gh-40574: Re-enable abs_integrate for Maxima integration
    
We used to have Maxima's `abs_integrate` package loaded by default, but
it caused more problems than it solved, and was disabled in
#12731. All known issues have
been resolved, however:

* https://sourceforge.net/p/maxima/code/ci/3ca4235b

This PR cleans up some existing tests, debt left over from when
`abs_integrate` was disabled, and then _re-enables_ it with a whole new
collection of tests for the old problems. We do not gain as much as we
used to (vanilla maxima has improved), but we do get some extra
integrals for "free" now that the problems appear to be resolved.
    
URL: #40574
Reported by: Michael Orlitzky
Reviewer(s): Dima Pasechnik

Author: Release Manager

build/pkgs/maxima/SPKG.rst

@@ -47,24 +47,5 @@ Special Update/Build Instructions
 All patch files in the patches/ directory are applied. Descriptions of
 these patches are either in the patch files themselves or below.
 
--  0001-taylor2-Avoid-blowing-the-stack-when-diff-expand-isn.patch:
-   Fix for Maxima bug #2520 (abs_integrate fails on abs(sin(x)) and
-   abs(cos(x))). Introduced in Issue #13364 (Upgrade Maxima to
-   5.29.1).
-
--  build-fasl.patch: Build a fasl library for ecl in addition to an
-   executable program. Introduced in Issue #16178 (Build maxima fasl
-   without asdf).
-
 -  infodir.patch: Correct the path to the Info directory. Introduced
    in Issue #11348 (maxima test fails when install tree is moved).
-
--  matrixexp.patch: Fix matrixexp(matrix([%i*%pi])), which broke after
-   Maxima 5.29.1. Introduced in Issue #13973.
-
--  maxima.system.patch: Set ``c::*compile-in-constants*`` to t.
-   Introduced in Issue #11966 (OS X 10.7 Lion: Maxima fails to build).
-
--  undoing_true_false_printing_patch.patch: Revert an upstream change
-   causing '?' to be printed around some words. Introduced in Trac
-   #13364 (Upgrade Maxima to 5.29.1).

src/sage/calculus/desolvers.py

@@ -541,7 +541,7 @@ def desolve(de, dvar, ics=None, ivar=None, show_method=False, contrib_ode=False,
         sage: forget()
         sage: y = function('y')(x)
         sage: desolve(diff(y, x) == sqrt(abs(y)), dvar=y, ivar=x)
-        integrate(1/sqrt(abs(y(x))), y(x)) == _C + x
+        sqrt(-y(x))*(sgn(y(x)) - 1) + (sgn(y(x)) + 1)*sqrt(y(x)) == _C + x
 
     AUTHORS:
 

src/sage/functions/piecewise.py

@@ -799,16 +799,21 @@ def integral(self, parameters, variable, x=None, a=None, b=None, definite=False,
                           y|-->1/2*y^2 + 3*y + 9/2 on (-3, 0),
                           y|-->3*y + 9/2 on (0, +oo); y)
 
-            ::
+            The output from this can change a bit depending on the
+            version of Maxima used, so we test for equality with a
+            known result on the sole piece::
 
+                sage: # long time
                 sage: f1(x) = e^(-abs(x))
                 sage: f = piecewise([[(-infinity, infinity), f1]])
                 sage: result = f.integral(definite=True)
                 ...
                 sage: result
                 2
-                sage: f.integral()
-                piecewise(x|-->-integrate(e^(-abs(x)), x, x, +Infinity) on (-oo, +oo); x)
+                sage: actual = f.integral().expression_at(0) # only one piece
+                sage: expected = -1/2*e^(-x)*sgn(x) - 1/2*e^x*sgn(x) - 1/2*e^(-x) + 1/2*e^x + sgn(x) - 1
+                sage: bool(actual == expected)
+                True
 
             ::
 

src/sage/interfaces/maxima_lib.py

@@ -218,7 +218,7 @@
 
 init_code = ['besselexpand : true', 'display2d : false', 'domain : complex', 'keepfloat : true',
              'load(to_poly_solve)', 'load(simplify_sum)',
-             'load(diag)']
+             'load(diag)', 'load(abs_integrate)']
 
 
 # Turn off the prompt labels, since computing them *very
@@ -726,58 +726,21 @@ def sr_integral(self, *args):
             sage: assumptions()  # Check the assumptions really were forgotten
             []
 
-        Make sure the abs_integrate package is being used,
-        :issue:`11483`. The following are examples from the Maxima
-        abs_integrate documentation::
+        An example from the maxima documentation involving the
+        absolute value::
 
