> > Wikipedia also has a few interesting remarks, e.g., that the Risch
> > algorithm isn't an algorithm, because it depends on being able to
> > check equality of general elementary functions, which is evidently an
> > open problem in general (so in practice you just fake it by evaluating
> > numerically at lots of points to decide if something is probably equal
> > to something else). It's also evidently not implemented anywhere,
> > e.g., a nice example on the Wikipedia page, is that if you let
> >
> > f = (x^2 + 2*x + 1 +
> > (3*x+1)*sqrt(x+log(x)))/(x*sqrt(x+log(x))*(x+sqrt(x+log(x))))
> >
> > then it has the antiderivative
> >
> > g = 2*(sqrt(x+log(x)) + log(x+sqrt(x+log(x))))
> >
> > since
> >
> > sage: h = g.derivative() - f
> > sage: h.full_simplify()
> > 0
> >
> > However, Sage, Maple, and Mathematica, all simply give "integral(f)"
> > back when asked to integrate f. (I just checked this with the latest
> > versions.)
>
> Curiously though, SymPy knows this particular integral.
>
> >>> from sympy import *
> >>> x=Symbol('x')
> >>> f=(x**2+2*x+1+(3*x+1)*sqrt(x+log(x)))/(x*sqrt(x+log(x))*(x+sqrt(x+log(x))))
> >>> integrate(f,x)
> 2*log(x + (x + log(x))**(1/2)) + 2*(x + log(x))**(1/2)
>
> Fredrik
A much shorter example is:
integrate(sqrt(x+log(x)),x)
to which Axiom replies:
integrate: implementation incomplete (constant residues)
For further information, Bronstein deals with constant residues
(written after his Axiom work) in his book [1] after page 147.
Tim Daly
[1] Bronstein, Manuel "Symbolic Integration I", Second Edition
Springer ISBN 3-540-21493-3
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