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tan5(x)=-sqrt(3)/3 equation

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Numerical solution:

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The solution

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             ___ 
   5      -\/ 3  
tan (x) = -------
             3
tan⁡5(x)=(−1)33\tan^{5}{\left(x \right)} = \frac{\left(-1\right) \sqrt{3}}{3}
Simplify
Detail solution
Given the equation
tan⁡5(x)=(−1)33\tan^{5}{\left(x \right)} = \frac{\left(-1\right) \sqrt{3}}{3}
transform
tan⁡5(x)+33=0\tan^{5}{\left(x \right)} + \frac{\sqrt{3}}{3} = 0
tan⁡5(x)−(−1)33=0\tan^{5}{\left(x \right)} - \frac{\left(-1\right) \sqrt{3}}{3} = 0
Do replacement
w=tan⁡(x)w = \tan{\left(x \right)}
Given the equation
w5−(−1)33=0w^{5} - \frac{\left(-1\right) \sqrt{3}}{3} = 0
Because equation degree is equal to = 5 - does not contain even numbers in the numerator, then
the equation has single real root.
Get the root 5-th degree of the equation sides:
We get:
w55=−335\sqrt[5]{w^{5}} = \sqrt[5]{- \frac{\sqrt{3}}{3}}
or
w=310⋅345⋅345⋅3910⋅3910−15⋅32581w = \frac{\sqrt[10]{3} \cdot 3^{\frac{4}{5}} \cdot 3^{\frac{4}{5}} \cdot 3^{\frac{9}{10}} \cdot 3^{\frac{9}{10}} \sqrt[5]{-1} \cdot 3^{\frac{2}{5}}}{81}
Expand brackets in the right part
w = 3^1/10*3^4/5*3^4/5*3^9/10*3^9/10-1^1/5*3^2/5)/81

We get the answer: w = (-1)^(1/5)*3^(9/10)/3

All other 4 root(s) is the complex numbers.
do replacement:
z=wz = w
then the equation will be the:
z5=−33z^{5} = - \frac{\sqrt{3}}{3}
Any complex number can presented so:
z=reipz = r e^{i p}
substitute to the equation
r5e5ip=−33r^{5} e^{5 i p} = - \frac{\sqrt{3}}{3}
where
r=39103r = \frac{3^{\frac{9}{10}}}{3}
- the magnitude of the complex number
Substitute r:
e5ip=−1e^{5 i p} = -1
Using Euler’s formula, we find roots for p
isin⁡(5p)+cos⁡(5p)=−1i \sin{\left(5 p \right)} + \cos{\left(5 p \right)} = -1
so
cos⁡(5p)=−1\cos{\left(5 p \right)} = -1
and
sin⁡(5p)=0\sin{\left(5 p \right)} = 0
then
p=2πN5+π5p = \frac{2 \pi N}{5} + \frac{\pi}{5}
where N=0,1,2,3,...
Looping through the values of N and substituting p into the formula for z
Consequently, the solution will be for z:
z1=−39103z_{1} = - \frac{3^{\frac{9}{10}}}{3}
z2=391012+3910512+3910i58−583z_{2} = \frac{3^{\frac{9}{10}}}{12} + \frac{3^{\frac{9}{10}} \sqrt{5}}{12} + \frac{3^{\frac{9}{10}} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{3}
z3=−3910512+391012−39105i58−586−3910i58−586z_{3} = - \frac{3^{\frac{9}{10}} \sqrt{5}}{12} + \frac{3^{\frac{9}{10}}}{12} - \frac{3^{\frac{9}{10}} \sqrt{5} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{6} - \frac{3^{\frac{9}{10}} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{6}
z4=391012+391058−5858+583−39105i58+5812−3910i58+5812−3910i58−5812+39105i58−5812z_{4} = \frac{3^{\frac{9}{10}}}{12} + \frac{3^{\frac{9}{10}} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{3} - \frac{3^{\frac{9}{10}} \sqrt{5} i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{12} - \frac{3^{\frac{9}{10}} i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{12} - \frac{3^{\frac{9}{10}} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{12} + \frac{3^{\frac{9}{10}} \sqrt{5} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{12}
z5=−391058−5858+583+391012−3910i58−5812+3910i58+5812+39105i58−5812+39105i58+5812z_{5} = - \frac{3^{\frac{9}{10}} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{3} + \frac{3^{\frac{9}{10}}}{12} - \frac{3^{\frac{9}{10}} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{12} + \frac{3^{\frac{9}{10}} i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{12} + \frac{3^{\frac{9}{10}} \sqrt{5} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{12} + \frac{3^{\frac{9}{10}} \sqrt{5} i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{12}
do backward replacement
z=wz = w
w=zw = z

