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

[chapoton]

Well, the primitives given by various algo are all different:

sage: f1 = (2*cos(2*pi*x) - cos(4*pi*x)) / (5 - 4*cos(2*pi*x))                  
sage: i1=f1.integral(x,algorithm='maxima').simplify_trig()                      
sage: i2=f1.integral(x,algorithm='giac').simplify_trig()                        
sage: i3=f1.integral(x,algorithm='fricas').simplify_trig()                      
sage: g1 = f1.simplify_trig()                                                   
sage: j1=g1.integral(x,algorithm='maxima').simplify_trig()                      
sage: j2=g1.integral(x,algorithm='giac').simplify_trig()                        
sage: j3=g1.integral(x,algorithm='fricas').simplify_trig()                      
sage: i1                                                                        
1/48*(30*pi*x + 24*cos(pi*x)*sin(pi*x) + 40*arctan(3*sin(pi*x)/cos(pi*x)) - 24*arctan(sin(pi*x)/cos(pi*x)) - 17*arctan2(2*cos(pi*x)*sin(pi*x), 2*cos(pi*x)^2 - 3/2) + 17*arctan2(2*cos(pi*x)*sin(pi*x), 2*cos(pi*x)^2 - 3))/pi
sage: i2                                                                        
1/8*(2*pi*x + 4*cos(pi*x)*sin(pi*x) + arctan(2*cos(pi*x)*sin(pi*x)/(2*sin(pi*x)^2 + 1)))/pi
sage: i3                                                                        
1/16*(2*pi*x + 8*cos(pi*x)*sin(pi*x) + arctan(1/6*(10*sin(pi*x)^2 - 1)/(cos(pi*x)*sin(pi*x))))/pi
sage: j1                                                                        
1/8*(pi*x + 4*cos(pi*x)*sin(pi*x) + arctan(3*sin(pi*x)/cos(pi*x)))/pi
sage: j2                                                                        
1/8*(2*pi*x + 4*cos(pi*x)*sin(pi*x) + arctan(2*cos(pi*x)*sin(pi*x)/(2*sin(pi*x)^2 + 1)))/pi
sage: j3                                                                        
1/16*(2*pi*x + 8*cos(pi*x)*sin(pi*x) + arctan(1/6*(10*sin(pi*x)^2 - 1)/(cos(pi*x)*sin(pi*x))))/pi
sage: (i1-j1).simplify_trig()       # maxima is not coherent with itself                                            
1/48*(24*pi*x + 34*arctan(3*sin(pi*x)/cos(pi*x)) - 24*arctan(sin(pi*x)/cos(pi*x)) - 17*arctan2(2*cos(pi*x)*sin(pi*x), 2*cos(pi*x)^2 - 3/2) + 17*arctan2(2*cos(pi*x)*sin(pi*x), 2*cos(pi*x)^2 - 3))/pi
sage: (i2-j2).simplify_trig()       # giac is coherent with itself                                          
0
sage: (i3-j3).simplify_trig()       # fricas is coherent with itself                                            
0

and only "giac" returns a continuous primitive ! All the others have a jump at 1/2

[chapoton]

display of the bad primitives

[chapoton]

Here is the picture, with mathematica_free result at the end, also not continuous