On Sun, Apr 19, 2009 at 1:23 PM, Fredrik Johansson
<fredrik.johans...@gmail.com> wrote:
>
> 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)
I was just about to point it out. It even works in Sage:
In [2]: f = (x^2 + 2*x + 1 +
...: (3*x+1)*sqrt(x+log(x)))/(x*sqrt(x+log(x))*(x+sqrt(x+log(x))))
In [5]: import sympy
In [6]: sympy.integrate(f, x)
Out[6]: 2*log(x + (x + log(x))**(1/2)) + 2*(x + log(x))**(1/2)
So we have a good start to implement the Risch algorithm in sympy already.
Ondrej
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