Hi,
On Wed, Dec 04, 2019 at 07:38:34AM -0800, Eric Gourgoulhon wrote:
> Hi,
>
> In Sage 9.0.beta8 we have
>
> sage: a = var('a')
> sage: integrate(1/(x^4 + x^2 + a), x)
> ...
> AttributeError: 'RootSum' object has no attribute '_sage_'
>
> The same error occurs in Sage 8.9, but not in Sage 8.8 (and below). In Sage
> 8.8, we have instead:
>
> sage: a = var('a')
> sage: integrate(1/(x^4 + x^2 + a), x)
> integrate(1/(x^4 + x^2 + a), x)
>
> Note that RootSum is a SymPy object. Actually, in Sage 9.0.beta8, forcing
> the algorithm to 'maxima' yields the same result as in Sage 8.8:
>
> sage: integrate(1/(x^4 + x^2 + a), x, algorithm='maxima')
> integrate(1/(x^4 + x^2 + a), x)
>
> So it seems that since Sage 8.9, when integrate() is not capable to find an
> answer via Maxima, it tries SymPy but is not capable to translate the
> result back to Sage. I could not find a ticket about this. Shall I open one?
For what it worth, the change was done at
https://trac.sagemath.org/ticket/27958
Ciao,
Thierry
> Eric.
>
> PS: for the record, a primitive of 1/(x^4 + x^2 + a) is
>
> sage: b = sqrt(1 - 4*a)
> sage: f = sqrt(2)/b*(arctan(sqrt(2)*x/sqrt(1 - b))/sqrt(1 - b) -
> arctan(sqrt(2)*x/sqrt(1 + b))/sqrt(1 + b))
>
> as we can check:
>
> sage: diff(f, x).simplify_full()
> 1/(x^4 + x^2 + a)
>
>
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