On Sun, Apr 19, 2009 at 10:14 PM, William Stein <wst...@gmail.com> wrote:
> 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
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