wrong result of integral

[dkrenn]

sage: f1 = (2*cos(2*pi*x) - cos(4*pi*x)) / (5 - 4*cos(2*pi*x))
sage: integrate(f1, x, 0, 1)
23/24

but it should be 1/4:

sage: sage: numerical_integral(f1,0,1)
(0.24999999999999997, 4.6160077311221225e-15)

and

sage: e(x)=exp(2*pi*I*x)
sage: f2=real(e(x)/(2-e(-x)))
sage: (f1-f2).simplify_trig()  # f1 equals f2
0
sage: integrate(f2,x,0,1)
1/4

This was reported by Lukas Spiegelhofer on 8/10/2015 16:00:13 via "Sage Notebooks Bugreports".

[mforets]

see: ​https://sourceforge.net/p/maxima/bugs/3295/

[SimonKing]

Is the following example an instance of the same bug, or a different problem?

sage: (cos(pi*x)*exp(-I*pi*x)).integral(x,-1/2,1/2)  # wrong
1
sage: F = (cos(pi*x)*exp(-I*pi*x)).integral(x); F(x=1/2)-F(x=-1/2)  # correct
1/2

[mforets]

Hmmm they look similar to me, although in your example the primitive computed by Maxima is correct, but for the case of the description also the primitive is wrong:

sage: f1 = (2*cos(2*pi*x) - cos(4*pi*x)) / (5 - 4*cos(2*pi*x))
sage: F1 = f1.integrate(x)
sage: F1(x=1) - F1(x=0)  # yet another result!
5/8
sage: F1_g = f1.integrate(x, algorithm="giac")
sage: F1_g(x=1) - F1_g(x=0)
1/4

[zimmerma]

I'm not sure integrate accepts inputs that take non-real values (it should be documented if yes or no):

sage: f(x)=cos(pi*x)*exp(-I*pi*x)
sage: f(1/4)
-1/2*I + 1/2

Note that:

sage: (cos(pi*x)*exp(-I*pi*x)).real().integral(x,-1/2,1/2) 
1/2