Wrong integral of sqrt(1-cos(x))

[tmonteil]

As reported on ​this ask question:

sage: integral(sqrt(1-cos(x)), x, 0, 2*pi)
0

This comes from the following wrong primitive

sage: integral(sqrt(1-cos(x)), x, algorithm='maxima')
-2*sqrt(2)/sqrt(sin(x)^2/(cos(x) + 1)^2 + 1)

where Fricas find a correct answer:

-2*(cos(x) + 1)*sqrt(-cos(x) + 1)/sin(x)

See ​Maxima bug 3659.

[mantepse]

#25220 fixes the problem with FriCAS.

[chapoton]

This should be upstreamed to maxima. Karl-Dieter, would you do so, please ?

[kcrisman]

Interestingly, this may not necessarily be a problem with domain:complex. Can you just confirm that the Fricas antiderivative makes more sense at ​this graphic? It's been a while since I've had to deal with branch cuts and I feel like that is part of the question here - but the way in which the graphs are different between pi and 2pi is definitely a problem. BUT Maxima gives the correct answer to the definite integral between pi and 2pi with either setting of domain.

[kcrisman]

Okay, it is domain:complex, but more subtly.

(%i1) integrate(sqrt(1-cos(x)), x, %pi, 2*%pi);
                                      3/2
(%o1)                                2
(%i2) integrate(sqrt(1-cos(x)), x, 0, 2*%pi);  
                                      5/2
(%o2)                                2
(%i3) domain:complex;
(%o3)                               complex
(%i4) integrate(sqrt(1-cos(x)), x, %pi, 2*%pi);
                                      3/2
(%o4)                                2
(%i5) integrate(sqrt(1-cos(x)), x, 0, 2*%pi);
(%o5)                                  0

This I am more confident about reporting upstream.