Sorry, but this is a bug (version 5.24). I'll reported it to the Maxima bug list.
Sadly, it seems that Maxima can find an antiderivative of acos(cos(t)) that is valid
on [0, %pi/2], but Maxima gives an incorrect value for a definite integral
(%i24) e : integrate(acos(cos(t)),t);
(%o24) (atan2(1,sin(t)/(cos(t)+1))^2-2*atan2(1,-sin(t)/(cos(t)+1))*atan2(1,sin(t)/(cos(t)+1))+atan2(1,-sin(t)/(cos(t)+1))^2)/2
Evidence that %o24 is correct:
(%i25) taylor(e,t,0,8);
(%o25)/T/ t^2/2+...
Also
(%i26) subst(t=%pi/2,e)-subst(t=0,e);
(%o26) %pi^2/8
A workaround for this particular integral is to use ldefint
(%i28) ldefint(acos(cos(t)),t,0,%pi/2);
(%o28) %pi^2/8
--Barton
-----maxima-bounces at math.utexas.edu wrote: -----
>To:?maxima at math.utexas.edu
>From:?Paul?Rey?<paul.rey82 at yahoo.fr>
>Sent?by:?maxima-bounces at math.utexas.edu
>Date:?05/14/2011?04:20PM
>Subject:?[Maxima]?Problem?with?acos
>
>Excuse?my?very?very?bad?english.
>I?have?a?proble?to?compute?integrate(acos(cos(t)),t,0,%pi/2),?maxima
>return?-3*%pi^2/8?instead?of?%pi^2/8.?More?genrally,?it?seems?that?for?x
>in?[0,%pi],?maxima?return?x^2/2-%pi*x?for?integrate(acos(cos(t)),t,0,x)
>instead?of?x^2/2.
>Regards
>Paul?Rey_______________________________________________
>Maxima?mailing?list
>Maxima at math.utexas.edu
>http://www.math.utexas.edu/mailman/listinfo/maxima