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):
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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