diff --git a/src/sage/calculus/calculus.py b/src/sage/calculus/calculus.py
index ef4854db676..64881aba812 100644
--- a/src/sage/calculus/calculus.py
+++ b/src/sage/calculus/calculus.py
@@ -387,6 +387,7 @@
 Ensure that :issue:`25626` is fixed. As the form of the answer is dependent of
 the giac version, we simplify it (see :issue:`34037`)::
 
+    sage: # needs sage.libs.giac
     sage: t = SR.var('t')
     sage: integrate(exp(t)/(t + 1)^2, t, algorithm='giac').full_simplify()
     ((t + 1)*Ei(t + 1) - e^(t + 1))/(t*e + e)
diff --git a/src/sage/functions/piecewise.py b/src/sage/functions/piecewise.py
index fc69057ef6f..2986e47960d 100644
--- a/src/sage/functions/piecewise.py
+++ b/src/sage/functions/piecewise.py
@@ -836,6 +836,7 @@ def integral(self, parameters, variable, x=None, a=None, b=None, definite=False,
 
             Check that the algorithm keyword can be used::
 
+                sage: # needs sage.libs.giac
                 sage: ex = piecewise([([0, 1], 1), ((1, oo), 1/x**2)])
                 sage: integral(ex, x, 0, 100, algorithm='sympy')
                 199/100
diff --git a/src/sage/libs/giac/giac.pyx b/src/sage/libs/giac/giac.pyx
index 91c42f3132c..65279b16504 100644
--- a/src/sage/libs/giac/giac.pyx
+++ b/src/sage/libs/giac/giac.pyx
@@ -102,6 +102,13 @@ TESTS::
     sage: libgiac(-11^1000)
     -2469932918005826334124088385085221477709733385238396234869182951830739390375433175367866116456946191973803561189036523363533798726571008961243792655536655282201820357872673322901148243453211756020067624545609411212063417307681204817377763465511222635167942816318177424600927358163388910854695041070577642045540560963004207926938348086979035423732739933235077042750354729095729602516751896320598857608367865475244863114521391548985943858154775884418927768284663678512441565517194156946312753546771163991252528017732162399536497445066348868438762510366191040118080751580689254476068034620047646422315123643119627205531371694188794408120267120500325775293645416335230014278578281272863450085145349124727476223298887655183167465713337723258182649072572861625150703747030550736347589416285606367521524529665763903537989935510874657420361426804068643262800901916285076966174176854351055183740078763891951775452021781225066361670593917001215032839838911476044840388663443684517735022039957481918726697789827894303408292584258328090724141496484460001
 
+Ensure that signed infinities get converted correctly::
+
+    sage: libgiac(+Infinity)
+    +infinity
+    sage: libgiac(-Infinity)
+    -infinity
+
 .. SEEALSO::
 
     ``libgiac``, ``giacsettings``, ``Pygen``,``loadgiacgen``
@@ -159,6 +166,7 @@ from sage.rings.integer_ring import ZZ
 from sage.rings.rational_field import QQ
 from sage.rings.finite_rings.integer_mod_ring import IntegerModRing
 from sage.rings.integer cimport Integer
+from sage.rings.infinity import AnInfinity
 from sage.rings.rational cimport Rational
 from sage.structure.element cimport Matrix
 
@@ -860,6 +868,8 @@ cdef class Pygen(GiacMethods_base):
                     s = s._giac_init_()
                 except AttributeError:
                     s = SRexpressiontoGiac(s)
+            elif isinstance(s, AnInfinity):
+                s = s._giac_init_()
             if not isinstance(s, str):
                 s = s.__str__()
             sig_on()
diff --git a/src/sage/symbolic/integration/external.py b/src/sage/symbolic/integration/external.py
index 5f647228564..1390c524300 100644
--- a/src/sage/symbolic/integration/external.py
+++ b/src/sage/symbolic/integration/external.py
@@ -214,70 +214,31 @@ def fricas_integrator(expression, v, a=None, b=None, noPole=True):
     return result
 
