rootsof/RootSum support in calculus

[dimpase]

Problem Description

There are symbolic calculations which involve summing over roots of a univariate polynomial, e.g.

$ maxima
[...]
(%i1) integrate(1/(x^3+a*x+1),x);
                               ⌠
                               ⎮      1
(%o1)                          ⎮ ──────────── dx
                               ⎮  3
                               ⌡ x  + a x + 1
(%i2) integrate_use_rootsof:true;
(%o2)                                true
(%i3)  integrate(1/(x^3+a*x+1),x);
              ____
              ╲                                     log(x - %r1)
(%o3)          ⟩                                    ────────────
              ╱                                               2
              ‾‾‾‾                                   a + 3 %r1
                                        3
              %r1 in rootsof(%r1 a + %r1  + 1, %r1)
(%i4) 

this example is taken from https://people.eecs.berkeley.edu/~fateman/papers/rootsum.pdf
-- where also a case for improving Maxima in this sense is advanced.

Can we have such a capability in Sage's calculus? Maxima is not the only system which supports this,
also FriCAS has RootSum which is the same thing more or less. Movever, FriCAS often returns results in this format. (For the integral above it returns an explicit sum using the ability to write roots of the cubics explicitly).

Proposed Solution

provide an implementation in Sage

Alternatives Considered

n/a

Additional Information

No response

Is there an existing issue for this?

• I have searched the existing issues for a bug report that matches the one I want to file, without success.

[dimpase]

@nbruin @mantepse