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 Submitted by : (unknown) at: 2007-11-17T22:30:42-08:00 (11 years ago) Name : Axiom Version : default friCAS-20090114 Axiom-20050901 OpenAxiom-20091012 OpenAxiom-20110220 OpenAxiom-Release-141 Category : Axiom Aldor Interface Axiom Compiler Axiom Library Axiom Interpreter Axiom Documentation Axiom User Interface building Axiom from source lisp system MathAction Doyen CD Reduce Axiom on Windows Axiom on Linux Severity : critical serious normal minor wishlist Status : open closed rejected not reproducible fix proposed fixed somewhere duplicate need more info Optional subject :   Optional comment :

This integral is elementary, but AXIOM returns the input as a formal integral. I thought this indicated a non-elementary result...

Messy, but works:

fricas
)set output algebra on
integrate((1 + tan(x))^(1/3), x)
(1)
1
2atan(--------)
+-+
6+-+          \|2  - 1
8\|2 sin(---------------)
3
*
atan
1
2atan(--------)
+-+
6+-+          \|2  - 1
\|2 cos(---------------)
3
/
ROOT
+---------------+2
|sin(x) + cos(x)
3|---------------
\|     cos(x)
+
1
2atan(--------)
+-+        +---------------+
6+-+          \|2  - 1    |sin(x) + cos(x)   6+-+2
2\|2 sin(---------------) 3|--------------- + \|2
3         \|     cos(x)
+
1
2atan(--------)
+---------------+                  +-+
|sin(x) + cos(x)    6+-+          \|2  - 1
3|---------------  + \|2 sin(---------------)
\|     cos(x)                       3
+
1                               1
2atan(--------)                 2atan(--------)
+-+                             +-+
6+-+          \|2  - 1      6+-+ +-+          \|2  - 1
(- 4\|2 sin(---------------) - 4\|2 \|3 cos(---------------))
3                               3
*
atan
1                          1
2atan(--------)            2atan(--------)
+-+                        +-+
6+-+ +-+          \|2  - 1     6+-+          \|2  - 1
\|2 \|3 sin(---------------) - \|2 cos(---------------)
3                          3
/
ROOT
+---------------+2
|sin(x) + cos(x)
4 3|---------------
\|     cos(x)
+
1
2atan(--------)
+-+
6+-+          \|2  - 1
- 4\|2 sin(---------------)
3
+
1
2atan(--------)
+-+
6+-+ +-+          \|2  - 1
- 4\|2 \|3 cos(---------------)
3
*
+---------------+
|sin(x) + cos(x)
3|---------------
\|     cos(x)
+
6+-+2
4\|2
+
1
2atan(--------)
+---------------+                 +-+
|sin(x) + cos(x)   6+-+          \|2  - 1
2 3|--------------- - \|2 sin(---------------)
\|     cos(x)                      3
+
1
2atan(--------)
+-+
6+-+ +-+          \|2  - 1
- \|2 \|3 cos(---------------)
3
+
1                               1
2atan(--------)                 2atan(--------)
+-+                             +-+
6+-+          \|2  - 1      6+-+ +-+          \|2  - 1
(4\|2 sin(---------------) - 4\|2 \|3 cos(---------------))
3                               3
*
atan
1                          1
2atan(--------)            2atan(--------)
+-+                        +-+
6+-+ +-+          \|2  - 1     6+-+          \|2  - 1
\|2 \|3 sin(---------------) + \|2 cos(---------------)
3                          3
/
ROOT
+---------------+2
|sin(x) + cos(x)
4 3|---------------
\|     cos(x)
+
1
2atan(--------)
+-+
6+-+          \|2  - 1
- 4\|2 sin(---------------)
3
+
1
2atan(--------)
+-+
6+-+ +-+          \|2  - 1
4\|2 \|3 cos(---------------)
3
*
+---------------+
|sin(x) + cos(x)
3|---------------
\|     cos(x)
+
6+-+2
4\|2
+
1
2atan(--------)
+---------------+                 +-+
|sin(x) + cos(x)   6+-+          \|2  - 1
2 3|--------------- - \|2 sin(---------------)
\|     cos(x)                      3
+
1
2atan(--------)
+-+
6+-+ +-+          \|2  - 1
\|2 \|3 cos(---------------)
3
+
1                          1
2atan(--------)            2atan(--------)
+-+                        +-+
6+-+ +-+          \|2  - 1     6+-+          \|2  - 1
(- \|2 \|3 sin(---------------) - \|2 cos(---------------))
3                          3
*
log
+---------------+2
|sin(x) + cos(x)
4 3|---------------
\|     cos(x)
+
1                               1
2atan(--------)                 2atan(--------)
+-+                             +-+
6+-+          \|2  - 1      6+-+ +-+          \|2  - 1
(- 4\|2 sin(---------------) + 4\|2 \|3 cos(---------------))
3                               3
*
+---------------+
|sin(x) + cos(x)
3|---------------
\|     cos(x)
+
6+-+2
4\|2
+
1                          1
2atan(--------)            2atan(--------)
+-+                        +-+
6+-+ +-+          \|2  - 1     6+-+          \|2  - 1
(\|2 \|3 sin(---------------) - \|2 cos(---------------))
3                          3
*
log
+---------------+2
|sin(x) + cos(x)
4 3|---------------
\|     cos(x)
+
1                               1
2atan(--------)                 2atan(--------)
+-+                             +-+
6+-+          \|2  - 1      6+-+ +-+          \|2  - 1
(- 4\|2 sin(---------------) - 4\|2 \|3 cos(---------------))
3                               3
*
+---------------+
|sin(x) + cos(x)
3|---------------
\|     cos(x)
+
6+-+2
4\|2
+
1
2atan(--------)
+-+
6+-+          \|2  - 1
2\|2 cos(---------------)
3
*
log
1
2atan(--------)
+---------------+2                   +-+        +---------------+
|sin(x) + cos(x)      6+-+          \|2  - 1    |sin(x) + cos(x)
3|---------------   + 2\|2 sin(---------------) 3|---------------
\|     cos(x)                         3         \|     cos(x)
+
6+-+2
\|2
/
4
 (1)
Type: Union(Expression(Integer),...)

