On Wed, Jan 25, 2023 at 4:29 PM Oscar Benjamin <oscar.j.benja...@gmail.com>
wrote:
> On Wed, 25 Jan 2023 at 21:15, emanuel.c...@gmail.com
> <emanuel.charpent...@gmail.com> wrote:
> >
> > There seems to be some funamental differences between Sympy's
> integration algorithm an other ones (say Sage, Maxima, Giac, Fricas or
> Mathematica). Sympy :
> >
> > ```
> > >>> from sympy import *
> > >>> x=symbols("x", positive=True)
> > >>> f=Lambda((x), (x**6+2)/(8*x**2))
> > >>> integrate(sqrt(f(x).diff(x)**2+1)*f(x))
> > (Integral(2*Abs(x**4 - x**2 + 1)/x**5, x) + Integral(2*Abs(x**4 - x**2 +
> 1)/x**3, x) + Integral(x*Abs(x**4 - x**2 + 1), x) + Integral(x**3*Abs(x**4
> - x**2 + 1), x))/16
> > ```
> >
> > Sage (either via Maxima's or Giac's integrators) :
> >
> > ```
> > sage: x=var("x", domain="positive")
> > sage: f(x)=(x^6+2)/(8*x^2)
> > sage: integrate((f(x).diff(x)^2+1).sqrt()*f(x),x)
> > 1/128*x^8 + 1/32*x^2 + 1/32*(2*x^6 - 1)/x^4
>
> Maybe it's because SymPy's integrate does not automatically factorise
> under the radical (I haven't checked the code to see if factorisation
> is attempted):
>
SymPy's integration algorithms that are currently implemented aren't
particularly good with algebraic functions. They do work sometimes, but
most of the time they don't. They are also quite sensitive to the exact
form an expression is written in, and a typical algebraic function can be
rewritten in multiple equivalent ways.
Aaron Meurer
>
> In [29]: integrand = sqrt(f(x).diff(x)**2+1)*f(x)
>
> In [30]: print(integrand)
> (x**6 + 2)*sqrt((3*x**3/4 - (x**6 + 2)/(4*x**3))**2 + 1)/(8*x**2)
>
> In [31]: print(integrand.factor())
> (x**2 + 1)*(x**6 + 2)*Abs(x**4 - x**2 + 1)/(16*x**5)
>
> In [32]: print(integrand.factor().integrate(x))
> x**8/128 + 3*x**2/32 - 1/(32*x**4)
>
> --
> Oscar
>
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