It is not Risch-complete, even in the purely transcendental
case (recursion problems in the logarithmic case). It is not
complete either in the purely algebraic case (see below),
and certainly not in the mixed algebraic-transcendental case.
Neither is Axiom (despite various rumors). The difference
is that Axiom issues an error message when an unimplemented
branch of the algorithm is hit, so an unevaluated integral
in Axiom is a proof that the integral is not elementary.
Neither is Mathematica, despite all the hype and ads
(source code is unavailable but "black box" experiments
show Mma's integrator to be in the, ahem, "low" category).
For the sceptics, here is an elementary integral of an
algebraic function, whose integral is a simple logarithm,
missed by the Risch integrators of both Maple and Mma:
g = x / Sqrt[x^4 + 10 x^2 - 96 x - 71]
f = -Log[(x^6+15 x^4-80 x^3+27 x^2-528 x+781) Sqrt[x^4+10 x^2-96 x-71]
- x^8 - 20 x^6 + 128 x^5 - 54 x^4 + 1408 x^3 - 3124 x^2 - 10001]/8
Simplify[D[f,x] - g]
Maple 5.5 and Maple 6:
g := x / sqrt(x^4 + 10*x^2 - 96*x - 71);
f := -log((x^6+15*x^4-80*x^3+27*x^2-528*x+781) *
- x^8 - 20*x^6 + 128*x^5 - 54*x^4 + 1408*x^3 - 3124*x^2 - 10001)/8;
For fairness, here is an easy one that Axiom cannot decide:
integrate(sqrt atan x,x)