#3340 wrong exponential integral of arccos

[Marcelo Forets]

Hello!
Maxima gets the following definite integral wrong:

(%i10) integrate(exp(acos(x)),x,0,1);

                                       - %pi/2
                                 1   %e
(%o10)                           - - ---------
                                 2       2

The correct answer is 1/2*e^(1/2*pi) + 1/2.

Downstream report: https://trac.sagemath.org/ticket/24008

[Marcelo Forets]

huh, sorry for the code misalignment... what you see is Not what you get :)

i don't find a button to edit the ticket's description.

[Robert Dodier]

Hmm, tried to fix formatting, didn't succeed ... try again.

[James R. Van Zandt]

Maxima is returning the answer for a different integral. E.g. in MATLAB:

(1-exp(-pi/2))/2

ans =

0.3961

quad(@(x) exp(-acos(x)),0,1)

ans =

0.3961

Another way to look at it is that Maxima has chosen a different branch.
Ordinarily, one would say that 0 < acos(x) < pi/2 if 0 < x < 1.
That's the branch that MATLAB uses:

acos(.9)

ans =

0.4510

cos(.4510)

ans =

0.9000

However, cos(x) is of course an even function:

cos(-.4510)

ans =

0.9000

...which means one could claim that acos(.9) = -.4510

That's essentially what Maxima has done.

I've no clue as to how one could persuade it to use a different branch.

On Wed, Oct 11, 2017 at 4:37 AM, Marcelo Forets via Maxima-bugs maxima-bugs@lists.sourceforge.net wrote:

---

• [bugs:#3340] https://sourceforge.net/p/maxima/bugs/3340/ wrong
  exponential integral of arccos *

Status: open
Group: None
Created: Wed Oct 11, 2017 08:37 AM UTC by Marcelo Forets
Last Updated: Wed Oct 11, 2017 08:37 AM UTC
Owner: nobody

Hello!
Maxima gets the following definite integral wrong:

(%i10) integrate(exp(acos(x)),x,0,1);
- %pi/2
1 %e
(%o10) - - ---------
2 2

The correct answer is 1/2e^(1/2pi) + 1/2.

Downstream report: https://trac.sagemath.org/ticket/24008

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