cosh integral incorrect because of abs_integrate

[kcrisman]

See this ask.sagemath question.

sage: a = integrate(sqrt(1-1/4*cosh(x)^2),x)
sage: a.full_simplify()
cosh(x)

Not that I'm an expert here, but at any rate, viewed as real functions this makes no sense, though perhaps on some branch it does. This happens because

Maxima 5.29.1 http://maxima.sourceforge.net
using Lisp ECL 12.12.1
Distributed under the GNU Public License. See the file COPYING.
Dedicated to the memory of William Schelter.
The function bug_report() provides bug reporting information.
(%i1) domain:complex;
(%o1)                               complex
(%i2) display2d:false;

(%o2) false
(%i3) B:integrate(sqrt(1-cosh(x)^2/4),x);

(%o3) 'integrate(sqrt(1-cosh(x)^2/4),x)
(%i4) trigsimp(B);

(%o4) 'integrate(sqrt(-(cosh(x)^2-4)/4),x)
(%i5) fullratsimp(B);

(%o5) 'integrate(sqrt(-(cosh(x)^2-4)/4),x)
(%i6) load(abs_integrate);

(%o6) "/Users/.../sage-5.9.rc1/local/share/maxima/5.29.1/share/contrib/integration/abs_integrate.mac"
(%i7) trigsimp(B);

(%o7) cosh(x)
(%i8) fullratsimp(B);

(%o8) cosh(x)

I'm not sure we want to unload abs_integrate but we need to figure something out here.

Upstream: Reported upstream. No feedback yet.

CC: @sagetrac-tmonteil @gagern

Component: calculus

Keywords: abs_integrate

Stopgaps: #12731

Issue created by migration from https://trac.sagemath.org/ticket/14591

[jdemeyer]

comment:3

Fixed by #13973.

[pjbruin]

comment:7

The answer cosh(x) is completely wrong, even in a neighbourhood of x = 0:

cosh(x) = integrate(sqrt(1 - 1/4*cosh(x)^2),x)
=> sinh(x) = sqrt(1 - 1/4*cosh(x)^2)
=> sinh(x)^2 = 1 - 1/4*cosh(x)^2
=> 0 = 3/4  (substitute x = 0)

According to the ask.sagemath.org link in the ticket description, the correct answer can be expressed in terms of an elliptic integral function, which I trust much more.

[kcrisman]

comment:8

As Thierry pointed out on the ask.sagemath question. So, does #13973 fix this or not?

[pjbruin]

comment:9

Replying to @kcrisman:

As Thierry pointed out on the ask.sagemath question.

Yes, I just wanted to give a clear reason here why the answer couldn't possibly be right.

So, does #13973 fix this or not?

No, it doesn't. I have been doing some debugging of the Maxima command

trigsimp(integrate(sqrt(4 - cosh(x)^2),x));

which returns cosh(x) when executed after loading abs_integrate. It seems that the problem arises in ratsubst, when substituting 1 + sinh(x)^2 for cosh(x)^2. What makes it more tricky is that calling ratsubst directly with the same arguments gives a correct answer. Possibly some flag is set at some intermediate point that triggers the bug.

[pjbruin]

comment:10

Reported upstream: https://sourceforge.net/p/maxima/bugs/2737/

[pjbruin]

Upstream: Reported upstream. No feedback yet.

[kcrisman]

comment:12

See also #12731.

[sagetrac-jakobkroeker]

Stopgaps: #12731

[fchapoton]

Changed keywords from none to abs_integrate

[fchapoton]

comment:15

Works (no wrong answer) in sage 8.9.b7 with py3:

sage: sage: a = integrate(sqrt(1-1/4*cosh(x)^2),x)
1/2*integrate(sqrt(-cosh(x)^2 + 4), x)

[pjbruin]

comment:16

Also in 8.8:

┌────────────────────────────────────────────────────────────────────┐
│ SageMath version 8.8, Release Date: 2019-06-26                     │
│ Using Python 2.7.15. Type "help()" for help.                       │
└────────────────────────────────────────────────────────────────────┘
sage: a = integrate(sqrt(1-1/4*cosh(x)^2),x)
sage: a.full_simplify()
integrate(sqrt(-1/4*cosh(x)^2 + 1), x)

This is most likely due to #12731. I'm not sure we should view this bug as being fixed; ideally, the answer should be given as an elliptic integral (see the ask.sagemath question).