Maxima -- GPL CAS based on DOE-MACSYMA

This looks like a good solution. But is it too broad? Are there cases in
trigint where triginverses set to $all would be wrong? Maybe a more
careful solution that only sets it when we use the tan(x/2) substitution
would be better?

I couldn't see any cases where it would produce incorrect results.

The problem comes from

(%i1) integrate(sqrt(sin(x)^2+cos(x)^2), x);
(%o1) atan(tan(x))

This integral is done with substitution (tan(x/2)) and we need to make
sure that atan(tan(x))=x when we back-substitute.

Here is a simple patch which fixes the problem:

Index: sin.lisp
===================================================================
RCS file: /cvsroot/maxima/maxima/src/sin.lisp,v
retrieving revision 1.51
diff -u -r1.51 sin.lisp
--- sin.lisp 18 Aug 2009 16:59:37 -0000 1.51
+++ sin.lisp 17 Oct 2009 21:33:54 -0000
@@ -1503,7 +1503,7 @@
getout
(setq y (list '(mtimes) *yy* *yz*))
get2 (setq y (simplify y))
- (return (substint repl 'x (integrator y 'x)))))
+ (return (let (($triginverses '$all)) (substint repl 'x (integrator y
'x))))))

(defmvar $integration_constant_counter 0)
(defmvar $integration_constant '$%c)

Perhaps this bug is not a new problem. The answer of Maxima depends on the
flag triginversers.

(%i36) integrate(sqrt(sin(t)^2+cos(t)^2),t,0,2*%pi),triginverses:true;
(%o36) %pi

(%i37) integrate(sqrt(sin(t)^2+cos(t)^2),t,0,2*%pi),triginverses:all;
(%o37) 2 %pi

The default value of triginverses has changed with revision 1.33 of
trigi.lisp from 'all to 'true. The bug was not visible in the past with the
standard value 'all of triginverses.

Dieter Kaiser