31873: replace giac by libgiac for integration

This also fixes a misplaced sig_on() call that otherwise leads to a
doctest failure in interfaces/sympy.py.

src/sage/libs/giac/giac.pyx

@@ -898,12 +898,12 @@ cdef class Pygen(GiacMethods_base):
 
          # Other types are converted with strings.
          else:
-           sig_on()
            if isinstance(s,Expression):
               # take account of conversions with key giac in the sage symbol dict
               s=SRexpressiontoGiac(s)
            if not(isinstance(s,str)):  #modif python3
              s=s.__str__()
+           sig_on()
            self.gptr = new gen(<string>encstring23(s),context_ptr)
            sig_off()
 

src/sage/symbolic/expression.pyx

@@ -874,7 +874,9 @@ cdef class Expression(CommutativeRingElement):
         EXAMPLES::
 
             sage: i = var('i')
-            sage: integral(exp(x + x^2)/(x+1), x)._sympy_character_art(False)
+            sage: f = integral(exp(x + x^2)/(x+1), x)
+            ...
+            sage: f._sympy_character_art(False)
             '  /          \n |           \n |   2       \n |  x  + x   \n | e...'
         """
         from sympy import pretty, sympify
@@ -11272,9 +11274,13 @@ cdef class Expression(CommutativeRingElement):
             integrate(f(x) + g(x), x)
             sage: integrate(f(x)+g(x),x).distribute()
             integrate(f(x), x) + integrate(g(x), x)
-            sage: integrate(f(x)+g(x),x,a,b)
+            sage: result = integrate(f(x)+g(x),x,a,b)
+            ...
+            sage: result
             integrate(f(x) + g(x), x, a, b)
-            sage: integrate(f(x)+g(x),x,a,b).distribute()
+            sage: result = integrate(f(x)+g(x),x,a,b).distribute()
+            ...
+            sage: result
             integrate(f(x), x, a, b) + integrate(g(x), x, a, b)
             sage: sum(X(j)+sum(Y(k)+Z(k),k,1,q),j,1,p)
             sum(X(j) + sum(Y(k) + Z(k), k, 1, q), j, 1, p)

src/sage/symbolic/integration/external.py

@@ -446,3 +446,48 @@ def giac_integrator(expression, v, a=None, b=None):
         return expression.integrate(v, a, b, hold=True)
     else:
         return result._sage_()
+
+def libgiac_integrator(expression, v, a=None, b=None):
+    r"""
+    Integration using libgiac
+
+    EXAMPLES::
+
+        sage: import sage.libs.giac
+        ...
+        sage: from sage.symbolic.integration.external import libgiac_integrator
+        sage: libgiac_integrator(sin(x), x)
+        -cos(x)
+        sage: libgiac_integrator(1/(x^2+6), x, -oo, oo)
+        No checks were made for singular points of antiderivative...
+        1/6*sqrt(6)*pi
+
+    TESTS::
+
+        sage: libgiac_integrator(e^(-x^2)*log(x), x)
+        integrate(e^(-x^2)*log(x), x)
+
+    The following integral fails with the Giac Pexpect interface, but works
+    with libgiac (:trac:`31873`)::
+
+        sage: a, x = var('a,x')
+        sage: f = sec(2*a*x)
+        sage: F = libgiac_integrator(f, x)
+        ...
+        sage: (F.derivative(x) - f).simplify_trig()
+        0
+    """
+    from sage.libs.giac import libgiac
+    from sage.libs.giac.giac import Pygen
+    # We call Pygen on first argument because otherwise some expressions
+    # involving derivatives result in doctest failures in interfaces/sympy.py
+    # -- related to the fixme note in sage.libs.giac.giac.GiacFunction.__call__
+    # regarding conversion of lists.
+    if a is None:
+        result = libgiac.integrate(Pygen(expression), v)
+    else:
+        result = libgiac.integrate(Pygen(expression), v, a, b)
+    if 'integrate' in format(result) or 'integration' in format(result):
+        return expression.integrate(v, a, b, hold=True)
+    else:
+        return result.sage()

src/sage/symbolic/integration/integral.py

@@ -28,6 +28,7 @@
 available_integrators['mathematica_free'] = external.mma_free_integrator
 available_integrators['fricas'] = external.fricas_integrator
 available_integrators['giac'] = external.giac_integrator
+available_integrators['libgiac'] = external.libgiac_integrator
 
 ######################################################
 #
@@ -71,15 +72,17 @@ def __init__(self):
 
         Check for :trac:`25119`::
 
-            sage: integrate(sqrt(x^2)/x,x)
+            sage: result = integrate(sqrt(x^2)/x,x)
+            ...
+            sage: result
             x*sgn(x)
         """
         # The automatic evaluation routine will try these integrators
         # in the given order. This is an attribute of the class instead of
         # a global variable in this module to enable customization by
         # creating a subclasses which define a different set of integrators
         self.integrators = [external.maxima_integrator,
-                            external.giac_integrator,
+                            external.libgiac_integrator,
                             external.sympy_integrator]
 
         BuiltinFunction.__init__(self, "integrate", nargs=2, conversions={'sympy': 'Integral',
@@ -189,7 +192,7 @@ def __init__(self):
         # a global variable in this module to enable customization by
         # creating a subclasses which define a different set of integrators
         self.integrators = [external.maxima_integrator,
-                            external.giac_integrator,
+                            external.libgiac_integrator,
                             external.sympy_integrator]
 
