> Could you enlarge on the Risch algorithm? I always assumed that Axiom
> had a complete implementation, but if not I'd want to be put right.
First, according to my research, the theory of integration, aka Risch
algorithm,
is still not complete.
Although Risch published his "algorithm" around 1970, it's very inefficient.
Bernoulli gives "algorithm" for rational function integration in 1703, but
the
efficient algorithm is given by Lazard, Rioboo and Trager around 1990.
Trager gives a better ("rational") algorithm for pure algebraic function
integration in 1984. Bronstein "extends the major techniques of Trager
to handle elementary functions. If all the algebraic extensions are n th
root adjunctions, then the algorithm is 'rational'. Otherwise, we use a
simplified version of Risch's (1968) technique.", quoted from "Integration
of Elementary Functions, 1990".
So I think the "not all the algebraic extensions are n th root adjunctions"
case still doesn't have an efficient "rational" algorithm.
Second, in FriCAS code, in intalg.spad, there are four cases failure of
"implementation incomplete": constant residues, irrational residues,
residue poly has multiple non-linear factors, has polynomial part.
Finally, there are lots of examples FriCAS can not handle and give
"implementation incomplete" error message, of course, these examples
all belong to the mixed algebraic-transcendental case:
integrate(sqrt atan x,x) -- example from Bronstein
-- examples from Symbolic Integration Tutorial by Bronstein
((log x)^2+2*x*log(x)+x^2+(x+1)*sqrt(x+log(x)))/(x*(log
x)^2+2*x^2*log(x)+x^3)
((x^2+2*x+1)*sqrt(x+log(x))+(3*x+1)*log(x)+3*x^2+x)/((x*log(x)+x^2)*sqrt(x+log(x))+x^2*log(x)+x^3)
(3*(x+exp x)^(1/3)+(2*x^2+3*x)*exp(x)+5*x^2)/(x*(x+exp x)^(1/3))
And there are more of this kind -- mixed algebraic-transcendental functions.
Please correct me if I'm wrong.
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