MathAction #346 Returns formal integration sign for (1 + tan(x))^(1/3), which is elementary

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  - #346 Returns formal integration sign for (1 + tan(x))^(1/3), which is elementary <-- You are here.

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

(1) -> )set output algebra on

integrate((1 + tan(x))^(1/3), x)

   (1)
                        +----------+
             +---+     3| +---+
         (- \|- 3  - 1)\|\|- 1  - 1
      *
                                                  +----------+
               3+----------+     +---+      +---+3| +---+
         log(2 \|tan(x) + 1  + (\|- 3  + 1)\|- 1 \|\|- 1  - 1 )
     + 
                      +------------+
           +---+     3|   +---+
         (\|- 3  - 1)\|- \|- 1  - 1
      *
                                                  +------------+
               3+----------+     +---+      +---+3|   +---+
         log(2 \|tan(x) + 1  + (\|- 3  - 1)\|- 1 \|- \|- 1  - 1 )
     + 
                        +------------+
             +---+     3|   +---+
         (- \|- 3  - 1)\|- \|- 1  - 1
      *
                                                    +------------+
               3+----------+       +---+      +---+3|   +---+
         log(2 \|tan(x) + 1  + (- \|- 3  - 1)\|- 1 \|- \|- 1  - 1 )
     + 
                      +----------+
           +---+     3| +---+
         (\|- 3  - 1)\|\|- 1  - 1
      *
                                                    +----------+
               3+----------+       +---+      +---+3| +---+
         log(2 \|tan(x) + 1  + (- \|- 3  + 1)\|- 1 \|\|- 1  - 1 )
     + 
          +------------+                           +------------+
         3|   +---+         3+----------+    +---+3|   +---+
       2 \|- \|- 1  - 1 log(\|tan(x) + 1  + \|- 1 \|- \|- 1  - 1 )
     + 
          +----------+                           +----------+
         3| +---+         3+----------+    +---+3| +---+
       2 \|\|- 1  - 1 log(\|tan(x) + 1  - \|- 1 \|\|- 1  - 1 )
  /
     4

$\label{eq1}\frac{{{\left(-{\sqrt{- 3}}- 1 \right)}\ {\root{3}\of{{\sqrt{- 1}}- 1}}\ {\log \left({{2 \ {\root{3}\of{{\tan \left({x}\right)}+ 1}}}+{{\left({\sqrt{- 3}}+ 1 \right)}\ {\sqrt{- 1}}\ {\root{3}\of{{\sqrt{- 1}}- 1}}}}\right)}}+{{\left({\sqrt{- 3}}- 1 \right)}\ {\root{3}\of{-{\sqrt{- 1}}- 1}}\ {\log \left({{2 \ {\root{3}\of{{\tan \left({x}\right)}+ 1}}}+{{\left({\sqrt{- 3}}- 1 \right)}\ {\sqrt{- 1}}\ {\root{3}\of{-{\sqrt{- 1}}- 1}}}}\right)}}+{{\left(-{\sqrt{- 3}}- 1 \right)}\ {\root{3}\of{-{\sqrt{- 1}}- 1}}\ {\log \left({{2 \ {\root{3}\of{{\tan \left({x}\right)}+ 1}}}+{{\left(-{\sqrt{- 3}}- 1 \right)}\ {\sqrt{- 1}}\ {\root{3}\of{-{\sqrt{- 1}}- 1}}}}\right)}}+{{\left({\sqrt{- 3}}- 1 \right)}\ {\root{3}\of{{\sqrt{- 1}}- 1}}\ {\log \left({{2 \ {\root{3}\of{{\tan \left({x}\right)}+ 1}}}+{{\left(-{\sqrt{- 3}}+ 1 \right)}\ {\sqrt{- 1}}\ {\root{3}\of{{\sqrt{- 1}}- 1}}}}\right)}}+{2 \ {\root{3}\of{-{\sqrt{- 1}}- 1}}\ {\log \left({{\root{3}\of{{\tan \left({x}\right)}+ 1}}+{{\sqrt{- 1}}\ {\root{3}\of{-{\sqrt{- 1}}- 1}}}}\right)}}+{2 \ {\root{3}\of{{\sqrt{- 1}}- 1}}\ {\log \left({{\root{3}\of{{\tan \left({x}\right)}+ 1}}-{{\sqrt{- 1}}\ {\root{3}\of{{\sqrt{- 1}}- 1}}}}\right)}}}{4}$

(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

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="816" height="1056"/>

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

... --test1, Sat, 03 Jan 2015 16:16:53 +0000 reply

Status: open => closed

Subject: (replying) Be Bold !!

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