On Sun, Apr 19, 2009 at 3:25 PM, Ondrej Certik <ond...@certik.cz> wrote:
>
> 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,
How does sympy compute the integral of
(x^2 + 2*x + 1 + (3*x+1)*sqrt(x+log(x)))/(x*sqrt(x+log(x))*(x+sqrt(x+log(x))))?
William
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