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Symbolic Integration

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Editor: mcd Time: 2009/06/26 00:23:36 GMT-7
Note: teste

added:

From mcd Fri Jun 26 00:23:35 -0700 2009
From: mcd
Date: Fri, 26 Jun 2009 00:23:35 -0700
Subject: teste
Message-ID: <20090626002335-0700@axiom-wiki.newsynthesis.org>

\begin{axiom}
integrate(a*x,x);
\end{axiom}

Errors in symbolic integration

AXIOM Examples

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$

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$

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}$

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)),...)

... --unknown, Sat, 21 May 2005 12:49:39 -0500 reply

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.

... --unknown, Sat, 21 May 2005 12:50:47 -0500 reply

axiom

integrate(x,x)

$\label{eq7}{1 \over 2}\ {x^2}$

(7)

Type: Polynomial(Fraction(Integer))

... --unknown, Sat, 21 May 2005 12:51:59 -0500 reply

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.

... --unknown, Sat, 21 May 2005 12:52:20 -0500 reply

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

... --unknown, Sat, 22 Oct 2005 19:04:53 -0500 reply

int(sqrt(x), x)

solvin --mcd, Fri, 26 Jun 2009 00:20:27 -0700 reply

int(sqrt(x), x);

teste --mcd, Fri, 26 Jun 2009 00:23:35 -0700 reply

axiom

integrate(a*x,x);

Type: Polynomial(Fraction(Integer))