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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:

    \begin{equation}
    -2\,{\it EllipticE}(\frac{\frac{\pi }{2} - x}{2},2)
    \end{equation}

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}

The answer should be:
\begin{equation}
15\,{\frac {\sqrt {\pi }}{\sqrt {\pi}}}
\end{equation}


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)

\label{eq1}\int^{
\displaystyle
x}{{\left({\sqrt{-{\%A^3}+ 1}}+{\sin \left({\%A}\right)}\right)}\ {d \%A}}(1)
Type: Union(Expression(Integer),...)

int(sin(x)+sqrt(1-x^3),x);
reduce
\displaylines{\qdd
\frac{-5\cdot \cos 
      \(x
       

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
\displaylines{\qdd
\int {\sqrt{
            -\ln 
            \(\sin 
              \(x
               

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
\displaylines{\qdd
\frac{2\cdot 
      \sqrt{\sin 
            \(\frac{1}{
                    x}
             

4)

axiom
integrate(sqrt(sin(x)),x)

\label{eq2}\int^{
\displaystyle
x}{{\sqrt{\sin \left({\%A}\right)}}\ {d \%A}}(2)
Type: Union(Expression(Integer),...)

int(sqrt(sin(x)),x);
reduce
\displaylines{\qdd
\int {\sqrt{\sin 
            \(x
             

For this Maple 9 gives the following result:


\label{eq3}
    -{\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)
    (3)

And Mathematica 4 gives:


\label{eq4}
    -2\,{\it EllipticE}(\frac{\frac{\pi }{2} - x}{2},2)
    (4)

symbolic integration
Tue, 22 Mar 2005 11:48:00 -0600 reply
axiom
integrate(exp(-x^2),x)

\label{eq5}{{\erf \left({x}\right)}\ {\sqrt{\pi}}}\over 2(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)

\label{eq6}{2 \ {\sqrt{\pi}}}\over{{\sqrt{2}}\ {\sqrt{2 \  \pi}}}(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 to learn more about the available operations.
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)

\label{eq7}{1 \over 2}\ {x^2}(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)

\label{eq8}\mbox{\tt "failed"}(8)
Type: Union(fail: failed,...)

The answer should be:


\label{eq9}
15\,{\frac {\sqrt {\pi }}{\sqrt {\pi}}}
(9)

integrate(exp(x)/x^2) --unknown, Thu, 25 Aug 2005 05:57:53 -0500 reply
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

int(sqrt(x), x)