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Edit detail for SandBox revision 12 of 171

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changed:
-[SandBox10]
[SandBox10] -- Miscellaneous

[SandBox11]

changed:
-From unknown Wed Dec 7 09:11:18 -0600 2005
-From: unknown
-Date: Wed, 07 Dec 2005 09:11:18 -0600
-Subject: szsz
-Message-ID: <20051207091118-0600@wiki.axiom-developer.org>
-
-Works with ASCII text output formatting.
-\begin{axiom}
-)set output tex off
-)set output algebra on
-\end{axiom}
-\begin{axiom}
-    solve([x^2 + y^2 - 2*(ax*x + ay*y) = l1, x^2 + y^2 - 2*(cx*x + cy*y) = l2],[x,y])
-\end{axiom}
-
-But fails with LaTeX.
-\begin{axiom}
-)set output tex on
-)set output algebra off
-\end{axiom}
-
-
-From kratt6 Tue Jan 3 05:22:52 -0600 2006
-From: kratt6
-Date: Tue, 03 Jan 2006 05:22:52 -0600
-Subject: 0**0
-Message-ID: <20060103052252-0600@wiki.axiom-developer.org>
-
-The result of '0**0' depends on the type of '0':
-
-\begin{axiom}
-  (0::Float)**(0::Float)
-\end{axiom}
-
-The idea was, that defining $0^0$ as 1 is ok whenever there is no notion of limit. However,
-
-\begin{axiom}
-  (0::EXPR INT)**(0::EXPR INT)
-\end{axiom}
-
-is not quite in line with this, I think.
-There has been some discussion on this subject on axiom-developer.
-
-It is easy to change this behaviour, if we know better...
-
-From unknown Sun Jan 29 13:09:07 -0600 2006
-From: unknown
-Date: Sun, 29 Jan 2006 13:09:07 -0600
-Subject: ruleset
-Message-ID: <20060129130907-0600@wiki.axiom-developer.org>
-
-Let's see if the same happens here:
-\begin{axiom}
-sinCosProducts := rule
-  sin (x) * sin (y) == (cos(x-y) - cos(x+y))/2
-  cos (x) * cos (y) == (cos(x-y) + cos(x+y))/2
-  sin (x) * cos (y) == (sin(x-y) + sin(x+y))/2
-\end{axiom}
-
-
-From BillPage Mon Jan 30 09:00:02 -0600 2006
-From: Bill Page
-Date: Mon, 30 Jan 2006 09:00:02 -0600
-Subject: when typing
-Message-ID: <20060130090002-0600@wiki.axiom-developer.org>
-
-When you are typing or when you cut-and-paste commands directly
-into the Axiom interpreter you must use an underscore character at
-the end of each incomplete line, and you must use the ( ) syntax
-
-  sinCosProducts := rule (_
-  sin (x) * sin (y) == (cos(x-y) - cos(x+y))/2; _
-  cos (x) * cos (y) == (cos(x-y) + cos(x+y))/2; _
-  sin (x) * cos (y) == (sin(x-y) + sin(x+y))/2)
-
-Alternatively, using a text editor you can enter the commands into a
-file called, for example 'sincos.input' exactly as in MathActon above
-and the use the command::
-
-
-
-From unknown Sat Mar 11 13:19:44 -0600 2006
-From: unknown
-Date: Sat, 11 Mar 2006 13:19:44 -0600
-Subject:
-Message-ID: <20060311131944-0600@wiki.axiom-developer.org>
-
-\begin{axiom}
-)lib RINTERPA RINTERP PCDEN GUESS GUESSINT GUESSP
-guess(n, [1, 5, 14, 34, 69, 135, 240, 416, 686, 1106], n+->n, [guessRat], [guessSum, guessProduct, guessOne],2)\$GuessInteger
-\end{axiom}
-
-
