than I.  However, I would bet that   
   > > only a finite number of rules are required to find elementary   
   > > antiderivatives for pseudo-elliptic integrands of the form   
   > >   
   > >     x^m * (a+b*x^3)^n * (c+d*x^3)^(p/2)   
   > >   
   > > where m, n and p are integers.   
   > >   
   > > Note that rather than finding optimal antiderivatives for all   
   > > possible expressions, the more modest goal of Rubi is to find such   
   > > antiderivatives for all elements of well-defined classes, like the   
   > > form given above.   
   > >   
      
   _Generic_ c and d in (c+d*x^3)^(p/2) give you _single elliptic curve.   
   Let me recall theory: integrals as above can be reduced to sum   
   of three different kinds:   
      
   (c+d*x^3)^(1/2)   
   x*(c+d*x^3)^(1/2)   
   (x - a)^(-1)*(c+d*x^3)^(1/2)   
      
   Note that rules present in Rubi are doing this, so there is no   
   point at looking triples different than (m, n, p) = (-1, -1, -1).   
      
   First forms above (first and second kind) are no problem: single pattern   
   for each.  Problem is with third kind.  First, to have any chance   
   of elementary integral one have to subract apropriate multiple   
   of integral of first kind.  Second, corrected single term is   
   elementary if (a, (c + d*a^3)^(1/2)) is a point of finite order   
   on elliptic curve.  Third, modulo derivatives of elementary functions   
   multiple terms of form   
      
   (c + d*a_i)^(1/2)*(x - a_i)^(-1)*(c+d*x^3)^(1/2)   
      
   can be reduced to single such term.  So if you split rational   
   part into partial fractions you are likely to loose elementary   
   integrability.   
      
   Generic integral doable by single logarithm is of form   
      
   f = D((a*log((p + q*del)/(p - q*del)), x)   
      
   where a is a constant, p and q are polynomials and del = sqrt(P)   
   where P is polynomial giving elliptic curve.   
      
   f can be written as R/del where R is rational function.  Now,   
   the first condition is that denominator d of R is of form   
      
   d = p^2 - q^2*P   
      
   When deg(P) = 3 the simples possible case is deg(d) = 3 which   
   leads to constant q and linear p.  So in this case checking   
   that d is of requested form is reasonably easy and gives you   
   one integrability condition.  Once you know p and q you can   
   compute numerator of R up to multiplicative constant.  In   
   particular for integrability your integrand must agree with the   
   computed form, so you get extra 3 integrbility conditions.   
   In principle one can give explicit formulas for all steps   
   and turn this into a rule or set of rules.   
      
   However, many practical examples are of nongeneric type,   
   where there is cancelation between numrator and denominator   
   of R.  As I wrote for "1/(x -a)" term you need torsion   
   points on elliptic curve.  Martin has posted formulas for   
   multiples a point on elliptic curve and for each k one can use   
   such formula to verify if "a" above corresponds to a torsion   
   point of order k.  But to integrate you also need to find p and   
   q above and that is more work.  Several examples include   
   multiple logaritms (pairs, triples, ...) and then you need   
   to determine residues and then check if each corresponding   
   points (divisors) is a tersion point.   
      
   Even for _very_ simple integrals corresponding rules are   
   likely to be incomplete: for curves defined over rationals   
   you may get points of order 12 which would imply rather   
   hairy rule.  In practice square roots can appear for   
   various reasons and than instead of 12 you need to consider   
   higher bound.   
      
   > 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.   
      
   --   
                                 Waldek Hebisch   
      
   --- SoupGate-Win32 v1.05   
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