MathAction #314 Wrong answer for integrate(2*atan(x)/(1+x^2),x)

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The antiderivative should be $\atan(x)^2$ , however:

axiom

integrate(2*atan(x)/(1+x^2),x)

$\label{eq1}{\arctan \left({x}\right)}^2$

Type: Union(Expression(Integer),...)

axiom

D(%,x)

$\label{eq2}{2 \ {\arctan \left({x}\right)}}\over{{x^2}+ 1}$

Type: Expression(Integer)

axiom

D(atan(x)^2,x)

$\label{eq3}{2 \ {\arctan \left({x}\right)}}\over{{x^2}+ 1}$

Type: Expression(Integer)

Maybe this is related to bug #293, i.e., Axiom choosing the wrong branch of $\atan$ , or #141?

Yes, it is related to #293 --kratt6, Tue, 29 Aug 2006 04:55:37 -0500 reply

Look at

axiom

f x == 2*atan(x)+atan(2*x/(x^2-1))

Type: Void

Differentiating, we see that is constant

axiom

D(f x, x)

axiom

Compiling function f with type Variable(x) -> Expression(Integer)

$\label{eq4}0$

Type: Expression(Integer)

However, is not continuous at 1, and indeed,

axiom

numeric f(%pi)

axiom

Compiling function f with type Pi -> Expression(Integer)

$\label{eq5}3.1415926535897932385$

Type: Float

Drawing the function makes the situation quite clear...

fixed in FriCAS --kratt6, Thu, 20 Dec 2007 00:56:46 -0800 reply

Category: Aldor Library Compiler => Axiom Library Status: open => fix proposed

... --gdr, Sun, 18 May 2008 13:42:27 -0700 reply

Status: fix proposed => fixed somewhere

no patch available

From Fricas --japp, Tue, 14 Oct 2008 06:03:22 -0700 reply

... --kratt6, Tue, 14 Oct 2008 06:20:07 -0700 reply

This was done in rev 418 of wh-sandbox, not FriCAS...

Patch Posting --japp, Tue, 14 Oct 2008 06:58:15 -0700 reply

Yes, but the link is to Fricas not the wh-sandbox.

... --japp, Thu, 16 Oct 2008 13:57:48 -0700 reply

Status: fixed somewhere => fix proposed

Fixed in OpenAxiom --japp, Thu, 16 Oct 2008 18:05:25 -0700 reply