-From BillPage Thu Mar 23 22:21:41 -0600 2006
-From: Bill Page
-Date: Thu, 23 Mar 2006 22:21:41 -0600
-Subject: conversion failed
-Message-ID: <20060323222141-0600@wiki.axiom-developer.org>
-
-Unknown wrote:
-
-> z:=sum(myfn(x),x=1..10) -- This fails, why?
-
-The reason this fails is because Axiom tries to evaluate
-'myfn(x)' **first**. But 'x' is not yet an 'Integer' so Axiom
-cannot compute 'myfn(x)'. I guess you were expecting Axiom
-to "wait" and not evaluate 'myfn(x)' until after 'x' has
-been assigned the value 1, right? But Axiom does not work
-this way.
-
-The solution is to write 'myfn(x)' so that is can be applied
-to something symbolic like 'x'. For example something this:
-
-\begin{axiom}
-myfn(i : Expression Integer) : Expression Integer == i
-myfn(x)
-z:=sum(myfn(x),x=1..10)
-\end{axiom}
-
-From Bill(Nameomitted) Fri Mar 24 21:45:25 -0600 2006
-From: Bill (Name omitted)
-Date: Fri, 24 Mar 2006 21:45:25 -0600
-Subject: Any hints for multivariate functions?
-Message-ID: <20060324214525-0600@wiki.axiom-developer.org>
-
-Hi Bill:
-
-Thanks for your quick response.  I tried to respond to this earlier, but didn't see it in the sand box, please forgive me if you get multiple copies.
-
-I tried to simplify the code from my original program, and generated a univariate function, however my actual code has a multivariate function,
-and your excellent hint on the use of the Expression qualifier on the parameter and return type which works great for the univariate function case appears to fail for multivarite functions.
-\begin{axiom}
-a(n : Expression Integer, k : Expression Integer, p : Expression Float) : Expression Float == binomial(n,k) * p**(k) * (1.0-p)**(n-k)
-output(a(4,3,0.25)) -- see that the function actually evaluates for sensible values
-z := sum(a(4,i,0.25), i=1..3) --- this fails
-output(z)
-\end{axiom}
-
-I did notice in the Axiom online book, chapter 6.6, around page 241, the recommendation to use untyped functions, which appears to allow Axiom to do
-inference on parameter and result type.
-\begin{axiom}
-b(n, k, p) == binomial(n,k) * p**(k) * (1.0-p)**(n-k)
-output(b(4,3,0.25)) -- see that the function actually evaluates for sensible values
-z := sum(b(4,i,0.25), i=1..3) --- this fails
-output(z)
-\end{axiom}
-For univariate functions the approach
-\begin{axiom}
-c(k) == binomial(4,k) * 0.25**k * (1.0 - 0.25)**(4-k) -- This approach is only a test, but is not suitable for my program
-output(c(3)) -- test to see if function can be evaluated for sensible arguments
-z := sum(c(i), i=1..3) -- still doesn't work
-output(z)
-\end{axiom}
-But interestingly something like
-\begin{axiom}
-d(k) == 1.5 * k -- coerce uotput to be a Float
-z := sum(d(i), i=1..3) -- This works!
-output(z)
-\end{axiom}
-Bill, thanks again for your quick help, unforutnatly I lack a local Axiom expert, any ideas would really be welcome here.
-