 
-def giac_integrator(expression, v, a=None, b=None):
-    r"""
-    Integration using Giac.
-
-    EXAMPLES::
-
-        sage: from sage.symbolic.integration.external import giac_integrator
-        sage: giac_integrator(sin(x), x)
-        -cos(x)
-        sage: giac_integrator(1/(x^2+6), x, -oo, oo)
-        1/6*sqrt(6)*pi
-
-    TESTS::
-
-        sage: giac_integrator(e^(-x^2)*log(x), x)
-        integrate(e^(-x^2)*log(x), x)
-
-    Check that :issue:`30133` is fixed::
-
-        sage: ee = SR.var('e')
-        sage: giac_integrator(ee^x, x)
-        e^x/log(e)
-        sage: y = SR.var('π')
-        sage: giac_integrator(cos(y), y)
-        sin(π)
-
-    Check that :issue:`29966` is fixed::
-
-        sage: giac_integrator(sqrt(x + sqrt(x)), x)
-        1/12*(2*sqrt(x)*(4*sqrt(x) + 1) - 3)*sqrt(x + sqrt(x))...
-    """
-    ex = expression._giac_()
-    if a is None:
-        result = ex.integrate(v._giac_())
-    else:
-        result = ex.integrate(v._giac_(), a._giac_(), b._giac_())
-    if 'integrate' in format(result) or 'integration' in format(result):
-        return expression.integrate(v, a, b, hold=True)
-    return result._sage_()
-
-
 def libgiac_integrator(expression, v, a=None, b=None):
     r"""
     Integration using libgiac.
 
     EXAMPLES::
 
+        sage: # needs sage.libs.giac
         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: # needs sage.libs.giac
         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 (:issue:`31873`)::
 
+        sage: # needs sage.libs.giac
         sage: a, x = var('a,x')
         sage: f = sec(2*a*x)
         sage: F = libgiac_integrator(f, x)
@@ -295,10 +256,9 @@ def libgiac_integrator(expression, v, a=None, b=None):
         return expression.integrate(v, a, b, hold=True)
 
     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.
+    # We call Pygen on first argument because otherwise some
+    # expressions involving derivatives result in doctest failures in
+    # sage/interfaces/sympy.py
     if a is None:
         result = libgiac.integrate(Pygen(expression), v)
     else:
diff --git a/src/sage/symbolic/integration/integral.py b/src/sage/symbolic/integration/integral.py
index 0877d132030..4a70d373516 100644
--- a/src/sage/symbolic/integration/integral.py
+++ b/src/sage/symbolic/integration/integral.py
@@ -28,7 +28,7 @@
 available_integrators['sympy'] = external.sympy_integrator
 available_integrators['mathematica_free'] = external.mma_free_integrator
 available_integrators['fricas'] = external.fricas_integrator
-available_integrators['giac'] = external.giac_integrator
+available_integrators['giac'] = external.libgiac_integrator
 available_integrators['libgiac'] = external.libgiac_integrator
 
 ######################################################
@@ -59,9 +59,10 @@ def __init__(self):
 
         Check for :issue:`28913`::
 
+            sage: # needs sage.libs.giac
             sage: Ex = (1-2*x^(1/3))^(3/4)/x
             sage: integrate(Ex, x, algorithm='giac')  # long time
-            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))
+            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))
 
         Check for :issue:`29833`::
 
@@ -207,10 +208,13 @@ def __init__(self):
 
         Check for :issue:`32354`::
 
+            sage: # needs sage.libs.giac
             sage: ex = 1/max_symbolic(x, 1)**2
             sage: integral(ex, x, 0, 2, algorithm='giac')
             3/2
-            sage: integral(1/max_symbolic(x, 1)**2, x, 0, oo, algorithm='giac')
+            sage: result = integral(1/max_symbolic(x, 1)**2, x, 0, oo, algorithm='giac')
+            ...
+            sage: result
             2
         """
         # The automatic evaluation routine will try these integrators
@@ -474,9 +478,9 @@ def integrate(expression, v=None, a=None, b=None, algorithm=None, hold=False):
 
       - ``'fricas'`` -- use FriCAS (the optional fricas spkg has to be installed)
 
-      - ``'giac'`` -- use Giac
+      - ``'giac'`` -- use libgiac
 
-      - ``'libgiac'`` -- use libgiac
+      - ``'libgiac'`` -- use libgiac (alias for ``'giac'``)
 
     To prevent automatic evaluation, use the ``hold`` argument.
 
@@ -695,9 +699,15 @@ def integrate(expression, v=None, a=None, b=None, algorithm=None, hold=False):
         sage: integrate(f(x), x, 1, 2, algorithm='sympy')                               # needs sympy
         -1/2*pi + arctan(8) + arctan(5) + arctan(2) + arctan(1/2)
 
-    Using Giac to integrate the absolute value of a trigonometric expression::
+    Using Giac to integrate the absolute value of a trigonometric
+    expression. If Giac is installed, this will be attempted
+    automatically in the event that Maxima is unable to integrate the
+    expression::
 
-        sage: integrate(abs(cos(x)), x, 0, 2*pi, algorithm='giac')
+        sage: # needs sage.libs.giac
+        sage: result = integrate(abs(cos(x)), x, 0, 2*pi, algorithm='giac')
+        ...
+        sage: result
         4
         sage: result = integrate(abs(cos(x)), x, 0, 2*pi)
         ...