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last edited 9 years ago by test1 |
Edit detail for #346 Returns formal integration sign for (1 + tan(x))^(1/3), which is elementary revision 2 of 4
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Editor: test1
Time: 2014/04/15 17:55:49 GMT+0 |
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changed: - \begin{axiom} integrate((1 + tan(x))^(1/3), x) \end{axiom}
This integral is elementary, but AXIOM returns the input as a formal integral. I thought this indicated a non-elementary result...
integrate((1 + tan(x))^(1/3),x)
![\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)}}") | (1) |
?) ] #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