-From billpage Sat Mar 25 16:01:07 -0600 2006
-From: billpage
-Date: Sat, 25 Mar 2006 16:01:07 -0600
-Subject: reduce(+,[...]) = sum(...)
-Message-ID: <20060325160107-0600@wiki.axiom-developer.org>
-
-Try this
-\begin{axiom}
-z := reduce(+,[b(4,i,0.25) for  i in 1..3])
-\end{axiom}
-
-
-From Bill(Nameomitted) Mon Mar 27 08:10:26 -0600 2006
-From: Bill (Name omitted)
-Date: Mon, 27 Mar 2006 08:10:26 -0600
-Subject: Handling the result from functions returning a matrix
-Message-ID: <20060327081026-0600@wiki.axiom-developer.org>
-
-Hi all:
-
-Thanks Bill Page for your help, it is much appreciated (although I used a for loop and not reduce :-)).
-
-I'm having a bit of difficulty getting a Function returning a matrix to work as expected,
-perhaps it is just cockpit error, but I don't see the error of my ways.
-\begin{axiom}
-CFM(Q : Matrix(Float)): Matrix(Float) ==
-   x := nrows(Q)
-   MyIdentityMatrix : Matrix(Float) := new(x, x, 0)
-   for i in 1..nrows(MyIdentityMatrix) repeat
-      MyIdnetityMatrix(i,i) := 1.0
-   Ninv := MyIdnetityMatrix - Q
-   N := inverse(Ninv)
-   N
-
---test ComputeFundamentalMatrix
-X := matrix[[0, 0.5, 0],[0.5, 0, 0.5],[0, 0.5, 0]]
-output(X)
-N := CFM(X)
-output(N)
-\end{axiom}
-Any ideas where I'm blowing it here?  I tried explicitly setting N to be a Matrix type but that failed too.
-\begin{axiom}
-CFM(Q : Matrix(Float)): Matrix(Float) ==
-   x := nrows(Q)
-   MyIdentityMatrix : Matrix(Float) := new(x, x, 0)
-   for i in 1..nrows(MyIdentityMatrix) repeat
-      MyIdnetityMatrix(i,i) := 1.0
-   Ninv := MyIdnetityMatrix - Q
-   N := inverse(Ninv)
-   N
-
---test ComputeFundamentalMatrix
-X := matrix[[0, 0.5, 0],[0.5, 0, 0.5],[0, 0.5, 0]]
-output(X)
-N : Matrix(Float) := CFM(X)
-output(N)
-\end{axiom}
-
-Thanks again for all your help.
-
-Regards:
-
-Bill M. (Sorry, my unique last name attracts too much spam).
-
-From billpage Mon Mar 27 09:34:35 -0600 2006
-From: billpage
-Date: Mon, 27 Mar 2006 09:34:35 -0600
-Subject: typo and identity
-Message-ID: <20060327093435-0600@wiki.axiom-developer.org>
-
-> although I used a for loop and not reduce :-)
-
-Good thinking. ;)
-
-You have a simple typographical error. You have written both::
-
-  MyIdentityMatrix
-
-and ::
-
-  MyIdnetityMatrix
-
-BTW, instead of the complicated construction of the identify matrix
-you should just write::
-
-  Ninv := 1 - Q
-
-For matrices '1' denotes the identity.
-
-
-From unknown Sat Apr 8 10:45:29 -0500 2006
-From: unknown
-Date: Sat, 08 Apr 2006 10:45:29 -0500
-Subject:
-Message-ID: <20060408104529-0500@wiki.axiom-developer.org>
-
-\begin{axiom}
-)set output tex off
-)set output algebra on
-FunFun := x**4 - 6* x**3 + 11* x*x + 2* x + 1
-)set output tex on
-)set output algebra off
-
-\end{axiom}
-
-Matthias
-
-\begin{axiom}
-t:=matrix ([[0,1,1],[1,-2,2],[1,2,-1]])
-\end{axiom}
-
-We cat diagonalise t by finding it's eigenvalues.
