Use the faster libgiac interface for "giac" integration

[orlitzky]

While working on #38668, I noticed that all of our examples that use giac for integration do so with algorithm='giac'. This uses the pexpect interface that is much slower (and simply sloppier) than the library interface that would be used with algorithm='libgiac'.

To fix this, I could just change those examples to algorithm='libgiac', but since the libgiac interface is better in every way, that seems kind of rude to me. Most users aren't going to know what lib-anything is, and if they just want to use giac, they're going to try algorithm='giac'. So instead this PR makes algorithm='giac' do the right thing, and use the better interface. I've left the algorithm='libgiac' alone for now to avoid breaking any code.

Afterwards I have removed the pexpect giac integrator entirely, because there's no reason to use it, and no other code in sagelib uses it.

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[orlitzky]

AFAICS the CI failure is an unrelated timeout:

2024-09-20T15:19:09.1450225Z ----------------------------------------------------------------------
2024-09-20T15:19:09.1451853Z sage -t --long --random-seed=286735480429121101562228604801325644303 src/sage/data_structures/bounded_integer_sequences.pyx  # Timed out
2024-09-20T15:19:09.1453353Z ----------------------------------------------------------------------

[mkoeppe]

This looks like a good idea, thanks.

[orlitzky]

I didn't see the label change, thanks for the review.

[vbraun]

I'm getting

sage -t --long --warn-long 45.2 --random-seed=123 src/sage/symbolic/integration/integral.py
**********************************************************************
File "src/sage/symbolic/integration/integral.py", line 64, in sage.symbolic.integration.integral.IndefiniteIntegral.__init__
Failed example:
    integrate(Ex, x, algorithm='giac')  # long time
Expected:
    4*(-2*x^(1/3) + 1)^(3/4) + 6*arctan((-2*x^(1/3) + 1)^(1/4)) - 3*log((-2*x^(1/3) + 1)^(1/4) + 1) + 3*log(abs((-2*x^(1/3) + 1)^(1/4) - 1))
Got:
    4*(-2*x^(1/3) + 1)^(3/4) + 6*arctan((-2*x^(1/3) + 1)^(1/4)) - 3*log(abs((-2*x^(1/3) + 1)^(1/4) + 1)) + 3*log(abs((-2*x^(1/3) + 1)^(1/4) - 1))
**********************************************************************
File "src/sage/symbolic/integration/integral.py", line 209, in sage.symbolic.integration.integral.DefiniteIntegral.__init__
Failed example:
    integral(1/max_symbolic(x, 1)**2, x, 0, oo, algorithm='giac')
Exception raised:
    Traceback (most recent call last):
      File "/home/release/Sage/src/sage/doctest/forker.py", line 714, in _run
        self.compile_and_execute(example, compiler, test.globs)
      File "/home/release/Sage/src/sage/doctest/forker.py", line 1144, in compile_and_execute
        exec(compiled, globs)
      File "<doctest sage.symbolic.integration.integral.DefiniteIntegral.__init__[4]>", line 1, in <module>
        integral(Integer(1)/max_symbolic(x, Integer(1))**Integer(2), x, Integer(0), oo, algorithm='giac')
      File "/home/release/Sage/src/sage/misc/functional.py", line 785, in integral
        return x.integral(*args, **kwds)
               ^^^^^^^^^^^^^^^^^^^^^^^^^
      File "sage/symbolic/expression.pyx", line 13222, in sage.symbolic.expression.Expression.integral
        return integral(self, *args, **kwds)
      File "/home/release/Sage/src/sage/symbolic/integration/integral.py", line 1073, in integrate
        return integrator(expression, v, a, b)
               ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
      File "/home/release/Sage/src/sage/symbolic/integration/external.py", line 259, in libgiac_integrator
        result = libgiac.integrate(Pygen(expression), v, a, b)
                 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
      File "sage/libs/giac/giac.pyx", line 1934, in sage.libs.giac.giac.GiacFunction.__call__
        return Pygen.__call__(self, *args)
      File "sage/libs/giac/giac.pyx", line 1109, in sage.libs.giac.giac.Pygen.__call__
        raise
      File "sage/libs/giac/giac.pyx", line 1099, in sage.libs.giac.giac.Pygen.__call__
        result = self.gptr[0](right.gptr[0], context_ptr)
    RuntimeError: Unable to check sign: 1>Infinity
**********************************************************************
File "src/sage/symbolic/integration/integral.py", line 697, in sage.symbolic.integration.integral.integrate
Failed example:
    integrate(abs(cos(x)), x, 0, 2*pi, algorithm='giac')
Expected:
    4
Got:
    Warning, integration of abs or sign assumes constant sign by intervals (correct if the argument is real):
    Check [abs(cos(sageVARx))]
    Discontinuities at zeroes of cos(sageVARx) were not checked
    4
**********************************************************************

[orlitzky]

Expected:
4*(-2x^(1/3) + 1)^(3/4) + 6arctan((-2x^(1/3) + 1)^(1/4)) - 3log((-2x^(1/3) + 1)^(1/4) + 1) + 3log(abs((-2x^(1/3) + 1)^(1/4) - 1))
Got:
4(-2x^(1/3) + 1)^(3/4) + 6arctan((-2x^(1/3) + 1)^(1/4)) - 3log(abs((-2x^(1/3) + 1)^(1/4) + 1)) + 3log(abs((-2*x^(1/3) + 1)^(1/4) - 1))

These are the same anwer. I updated the expected output.

Expected:
4
Got:
Warning, integration of abs or sign assumes constant sign by intervals (correct if the argument is real):
Check [abs(cos(sageVARx))]
Discontinuities at zeroes of cos(sageVARx) were not checked
4

Nothing too scary here. This doctest already looks for the warning output on the previous line, where integrate is used without an algorithm argument. I tweaked the description a bit and added some ....

File "src/sage/symbolic/integration/integral.py", line 209, in sage.symbolic.integration.integral.DefiniteIntegral.init
Failed example:
integral(1/max_symbolic(x, 1)**2, x, 0, oo, algorithm='giac')
...
RuntimeError: Unable to check sign: 1>Infinity

This one on the other hand is a treat. It's reproducible using libgiac_integrator, but not through integrate(), which (if maxima fails) calls libgiac_integrator with exactly the same arguments. My guess is hidden state within sage.libs.giac.

In any case, I was able to work around it. Don't ask me what the difference is, but giac likes +infinity better than Infinity. You get the latter passing Sage's infinity directly to libgiac, and the former by calling _giac_() on Sage's infinity. So I added a few lines to the integrator to do that when the endpoints are plus/minus infinity.