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Submitted by : (unknown) at: 2007-11-17T22:30:42-08:00 (9 years ago)
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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

\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)
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


In[1]:= Integrate[(1 + Tan[x])^(1/3),x]<p>                                                          1/3
                        3     6     Log[-#1 + (1 + Tan[x]<a class=?) ] #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




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