Is it really elementary? --kratt6, Thu, 05 Apr 2007 15:04:01 -0500 reply
I'm not so sure whether this integral really is elementary. Mathematica gives

?) ] #1 RootSum?[2 - 2 #1 + #1 & , ----------------------------- & ]? 3 -1 + #1 Out[1]?= ------------------------------------------------------------- 2

In[2]:= ?RootSum? RootSum?[f, form]? represents the sum of form[x]? for all x that satisfy the polynomial equation f[x]? == 0. " title=" In[1]:= Integrate[(1 + Tan[x]?)^(1/3),x]

1/3 3 6 Log[-#1 + (1 + Tan[x]?) ] #1 RootSum?[2 - 2 #1 + #1 & , ----------------------------- & ]? 3 -1 + #1 Out[1]?= ------------------------------------------------------------- 2

In[2]:= ?RootSum? RootSum?[f, form]? represents the sum of form[x]? for all x that satisfy the polynomial equation f[x]? == 0. " class="equation" src="images/585158006443671166-16.0px.png" align="bottom" Style="vertical-align:text-bottom" width="793" height="1123"/>

Is the function "n-th root of some univariate polynomial" elementary?

Martin

typo in title --kratt6, Thu, 05 Apr 2007 15:04:40 -0500 reply
Name: #346 Returns format integration sign for (1 + tan(x))^(1/3), which is elementary => #346 Returns formal integration sign for (1 + tan(x))^(1/3), which is elementary

Status: open => closed

 Subject:   Be Bold !! ( 14 subscribers )