-\begin{axiom}
-)set output tex off
-)set output algebra on
-\end{axiom}
-
-Now prove it by constructing the simularity transformation
-from the eigenvectors:
-\begin{axiom}
-p*d*inverse(p)
-)set output tex on
-)set output algebra off
-\end{axiom}
-\end{axiom}
-
-
-From unknown Fri Apr 28 14:03:28 -0500 2006
-From: unknown
-Date: Fri, 28 Apr 2006 14:03:28 -0500
-Subject: Axiom can't integrame exp(x^4) ;(
-Message-ID: <20060428140328-0500@wiki.axiom-developer.org>
-
-Axiom can't integrame exp(x^4) ;(
-
-\begin{axiom}
-integrate(exp(x**4),x)
-\end{axiom}
-
-But Maple can...
-
-\begin{axiom}
-f(x) == (1/4)*x*(-Gamma(1/4,-x**4)*Gamma(3/4)+%pi*sqrt(2))/((-x**4)**(1/4)*Gamma(3/4))
-D(f(x),x)
-\end{axiom}
-
-From kratt6 Fri Apr 28 16:34:16 -0500 2006
-From: kratt6
-Date: Fri, 28 Apr 2006 16:34:16 -0500
-Subject: Axiom cannot integrate 'e^(4*x)'
-Message-ID: <20060428163416-0500@wiki.axiom-developer.org>
-
-This is not a big surprise: note that 'Gamma(x,y)' is not an elementary function.
-
-Martin
-
-
-From unknown Thu May 18 11:30:21 -0500 2006
-From: unknown
-Date: Thu, 18 May 2006 11:30:21 -0500
-Subject:
-Message-ID: <20060518113021-0500@wiki.axiom-developer.org>
-
-This is both obviously wrong since the integrand is a positive function:
-\begin{axiom}
-integrate(1/(1+x^4),x=%minusInfinity..%plusInfinity)
-numeric(integrate(1/(1+x^4),x=0..1))
-\end{axiom}
-
-
-From unknown Wed May 24 04:31:40 -0500 2006
-From: unknown
-Date: Wed, 24 May 2006 04:31:40 -0500
-Subject:
-Message-ID: <20060524043140-0500@wiki.axiom-developer.org>
-
-\begin{axiom}
-)clear co
-n := 32
-
-y : FARRAY INT := new(n,1)
-
-n0 := n
-n1 := sum(x^1, x=0..n-1)
-n2 := sum(x^2, x=0..n-1)
-n3 := sum(x^3, x=0..n-1)
-n4 := sum(x^4, x=0..n-1)
-
-A := matrix([[n4, n3, n2],_
-            [n3, n2, n1],_
-            [n2, n1, n0]])
-
-X := vector([x1, x2, x3])
-B := vector([sum(x^2* u, x=0..n-1),_
-       sum(x*   v, x=0..n-1),_
-       sum(     w, x=0..n-1)])
-
-solve([A * X = B], [x1, x2, x3])
-\end{axiom}
-
-
-From unknown Tue May 30 23:51:26 -0500 2006
-From: unknown
-Date: Tue, 30 May 2006 23:51:26 -0500
-Subject: can this be correct?
-Message-ID: <20060530235126-0500@wiki.axiom-developer.org>
-
-\begin{axiom}
-integrate(1/((x+t)*sqrt(1+(x*t)**2)),t=0..%plusInfinity,"noPole")
-subst(%,x=1)
-integrate(1/((1+t)*sqrt(1+(1*t)**2)),t=0..%plusInfinity,"noPole")
-simplify(%-subst((asinh(x^2)+asinh(1/x^2))/sqrt(1+x^4),x=1))
-%::Expression Float
-\end{axiom}
-
-
-From unknown Mon Jul 3 02:07:02 -0500 2006
-From: unknown
-Date: Mon, 03 Jul 2006 02:07:02 -0500
-Subject:
-Message-ID: <20060703020702-0500@wiki.axiom-developer.org>
-
-\begin{axiom}
-a := matrix([ [-1,0,0,0,1,0], [0,1,0,0,0,0], [0,0,2,0,0,-2], [0,0,0,4,0,0], [0,0,0,0,3,0], [0,0,-3,0,0,3]])
-determinant(a)
-inverse(a)
-\end{axiom}
-
-
-From unknown Fri Jul 7 11:54:52 -0500 2006
-From: unknown
-Date: Fri, 07 Jul 2006 11:54:52 -0500
