[Maxima-bugs] [ maxima-Bugs-1374700 ] integrate((1+tan(x)^2)/tan(x),x);

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Bugs item #1374700, was opened at 2005-12-06 13:41
Message generated for change (Comment added) made by rtoy
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Category: Lisp Core - Integration
Group: None
Status: Open
Resolution: None
Priority: 5
Submitted By: Nobody/Anonymous (nobody)
Assigned to: Nobody/Anonymous (nobody)
Summary: integrate((1+tan(x)^2)/tan(x),x);

Initial Comment:
Non-real result

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>Comment By: Raymond Toy (rtoy)
Date: 2006-08-31 16:40

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Maxima 5.9.3.99rc2 and cmucl 2006-09 says

log(sin(x)) - log(sin(x)^2-1)/2

But the same maxima with clisp 2.35 says log(tan(x)). 

I don't know why.

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Comment By: Robert Dodier (robert_dodier)
Date: 2006-08-14 22:46

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Maxima 5.9.3.99rc1 / Clisp 2.38:

 integrate((1+tan(x)^2)/tan(x),x);
  => log(tan(x))

which seems right. Maybe if someone else wants to weigh in
here. If someone else agrees this result is OK, we can close
this report.

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Comment By: Raymond Toy (rtoy)
Date: 2006-02-13 13:07

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This integral is transformed to
cos(x)/sin(x)*(sin(x)^2/cos(x)^2+1).  Then maxima uses the
substitution y=sin(x) to get 1/y*(y^2/(1-y^2)+1.

However:

integrate(1/y*(y^2/(1-y^2)+1),y) ->
log(y)-log(y^2-1)/2.

But
integrate(expand(1/y*(y^2/(1-y^2)+1)),y) ->
log(y)-log(1-y^2)/2.

The former is wrong for our integration problem; the latter
would produce the desired answer.


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