Get integral of abs(sin(x)) and abs(cos(x)) right

[kcrisman]

See this comment, discussion in #13364, the supposed fix at ​https://sourceforge.net/p/maxima/bugs/2520/, ​this ask.sagemath question, and so forth.

E.g. this is annoying (but apparently right)

(%i1) load(abs_integrate);
(%i3) display2d:false;

(%o3) false
(%i4) integrate(abs(sin(x)),x);

(%o4) 4*(abs(sin(x))*(atan(sin(x)/(cos(x)+1))/2
                     +sin(x)/((cos(x)+1)*(2*sin(x)^2/(cos(x)+1)^2+2)))
        /abs(cos(x)+1)
        -(signum(1/(cos(x)+1))*signum(sin(x))*log(2*sin(x)^2/(cos(x)+1)^2+2)
         -signum(1/(cos(x)+1))*signum(sin(x))*log(sin(x)^2/(cos(x)+1)^2+1)
         +signum(1/(cos(x)+1))*signum(sin(x))
                              *(2*sin(x)*atan(sin(x)/(cos(x)+1))/(cos(x)+1)
                               -log(2)))
         /4)

but it doesn't do the definite integral at all, even the easy ones where abs does nothing on the interval!

[kcrisman]

Worse,

sage: integrate(abs(cos(x)),x,0,pi)
-1

[kcrisman]

​https://sourceforge.net/p/maxima/bugs/2859/

[kcrisman]

Reported again at ​https://groups.google.com/forum/#!topic/sage-support/1Hcv29UyqVo

[chapoton]

Everything works in 8.9.b7 after #27958. Some doctests were added there.

Maybe one could add one doctest for

sage: integrate(abs(sin(x)),x)
-cos(x)*sgn(sin(x)) + sgn(sin(x))

[chapoton]

I have added a doctest.

---

New commits:

​4a8dff9

trac 17511 adding a doctest

[vdelecroix]

This answer

sage: integrate(abs(sin(x)),x)
-cos(x)*sgn(sin(x)) + sgn(sin(x))

is *very* wrong (outside of [-pi, pi])! It takes negative values

sage: f = integrate(abs(sin(x)), x)
sage: (f(5) - f(0)).n()
-0.716337814536774

and the integral is supposed to go to +oo as x -> +oo/-oo...

[vdelecroix]

But this is correct

sage: integrate(abs(sin(x)), x, 0, 10)
cos(10) + 7

[vdelecroix]

And I like very much

sage: %time integrate(abs(sin(x)), x, algorithm='sympy')
CPU times: user 8.09 s, sys: 99 µs, total: 8.09 s
Wall time: 8.1 s
integrate(abs(sin(x)), x)

[chapoton]

Indeed. Giac is responsible for the wrong answer. This should be reported upstream.

sage: integrate(abs(sin(x)),x,algorithm='maxima')
integrate(abs(sin(x)), x)
sage: integrate(abs(sin(x)),x,algorithm='sympy')
integrate(abs(sin(x)), x)
sage: integrate(abs(sin(x)),x,algorithm='giac')
-cos(x)*sgn(sin(x)) + sgn(sin(x))
sage: integrate(abs(sin(x)),x,algorithm='fricas')
integral(abs(sin(x)), x)
sage: integrate(abs(sin(x)),x,algorithm='mathematica_free')
-cos(x)*sgn(sin(x))

[chapoton]

But giac is also providing the correct answer for the definite integral, where sympy fails to deliver:

sage: integrate(abs(cos(x)),x,0,44,algorithm='maxima')
integrate(abs(cos(x)), x, 0, 44)
sage: integrate(abs(cos(x)),x,0,44,algorithm='sympy')
-sin(44) + 4
sage: integrate(abs(cos(x)),x,0,44,algorithm='giac')
sin(44) + 28
sage: integrate(abs(cos(x)),x,0,44,algorithm='fricas')
integrate(abs(cos(x)), x, 0, 44)
sage: integrate(abs(cos(x)),x,0,44,algorithm='mathematica_free')
sin(44) + 28

[vdelecroix]

Replying to chapoton:

But giac is also providing the correct answer for the definite integral, where sympy fails to deliver:

[...]

which should also be reported upstream I guess...

[chapoton]

Samuel, would you please report to giac the failure of comment:12 ?

[slelievre]

I emailed Bernard Parisse, here is his reply:

Ce n'est pas une erreur, au sens ou un avertissement est renvoyé
0>> integrate(abs(sin(x))
Warning adding 1 ) at end of input
Attention, l'intégration de abs() ou sign() suppose un signe constant
par intervalles (vérifié si l'argument est réel):
  Verifiez [abs(sin(x))]
Discontinuités aux zeros de sin(x) were not checked
sign(sin(x))-cos(x)*sign(sin(x))
// Time 0.04

C'est à l'utilisateur de prendre garde aux discontinuites de la
primitive pour calculer une intégrale définie s'il utilise la primitive
renvoyée. Lorsqu'on demande à giac une intégrale définie, il essaie de
tenir compte des discontinuités, et il fait le calcul numérique en
parallèle pour verifier. En cas de non concordance, les 2 valeurs sont
renvoyées (la plupart du temps c'est la valeur numérique qui est la bonne).

which roughly translates as

This is not an error, in the sense that a warning is issued:
0>> integrate(abs(sin(x))
Warning adding 1 ) at end of input
Caution, integrating abs() or sign() assumes piecewise constant sign
(check if the argument is real):
Check [abs(sin(x))]
Discontinuities at zeros of sin(x) were not checked
sign(sin(x))-cos(x)*sign(sin(x))
// Time 0.04

It is up to the user to check for discontinuities of the antiderivative
to compute an integral using the antiderivative returned by giac.
When asking giac for an integral on an interval, it tries to take
discontinuities into account, and does the numerical integration in parallel
to check. In case of mismatch, both values are returned (most of the time
the numerical value is the one that is correct).