-Subject:
-Message-ID: <20060707115452-0500@wiki.axiom-developer.org>
-
-a := matrix([ [-3,1,1,1], [1,1,1,1], [1,1,1,1], [1,1,1,1]])
-
-
-From unknown Fri Jul 7 13:24:57 -0500 2006
-From: unknown
-Date: Fri, 07 Jul 2006 13:24:57 -0500
-Subject:
-Message-ID: <20060707132457-0500@wiki.axiom-developer.org>
-
-\begin{axiom}
-As := matrix([ [-3,1,1,1], [1,1,1,1], [1,1,1,1], [1,1,1,1]])
-A := subMatrix(As, 2,4,2,4)
-ob := orthonormalBasis(A)
-P : Matrix(Expression Integer) := new(3,3,0)
-setsubMatrix!(P,1,1,ob.3)
-setsubMatrix!(P,1,2,ob.1)
-setsubMatrix!(P,1,3,ob.2)
-Pt := transpose(P)
-Ps : Matrix(Expression Integer) := new(4,4,0)
-Ps(1,1) := 1
-setsubMatrix!(Ps,2,2,P)
-PsT := transpose(Ps)
-PsTAsPs := PsT * As * Ps
-b1 := PsTAsPs(2,1)
-l1 := PsTAsPs(2,2)
-Us : Matrix(Expression Integer) := new(4,4,0)
-Us(1,1) := 1
-Us(2,2) := 1
-Us(3,3) := 1
-Us(4,4) := 1
-Us(2,1) := -b1 / l1
-PsUs := Ps * Us
-PsUsT := transpose(PsUs)
-PsUsTAsPsUs := PsUsT * As * PsUs
-C := inverse(PsUs)
-c := PsUsTAsPsUs(1,1)
-gQ := PsUsTAsPsUs / c
-x1 := transpose(matrix([[1,2,3,4]]))
-v1 := transpose(x1) * As * x1
-x2 := C * x1
-v2 := transpose(x2) * PsUsTAsPsUs * x2
-\end{axiom}
-
-
-From unknown Tue Aug 1 02:17:12 -0500 2006
-From: unknown
-Date: Tue, 01 Aug 2006 02:17:12 -0500
-Subject: graphics
-Message-ID: <20060801021712-0500@wiki.axiom-developer.org>
-
-\begin{axiom}
-draw(y**2/2+(x**2-1)**2/4-1=0, x,y, range ==[-2..2, -1..1])
-\end{axiom}
-
-
-From greg Sat Feb 3 09:35:50 -0600 2007
-From: greg
-Date: Sat, 03 Feb 2007 09:35:50 -0600
-Subject: series test
-Message-ID: <20070203093550-0600@wiki.axiom-developer.org>
-
-\begin{axiom}
-f1 := taylor(1 - x**2,x = 0)
-asin f1
-sin %
-\end{axiom}
-
-SandboxMSkuce
-\begin{axiom}
-1+1
-\end{axiom}
-
-From jhnbk Fri Jun 8 03:50:35 -0500 2007
-From: jhnbk
-Date: Fri, 08 Jun 2007 03:50:35 -0500
-Subject: integration
-Message-ID: <20070608035035-0500@wiki.axiom-developer.org>
-
-\begin{axiom}
-integrate((x-1)/log(x), x)
-integrate(x*exp(x)*sin(x),x)
-\end{axiom}
-
-From daneshpajouh Sat Jun 16 07:00:00 -0500 2007
-From: daneshpajouh
-Date: Sat, 16 Jun 2007 07:00:00 -0500
-Subject: Working With Lists
-Message-ID: <20070616070000-0500@wiki.axiom-developer.org>
-
-\begin{axiom}
-[p for p in primes(2,1000)|(p rem 16)=1]
-[p**2+1 for p in primes(2,100)]
-\end{axiom}
-
-\begin {axiom}
-integrate (2x^2 + 2x, x)
-\end {axiom}
-
-
-From vv Sat Jul 28 14:00:27 -0500 2007
-From: vv
-Date: Sat, 28 Jul 2007 14:00:27 -0500
-Subject: is it error?
-Message-ID: <20070728140027-0500@wiki.axiom-developer.org>
-
-\begin{axiom}
-\end{axiom}
-
-Is it error?
-
-
-From pbwagner Mon Sep 10 13:01:48 -0500 2007
-From: pbwagner
-Date: Mon, 10 Sep 2007 13:01:48 -0500
-Subject: (better) example (with axiom markers this time) ;-)
-Message-ID: <20070910130148-0500@wiki.axiom-developer.org>
-
-\begin{axiom}
-integrate(log(log(x)),x)
-\end{axiom}
-