         BuiltinFunction.__init__(self, "integrate", nargs=4, conversions={'sympy': 'Integral',
@@ -268,6 +271,7 @@ def _tderivative_(self, f, x, a, b, diff_param=None):
             sage: from sage.symbolic.integration.integral import definite_integral
             sage: f = function('f'); a,b=var('a,b')
             sage: h = definite_integral(f(x), x,a,b)
+            ...
             sage: h.diff(x) # indirect doctest
             0
             sage: h.diff(a)
@@ -425,9 +429,9 @@ def integrate(expression, v=None, a=None, b=None, algorithm=None, hold=False):
 
     - ``b`` - (optional) upper endpoint of definite integral
 
-    - ``algorithm`` - (default: 'maxima') one of
+    - ``algorithm`` - (default: 'maxima', 'libgiac' and 'sympy') one of
 
-       - 'maxima' - use maxima (the default)
+       - 'maxima' - use maxima
 
        - 'sympy' - use sympy (also in Sage)
 
@@ -437,6 +441,8 @@ def integrate(expression, v=None, a=None, b=None, algorithm=None, hold=False):
 
        - 'giac' - use Giac
 
+       - 'libgiac' - use libgiac
+
     To prevent automatic evaluation use the ``hold`` argument.
 
     .. SEEALSO::
@@ -623,7 +629,9 @@ def integrate(expression, v=None, a=None, b=None, algorithm=None, hold=False):
 
     where the default integrator obtains another answer::
 
-        sage: integrate(f(x), x)
+        sage: result = integrate(f(x), x)
+        ...
+        sage: result
         1/8*sqrt(x)*gamma(1/4)*gamma(-1/4)^2*hypergeometric((-1/4, -1/4, 1/4), (1/2, 3/4), -1/x^2)/(pi*gamma(3/4))
 
     The following definite integral is not found by maxima::
@@ -648,7 +656,9 @@ def integrate(expression, v=None, a=None, b=None, algorithm=None, hold=False):
 
         sage: integrate(abs(cos(x)), x, 0, 2*pi, algorithm='giac')
         4
-        sage: integrate(abs(cos(x)), x, 0, 2*pi)
+        sage: result = integrate(abs(cos(x)), x, 0, 2*pi)
+        ...
+        sage: result
         4
 
     ALIASES: integral() and integrate() are the same.
@@ -795,7 +805,9 @@ def integrate(expression, v=None, a=None, b=None, algorithm=None, hold=False):
     the previous (wrong) answer of zero. See :trac:`10914`::
 
         sage: f = abs(sin(x))
-        sage: integrate(f, x, 0, 2*pi)
+        sage: result = integrate(f, x, 0, 2*pi)
+        ...
+        sage: result
         4
 
     Another incorrect integral fixed upstream in Maxima, from
@@ -891,42 +903,63 @@ def integrate(expression, v=None, a=None, b=None, algorithm=None, hold=False):
 
     Some integrals are now working (:trac:`27958`, using giac or sympy)::
 
-        sage: integrate(1/(1 + abs(x)), x)
+        sage: result = integrate(1/(1 + abs(x)), x)
+        ...
+        sage: result
         log(abs(x*sgn(x) + 1))/sgn(x)
 
-        sage: integrate(cos(x + abs(x)), x)
+        sage: result = integrate(cos(x + abs(x)), x)
+        ...
+        sage: result
         sin(x*sgn(x) + x)/(sgn(x) + 1)
 
-        sage: integrate(abs(x^2 - 1), x, -2, 2)
+        sage: result = integrate(abs(x^2 - 1), x, -2, 2)
+        ...
+        sage: result
         4
 
         sage: f = sqrt(x + 1/x^2)
         sage: actual = integrate(f, x)
+        ...
         sage: expected = (1/3*(2*sqrt(x^3 + 1) - log(sqrt(x^3 + 1) + 1)
         ....:             + log(abs(sqrt(x^3 + 1) - 1)))*sgn(x))
         sage: bool(actual == expected)
         True
 
         sage: g = abs(sin(x)*cos(x))
-        sage: g.integrate(x, 0, 2*pi)
+        sage: result = g.integrate(x, 0, 2*pi)
+        ...
+        sage: result
         2
 
-        sage: integrate(1/sqrt(abs(x)), x)
+        sage: result = integrate(1/sqrt(abs(x)), x)
+        ...
+        sage: result
         2*sqrt(x*sgn(x))/sgn(x)
 
-        sage: integrate(sgn(x) - sgn(1-x), x)
+        sage: result = integrate(sgn(x) - sgn(1-x), x)
+        ...
+        sage: result
         x*(sgn(x) - sgn(-x + 1)) + sgn(-x + 1)
 
-        sage: integrate(1 / (1 + abs(x-5)), x, -5, 6)
+        sage: result = integrate(1 / (1 + abs(x-5)), x, -5, 6)
+        ...
+        sage: result
         log(11) + log(2)
 
-        sage: integrate(1/(1 + abs(x)), x)
+        sage: result = integrate(1/(1 + abs(x)), x)
+        ...
+        sage: result
         log(abs(x*sgn(x) + 1))/sgn(x)
 
-        sage: integrate(cos(x + abs(x)), x)
+        sage: result = integrate(cos(x + abs(x)), x)
+        ...
+        sage: result
         sin(x*sgn(x) + x)/(sgn(x) + 1)
 
-        sage: integrate(abs(x^2 - 1), x, -2, 2)
+        sage: result = integrate(abs(x^2 - 1), x, -2, 2)
+        ...
+        sage: result
         4
 
     Some tests for :trac:`17468`::