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# Edit detail for Symbolic Integration revision 1 of 12

 1 2 3 4 5 6 7 8 9 10 11 12 Editor: 127.0.0.1 Time: 2007/11/15 20:25:38 GMT-8 Note: transferred from axiom-developer

changed:
-
Errors in symbolic integration

AXIOM Examples

1)

\begin{axiom}
integrate(sin(x)+sqrt(1-x^3),x)
\end{axiom}

\begin{reduce}
int(sin(x)+sqrt(1-x^3),x);
\end{reduce}

2)

\begin{axiom}
integrate(sqrt(1-log(sin(x)^2)),x)
\end{axiom}

\begin{reduce}
int(sqrt(1-log(sin(x)^2)),x);
\end{reduce}

3)

\begin{axiom}
integrate(sqrt(sin(1/x)),x)
\end{axiom}

That seems strange given the claims about the "completeness" of
Axiom's integration algorithm! But to be fair, Maple also returns
this integral unevaluated.

\begin{reduce}
int(sqrt(sin(1/x)),x);
\end{reduce}

4)

\begin{axiom}
integrate(sqrt(sin(x)),x)
\end{axiom}

\begin{reduce}
int(sqrt(sin(x)),x);
\end{reduce}

For this Maple 9 gives the following result:

\begin{eqnarray}
-{\frac {\sqrt {1+\sin \left( x \right) }\sqrt {-2\,\sin \left( x
\right) +2}\sqrt {-\sin \left( x \right) }}{\cos \left( x \right) \sqrt {\sin \left( x
\right) }}} \times
\\
\left( 2\,{\it EllipticE}
\left( \sqrt {1+\sin \left( x \right) },1/2\,\sqrt {2} \right) -{\it
EllipticF} \left( \sqrt {1+\sin \left( x \right) },1/2\,\sqrt {2}
\right)  \right)
\end{eqnarray}

And Mathematica 4 gives:

$$-2\,{\it EllipticE}(\frac{\frac{\pi }{2} - x}{2},2)$$

From unknown Tue Mar 22 11:48:00 -0600 2005
From:
Date: Tue, 22 Mar 2005 11:48:00 -0600
Subject: symbolic integration
Message-ID: <20050322114800-0600@page.axiom-developer.org>

\begin{axiom}
integrate(exp(-x^2),x)
\end{axiom}

From unknown Wed Mar 23 08:23:21 -0600 2005
From:
Date: Wed, 23 Mar 2005 08:23:21 -0600
Subject: Errorfunction
Message-ID: <20050323082321-0600@page.axiom-developer.org>

\begin{axiom}
integrate(exp(-x^2/2)/sqrt(%pi*2),x=%minusInfinity..%plusInfinity)
\end{axiom}

From unknown Sat May 21 12:49:39 -0500 2005
From: unknown
Date: Sat, 21 May 2005 12:49:39 -0500
Subject:
Message-ID: <20050521124939-0500@page.axiom-developer.org>

\begin{axiom}
int(x,x)
\end{axiom}

From unknown Sat May 21 12:50:47 -0500 2005
From: unknown
Date: Sat, 21 May 2005 12:50:47 -0500
Subject:
Message-ID: <20050521125047-0500@page.axiom-developer.org>

\begin{axiom}
integrate(x,x)
\end{axiom}

From unknown Sat May 21 12:51:59 -0500 2005
From: unknown
Date: Sat, 21 May 2005 12:51:59 -0500
Subject:
Message-ID: <20050521125159-0500@page.axiom-developer.org>

\begin{axiom}
axiomintegrate(x^6*exp(-x^2/2)/sqrt(%pi*2),x=%minusInfinity..%plusInfinity)
\end{axiom}

From unknown Sat May 21 12:52:20 -0500 2005
From: unknown
Date: Sat, 21 May 2005 12:52:20 -0500
Subject:
Message-ID: <20050521125220-0500@page.axiom-developer.org>

\begin{axiom}
integrate(x^6*exp(-x^2/2)/sqrt(%pi*2),x=%minusInfinity..%plusInfinity)
\end{axiom}

$$15\,{\frac {\sqrt {\pi }}{\sqrt {\pi}}}$$

From unknown Thu Aug 25 05:57:53 -0500 2005
From: unknown
Date: Thu, 25 Aug 2005 05:57:53 -0500
Subject: integrate(exp(x)/x^2)
Message-ID: <20050825055753-0500@page.axiom-developer.org>

Axiom does not perform the integration (while it perform the integration of exp(x)/x ), but the integration can be given in terms of Ei(x)

integrate(exp(x)/x^2,x)  -->  Ei(x)-exp(x)/x

From unknown Sat Oct 22 19:04:53 -0500 2005
From: unknown
Date: Sat, 22 Oct 2005 19:04:53 -0500
Subject:
Message-ID: <20051022190453-0500@page.axiom-developer.org>

int(sqrt(x), x)


## Errors in symbolic integration

AXIOM Examples

1)

axiom
integrate(sin(x)+sqrt(1-x^3),x)
 (1)
Type: Union(Expression(Integer),...)

 int(sin(x)+sqrt(1-x^3),x); reduce

2)

axiom
integrate(sqrt(1-log(sin(x)^2)),x)
>> Error detected within library code:
integrate: implementation incomplete (constant residues)

 int(sqrt(1-log(sin(x)^2)),x); reduce

3)

axiom
integrate(sqrt(sin(1/x)),x)
>> Error detected within library code:
integrate: implementation incomplete (constant residues)

That seems strange given the claims about the "completeness" of Axiom's integration algorithm! But to be fair, Maple also returns this integral unevaluated.

 int(sqrt(sin(1/x)),x); reduce

4)

axiom
integrate(sqrt(sin(x)),x)
 (2)
Type: Union(Expression(Integer),...)

 int(sqrt(sin(x)),x); reduce

For this Maple 9 gives the following result:

 (3)

And Mathematica 4 gives:

 (4)

symbolic integration
Tue, 22 Mar 2005 11:48:00 -0600 reply
axiom
integrate(exp(-x^2),x)
 (5)
Type: Union(Expression(Integer),...)
Errorfunction
Wed, 23 Mar 2005 08:23:21 -0600 reply
axiom
integrate(exp(-x^2/2)/sqrt(%pi*2),x=%minusInfinity..%plusInfinity)
 (6)
Type: Union(f1: OrderedCompletion?(Expression(Integer)),...)

axiom
int(x,x)
There are no exposed library operations named int but there are 5
unexposed operations with that name. Use HyperDoc Browse or issue
)display op int
Cannot find a definition or applicable library operation named int
with argument type(s)
Variable(x)
Variable(x)
Perhaps you should use "@" to indicate the required return type,
or "$" to specify which version of the function you need. axiom integrate(x,x)  (7) Type: Polynomial(Fraction(Integer)) axiom axiomintegrate(x^6*exp(-x^2/2)/sqrt(%pi*2),x=%minusInfinity..%plusInfinity) There are no library operations named axiomintegrate Use HyperDoc Browse or issue )what op axiomintegrate to learn if there is any operation containing " axiomintegrate " in its name. Cannot find a definition or applicable library operation named axiomintegrate with argument type(s) Expression(Integer) SegmentBinding(OrderedCompletion(Integer)) Perhaps you should use "@" to indicate the required return type, or "$" to specify which version of the function you need.

axiom
integrate(x^6*exp(-x^2/2)/sqrt(%pi*2),x=%minusInfinity..%plusInfinity)
 (8)
Type: Union(fail: failed,...)