On Sun, Apr 19, 2009 at 3:33 PM, root <d...@axiom-developer.org> 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
>
> A much shorter example is:
>
> integrate(sqrt(x+log(x)),x)
>
> to which Axiom replies:
>
> integrate: implementation incomplete (constant residues)
>
What is f(x) = sqrt(x+log(x)) supposed to be an example of? Does f
has an antiderivative that can be expressed in terms of elementary
functions? If so, what is it?
-- William
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