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

[sagetrac-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.

Upstream: Reported upstream. No feedback yet.

CC: @EmmanuelCharpentier @sagetrac-mafra @kcrisman

Component: symbolics

Keywords: integrate

Issue created by migration from https://trac.sagemath.org/ticket/24587

[mantepse]

comment:2

#25220 fixes the problem with FriCAS.

[fchapoton]

Changed keywords from none to integrate

[fchapoton]

comment:6

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

[kcrisman]

comment:7

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]

comment:8

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.

[kcrisman]

Upstream: Reported upstream. No feedback yet.

[fchapoton]

comment:10

No reaction upstream, it seems. Their primitive is plain wrong ; Fricas is right.

[EmmanuelCharpentier]

comment:11

I forgot this one...

Replying to @fchapoton:

No reaction upstream, it seems. Their primitive is plain wrong ; Fricas is right.

More or less :

• None of maxima, giac, fricas, mathematica has a term proportional to the length if the interval of integration ; all of them are periodic, which is wrong for the integral of a non-negative function positive almost everywhere on its period.

• sympy doesn't integrate this.

• fricas' result can be differentiated and simplified back to sqrt(1-cos(x)) ; others' can't.

• giac, fricas and mathematica give results numerically close to the numerical integration up to a constant plus a discontinuity at 2kpi ;maxima is just plain wrong.

To summarize, contemplate :

sage: Num=lambda t:numerical_integral(lambda u:sqrt(1-cos(u)),0,t)[0]
sage: Max=sqrt(1-cos(x)).integrate(x, algorithm="maxima").function(x)
sage: Max=sqrt(1-cos(x)).integrate(x, algorithm="maxima").function(x)
sage: Giac=sqrt(1-cos(x)).integrate(x, algorithm="giac").function(x)
sage: Fricas=sqrt(1-cos(x)).integrate(x, algorithm="fricas").function(x)
sage: Mathematica=sqrt(1-cos(x)).integrate(x, algorithm="mathematica_free").function(x)
sage: plot([Num, Max, Giac, Fricas, Mathematica], (0, 4*pi), legend_label=["Numerical", "Maxima", "Giac", "Fricas", "Mathematica"])

It seems that up to the "branch cut", fricas, giac and mathematica give numerically correctly answer, only fricas' being (easily) checkable by differentiation and simplification.

BTW : when the result of an integration is a periodic function, shouldn't a term adding (floor(number of periods in the integration interval)*(integral on one period) be added to the result ? If so, how to implement this ?