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Symbolic Integration

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Editor: mcd Time: 2009/06/26 00:20:27 GMT-7
Note: solvin

added:

From mcd Fri Jun 26 00:20:27 -0700 2009
From: mcd
Date: Fri, 26 Jun 2009 00:20:27 -0700
Subject: solvin
Message-ID: <20090626002027-0700@axiom-wiki.newsynthesis.org>

int(sqrt(x), x);

Errors in symbolic integration

AXIOM Examples

axiom

integrate(sin(x)+sqrt(1-x^3),x)

   >> Error detected within library code:
   failed - cannot handle that integrand

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

reduce

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

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

axiom

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

(1)

Type: Union(Expression Integer,...)

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

reduce

For this Maple 9 gives the following result:

(2)

And Mathematica 4 gives:

(3)

symbolic integration: Tue, 22 Mar 2005 11:48:00 -0600 reply

axiom

integrate(exp(-x^2),x)

(4)

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)

(5)

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)

(6)

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)

(7)

Type: Union(fail: failed,...)

The answer should be:

(8)

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)