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
Type: Union(Expression(Integer),...)
I'm not so sure whether this integral really is elementary. Mathematica gives
![In[1]:= Integrate[(1 + Tan[x])^(1/3),x]<p> 1/3
3 6 Log[-#1 + (1 + Tan[x]<a class=]( "create this page")
?) ] #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
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
![\label{eq1}{\left(
\begin{array}{@{}l}
\displaystyle
{
\begin{array}{@{}l}
\displaystyle
8 \ {\root{6}\of{2}}\ {\sin \left({{2 \ {\arctan \left({1 \over{{\sqrt{2}}- 1}}\right)}}\over 3}\right)}\ \cdot
\
\
\displaystyle
{\arctan{\left({{{\root{6}\of{2}}\ {\cos \left({{2 \ {\arctan \left({1 \over{{\sqrt{2}}- 1}}\right)}}\over 3}\right)}}\over{{\sqrt{{{\root{3}\of{{{\sin \left({x}\right)}+{\cos \left({x}\right)}}\over{\cos \left({x}\right)}}}^{2}}+{2 \ {\root{6}\of{2}}\ {\sin \left({{2 \ {\arctan \left({1 \over{{\sqrt{2}}- 1}}\right)}}\over 3}\right)}\ {\root{3}\of{{{\sin \left({x}\right)}+{\cos \left({x}\right)}}\over{\cos \left({x}\right)}}}}+{{\root{6}\of{2}}^{2}}}}+{\root{3}\of{{{\sin \left({x}\right)}+{\cos \left({x}\right)}}\over{\cos \left({x}\right)}}}+{{\root{6}\of{2}}\ {\sin \left({{2 \ {\arctan \left({1 \over{{\sqrt{2}}- 1}}\right)}}\over 3}\right)}}}}\right)}}](images/2769270408676747895-16.0px.png "\label{eq1}{\left(
\begin{array}{@{}l}
\displaystyle
{
\begin{array}{@{}l}
\displaystyle
8 \ {\root{6}\of{2}}\ {\sin \left({{2 \ {\arctan \left({1 \over{{\sqrt{2}}- 1}}\right)}}\over 3}\right)}\ \cdot
\
\
\displaystyle
{\arctan{\left({{{\root{6}\of{2}}\ {\cos \left({{2 \ {\arctan \left({1 \over{{\sqrt{2}}- 1}}\right)}}\over 3}\right)}}\over{{\sqrt{{{\root{3}\of{{{\sin \left({x}\right)}+{\cos \left({x}\right)}}\over{\cos \left({x}\right)}}}^{2}}+{2 \ {\root{6}\of{2}}\ {\sin \left({{2 \ {\arctan \left({1 \over{{\sqrt{2}}- 1}}\right)}}\over 3}\right)}\ {\root{3}\of{{{\sin \left({x}\right)}+{\cos \left({x}\right)}}\over{\cos \left({x}\right)}}}}+{{\root{6}\of{2}}^{2}}}}+{\root{3}\of{{{\sin \left({x}\right)}+{\cos \left({x}\right)}}\over{\cos \left({x}\right)}}}+{{\root{6}\of{2}}\ {\sin \left({{2 \ {\arctan \left({1 \over{{\sqrt{2}}- 1}}\right)}}\over 3}\right)}}}}\right)}}")