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[SandBox Aldor Foreign]?
Using Aldor to call external C routines
[SandBox Aldor Generator]?
Aldor defines a generator for type Vector
[SandBox Aldor Semantics]?
exports and constants
[SandBox Aldor Sieve]?
A prime number sieve in Aldor to count primes <= n.
[SandBox Aldor Testing]?
Using Aldor to write Axiom library routines
[SandBox Arrays]?
How fast is array access in Axiom?
[SandBox Axiom Syntax]?
Syntax of if then else

[SandboxBiblography]?

[SandBox Boolean]?
evaluating Boolean expressions and conditions
[SandBox Cast]?
Meaning and use of pretend vs. strong typing
[SandBox Categorical Relativity]?
Special relativity without the Lorentz group
[SandBox Category of Graphs]?
Graph theory in Axiom
[SandBox CL-WEB]?
Tangle operation for literate programming implemented in Common Lisp
[SandBox Combinat]?
A{ld,xi}o{r,m}Combinat
[SandBox Content MathML]?
Content vs. presentation MathML?

SandBoxCS224?

[SandBox Direct Product]?
A x B
[SandBox DistributedExpression]?
expression in sum-of-products form
[SandBox Domains and Types]?
What is the difference?
[AxiomEmacsMode]?
Beginnings of an Emacs mode for Axiom based off of Jay's work and others
[SandBox Embeded PDF]?
pdf format documents can be displayed inline
[SandBox EndPaper]?
Algebra and Data Structure Hierarchy (lattice) diagrams
[SandBox Folding]?
experiments with DHTML, javascript, etc.
[SandBox Functions]?
How do they work?
[SandBox Functors]?
What are they? In Axiom functors are also called domain constructors.
[SandBox Gamma]?
Numerical evaluation of the incomplete Gamma function
[SandBox GuessingSequence]?
Guessing integer sequences
[SandBox Integration]?
Examples of integration in Axiom and Reduce
[SandBox Kernel]?
What is a "kernel"?

[SandBox kaveh]?

[SandBox LaTeX]?
LaTeX? commands allowed in MathAction?
[SandBox Lisp]?
Using Lisp in Axiom
[SandBox Manip]?
expression manipulations
[SandBox Manipulating Domains]?
testing the domain of an expression
[SandBox Mapping]?
A->B is a type in Axiom

[MathMLFormat]?

[SandBox Matrix]?
Examples of working with matrices in Axiom
[SandBox Maxima]?
Testing the Maxima interface
[SandBox Monoid]?
Rings and things
[SandBox Monoid Extend]?
Martin Rubey's beautiful idea about using extend to add a category to a previously defined domain.
[SandBox Noncommutative Polynomials]?
XPOLY and friends
[SandBox Numerical Integration]?
Simpson method
[SandBox NNI]?
NonNegative? Integer without using SubDomain?
[SandBox Pamphlet]?
[Literate Programming]? support on MathAction?
[SandBoxPartialFraction]?
Trigonometric expansion example
SandBoxPfaffian?
Computing the Pfaffian of a square matrix
[SandBox Polymake]?
an interface between Axiom and PolyMake?
[SandBox Polynomials]?
Axiom's polynomial domains are certainly rich and complex!
[SandBox ProblemSolving]?
Test page for educational purposes
[SandBox Qubic]?
Solving cubic polynomials
[SandBox Reduce And MathML]?
Reduce can use MathML? for both input and output
[SandBox Reflection in Aldor]?
a reflection framework
[SandBoxRelativeVelocity]?
Slides for IARD 2006: Addition of Relative Velocites is Associative
[SandBox RenameTitle]?
trying to re-create a crash due to renaming pages
[SandBox Sage]?
This is a test of Sage in MathAction?
[SandBox Shortcoming]?
Implementation of solve
[SandBox Solve]?
Solving equations
[SandBox Statistics]?
calculating statistics in Axiom
[SandBox SubDomain]?
What is a SubDomain??
[SandBox Tail Recursion]?
When does Axiom replace recursion with iteration?
[SandBox Text Files]?
How to access text files in Axiom
[SandBox Trace Analysed]?
Tracing can affect output of 1::EXPR INT or 1::FRAC INT
[SandBox Tuples Products and Records]?
Basic structured data types in Axiom
[SandBox Units and Dimensions]?
Scientific units and dimensions
Domain construction
[SandBox Speed]?
Compilation speed

[SandBox Zero]?

[SandBox Axiom Strengths]?

SandBoxJohn2?
Experiments with matrices and various other stuff
SandBox2?
Experiments
SandBox3?
Experiments
SandBox4?
Experiments
SandBox5?
Experiments with GraphViz? and StructuredTables?
SandBox6?
Differential Equations etc.
[SandBox7]?
[SandBox8]?
Here you can create your own SandBox?.
[SandBox9]?
Experiments with JET Bundles
[SandBox10]?
Miscellaneous

[SandBox11]?

[SandBox DoOps]?
used to run Axiom without actually have to have it installed!

[SandBoxKMG]?

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Examples

Here is a simple Axiom command:

    \begin{axiom}
integrate(1/(a+z^3), z=0..1,"noPole")
\end{axiom}


axiomintegrate(1/(a+z^3), z=0..1,"noPole")
 (1)
Type: Union(f1: OrderedCompletion? Expression Integer,...)

And here is a REDUCE command:

  \begin{reduce}
int(1/(a+z^3), z,0,1);
\end{reduce}

\begin{reduce}