Trac #12947: Bug in integrating x*cos(x^3)

From this [https://groups.google.com/forum/?fromgroups#!topic/sage-
support/-Dzaion3VrA sage-support thread]:
{{{
sage: numerical_integral(x*cos(x^3),0,.5)
(0.1247560409610376, 1.3850702913602309e-15)
sage: integrate(x*cos(x^3),(x,0,1/2)).n()
-0.0677842754623305
}}}
Given
{{{
sage: integrate(x*cos(x^3),x)
1/12*((-I*sqrt(3) - 1)*gamma(2/3, -I*x^3) + (I*sqrt(3) - 1)*gamma(2/3,
I*x^3))*(x^3)^(1/3)/x
sage: integrate(x*cos(x^3),(x,0,1/2))
-1/3*gamma(2/3) + 1/6*gamma(2/3, -1/8*I) + 1/6*gamma(2/3, 1/8*I)
}}}
this is probably something where Maxima is returning too many imaginary
things and something isn't cancelling right either there or in Sage
proper?  We've had trouble with either side of this with incomplete
gamma functions before.

URL: http://trac.sagemath.org/12947
Reported by: kcrisman
Ticket author(s): Peter Bruin
Reviewer(s): Ralf Stephan

src/sage/interfaces/maxima_lib.py

@@ -742,7 +742,7 @@ def sr_integral(self,*args):
             4
 
         This definite integral returned zero (incorrectly) in at least
-        maxima-5.23. The correct answer is now given (:trac:`11591`)::
+        Maxima 5.23. The correct answer is now given (:trac:`11591`)::
 
             sage: f = (x^2)*exp(x) / (1+exp(x))^2
             sage: integrate(f, (x, -infinity, infinity))
@@ -766,6 +766,15 @@ def sr_integral(self,*args):
             sage: maxima('radexpand: true')
             true
 
+        The following integral was computed incorrectly in versions of
+        Maxima before 5.27 (see :trac:`12947`)::
+
+            sage: a = integrate(x*cos(x^3),(x,0,1/2)).n()
+            sage: a.real()
+            0.124756040961038
+            sage: a.imag().abs() < 3e-17
+            True
+
         """
         try:
             return max_to_sr(maxima_eval(([max_integrate],[sr_to_max(SR(a)) for a in args])))