-            sage: integrate(abs(x), x)
-            1/2*x*abs(x)
-
-        ::
-
-            sage: integrate(sgn(x) - sgn(1-x), x)  # known bug
+            sage: integrate(sgn(x) - sgn(1-x), x)
             abs(x - 1) + abs(x)
 
-        This is a known bug in Sage symbolic limits code, see
+        This is a fixed bug in Sage symbolic limits code, see
         :issue:`17892` and https://sourceforge.net/p/maxima/bugs/3237/ ::
 
-            sage: integrate(1 / (1 + abs(x-5)), x, -5, 6) # not tested -- known bug
+            sage: integrate(1 / (1 + abs(x-5)), x, -5, 6)
             log(11) + log(2)
 
-        ::
-
-            sage: integrate(1/(1 + abs(x)), x)  # known bug
-            1/2*(log(x + 1) + log(-x + 1))*sgn(x) + 1/2*log(x + 1) - 1/2*log(-x + 1)
-
-        ::
-
-            sage: integrate(cos(x + abs(x)), x)  # known bug
-            -1/2*x*sgn(x) + 1/4*(sgn(x) + 1)*sin(2*x) + 1/2*x
-
-        The last example relies on the following simplification::
-
-            sage: maxima("realpart(signum(x))")
-            signum(x)
-
-        An example from sage-support thread e641001f8b8d1129::
-
-            sage: f = e^(-x^2/2)/sqrt(2*pi) * sgn(x-1)
-            sage: integrate(f, x, -Infinity, Infinity)  # known bug
-            -erf(1/2*sqrt(2))
-
-        From :issue:`8624`::
-
-            sage: integral(abs(cos(x))*sin(x),(x,pi/2,pi))
-            1/2
-
-        ::
-
-            sage: integrate(sqrt(x + sqrt(x)), x).canonicalize_radical()  # known bug
-            1/12*((8*x - 3)*x^(1/4) + 2*x^(3/4))*sqrt(sqrt(x) + 1) + 1/8*log(sqrt(sqrt(x) + 1) + x^(1/4)) - 1/8*log(sqrt(sqrt(x) + 1) - x^(1/4))
-
         And :issue:`11594`::
 
-            sage: integrate(abs(x^2 - 1), x, -2, 2)  # known bug
+            sage: integrate(abs(x^2 - 1), x, -2, 2)
             4
 
         This definite integral returned zero (incorrectly) in at least
@@ -795,6 +758,45 @@ def sr_integral(self, *args):
             0.124756040961038
             sage: a.imag().abs() < 3e-17
             True
+
+        The following examples require Maxima's ``abs_integrate``
+        package. Enabling ``abs_integrate`` globally caused several
+        bugs (catalogued in :issue:`12731`) but most of these have
+        been fixed, and ``abs_integrate`` has been re-enabled::
+
+            sage: integrate(1/(abs(x) + 1), x, algorithm="maxima")
+            1/2*(log(x + 1) + log(-x + 1))*sgn(x) + 1/2*log(x + 1) - 1/2*log(-x + 1)
+            sage: integrate(cos(x + abs(x)), x, algorithm="maxima")
+            -1/4*(2*x - sin(2*x))*sgn(x) + 1/2*x + 1/4*sin(2*x)
+
+        Several examples where ``abs_integrate`` previously lead to
+        incorrect results. This was once reported to be divergent in
+        :issue:`13733`, and only in maxima-5.48 does it give the
+        correct answer::
+
+            sage: # long time, not tested until maxima-5.48 is widespread
+            sage: integral(log(cot(x)-1), x, 0, pi/4, algorithm="maxima")
+            catalan + 1/2*I*dilog(1/2*I + 1/2) - 1/2*I*dilog(-1/2*I + 1/2)
+
+        This used to return ``1/2`` in :issue:`11590`::
+
+            sage: integrate(x * sgn(x^2 - 1/4), x, -1, 0, algorithm="maxima")
+            -1/4
+
+        In :issue:`14591`, this incorrectly simplified to ``cosh(x)``::
+
+            sage: integrate(sqrt(1-1/4*cosh(x)^2), x, algorithm="maxima")
+            integrate(sqrt(-1/4*cosh(x)^2 + 1), x)
+
+        In :issue:`17468`, this integral hangs::
+
+            sage: integral(log(abs(2*sin(x))), x, 0, pi/3, algorithm="maxima")
+            1/36*I*pi^2 + I*dilog(1/2*I*sqrt(3) + 1/2) + I*dilog(-1/2*I*sqrt(3) - 1/2)
+
+        This used to return a *negative* answer in :issue:`17511`::
+
+            sage: integrate(abs(cos(x)), x, 0, pi, algorithm="maxima")
+            2
         """
         try:
             return max_to_sr(maxima_eval(([max_integrate],