The final answer:
w1=−39103w_{1} = - \frac{3^{\frac{9}{10}}}{3}
w2=391012+3910512+3910i58−583w_{2} = \frac{3^{\frac{9}{10}}}{12} + \frac{3^{\frac{9}{10}} \sqrt{5}}{12} + \frac{3^{\frac{9}{10}} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{3}
w3=−3910512+391012−39105i58−586−3910i58−586w_{3} = - \frac{3^{\frac{9}{10}} \sqrt{5}}{12} + \frac{3^{\frac{9}{10}}}{12} - \frac{3^{\frac{9}{10}} \sqrt{5} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{6} - \frac{3^{\frac{9}{10}} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{6}
w4=391012+391058−5858+583−39105i58+5812−3910i58+5812−3910i58−5812+39105i58−5812w_{4} = \frac{3^{\frac{9}{10}}}{12} + \frac{3^{\frac{9}{10}} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{3} - \frac{3^{\frac{9}{10}} \sqrt{5} i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{12} - \frac{3^{\frac{9}{10}} i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{12} - \frac{3^{\frac{9}{10}} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{12} + \frac{3^{\frac{9}{10}} \sqrt{5} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{12}
w5=−391058−5858+583+391012−3910i58−5812+3910i58+5812+39105i58−5812+39105i58+5812w_{5} = - \frac{3^{\frac{9}{10}} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{3} + \frac{3^{\frac{9}{10}}}{12} - \frac{3^{\frac{9}{10}} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{12} + \frac{3^{\frac{9}{10}} i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{12} + \frac{3^{\frac{9}{10}} \sqrt{5} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{12} + \frac{3^{\frac{9}{10}} \sqrt{5} i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{12}
do backward replacement
tan⁡(x)=w\tan{\left(x \right)} = w
Given the equation
tan⁡(x)=w\tan{\left(x \right)} = w
- this is the simplest trigonometric equation
This equation is transformed to
x=πn+atan⁡(w)x = \pi n + \operatorname{atan}{\left(w \right)}
Or
x=πn+atan⁡(w)x = \pi n + \operatorname{atan}{\left(w \right)}
, where n - is a integer
substitute w:
x1=πn+atan⁡(w1)x_{1} = \pi n + \operatorname{atan}{\left(w_{1} \right)}
x1=πn+atan⁡(−15⋅39103)x_{1} = \pi n + \operatorname{atan}{\left(\frac{\sqrt[5]{-1} \cdot 3^{\frac{9}{10}}}{3} \right)}
x1=πn+atan⁡(−15⋅39103)x_{1} = \pi n + \operatorname{atan}{\left(\frac{\sqrt[5]{-1} \cdot 3^{\frac{9}{10}}}{3} \right)}
The graph
0-80-60-40-2020406080-100100-5000000000000000000001e21
Plot the graph f1(x)
Plot the graph f2(x)
Rapid solution [src] 
          / 9/10\
          |3    |
x1 = -atan|-----|
          \  3  /
x1=−atan⁡(39103)x_{1} = - \operatorname{atan}{\left(\frac{3^{\frac{9}{10}}}{3} \right)} 
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         |    |   9/10    9/10   ___   I*3    *  /   - - -----    I*3    *\/ 5 *  /   - - ----- ||       |    |   9/10    9/10   ___   I*3    *  /   - - -----    I*3    *\/ 5 *  /   - - ----- ||
         |    |  3       3    *\/ 5            \/    8     8                    \/    8     8   ||       |    |  3       3    *\/ 5            \/    8     8                    \/    8     8   ||
x2 = - re|atan|- ----- + ----------- + ------------------------ + ------------------------------|| - I*im|atan|- ----- + ----------- + ------------------------ + ------------------------------||
         \    \    12         12                  6                             6               //       \    \    12         12                  6                             6               //
x2=−re⁡(atan⁡(−391012+3910512+3910i58−586+39105i58−586))−iim⁡(atan⁡(−391012+3910512+3910i58−586+39105i58−586))x_{2} = - \operatorname{re}{\left(\operatorname{atan}{\left(- \frac{3^{\frac{9}{10}}}{12} + \frac{3^{\frac{9}{10}} \sqrt{5}}{12} + \frac{3^{\frac{9}{10}} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{6} + \frac{3^{\frac{9}{10}} \sqrt{5} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{6} \right)}\right)} - i \operatorname{im}{\left(\operatorname{atan}{\left(- \frac{3^{\frac{9}{10}}}{12} + \frac{3^{\frac{9}{10}} \sqrt{5}}{12} + \frac{3^{\frac{9}{10}} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{6} + \frac{3^{\frac{9}{10}} \sqrt{5} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{6} \right)}\right)} 
         /    /                     ___________      ___________                ___________                ___________                      ___________                      ___________\\       /    /                     ___________      ___________                ___________                ___________                      ___________                      ___________\\
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         |    |   9/10   3    *  /   - - ----- *  /   - + -----    I*3    *  /   - - -----    I*3    *  /   - + -----    I*3    *\/ 5 *  /   - - -----    I*3    *\/ 5 *  /   - + ----- ||       |    |   9/10   3    *  /   - - ----- *  /   - + -----    I*3    *  /   - - -----    I*3    *  /   - + -----    I*3    *\/ 5 *  /   - - -----    I*3    *\/ 5 *  /   - + ----- ||
         |    |  3             \/    8     8    \/    8     8              \/    8     8              \/    8     8                    \/    8     8                    \/    8     8   ||       |    |  3             \/    8     8    \/    8     8              \/    8     8              \/    8     8                    \/    8     8                    \/    8     8   ||
x3 = - re|atan|- ----- - --------------------------------------- + ------------------------ + ------------------------ - ------------------------------ + ------------------------------|| - I*im|atan|- ----- - --------------------------------------- + ------------------------ + ------------------------ - ------------------------------ + ------------------------------||
         \    \    12                       3                                 12                         12                            12                               12              //       \    \    12                       3                                 12                         12                            12                               12              //
x3=−re⁡(atan⁡(−391058−5858+583−391012−39105i58−5812+3910i58−5812+3910i58+5812+39105i58+5812))−iim⁡(atan⁡(−391058−5858+583−391012−39105i58−5812+3910i58−5812+3910i58+5812+39105i58+5812))x_{3} = - \operatorname{re}{\left(\operatorname{atan}{\left(- \frac{3^{\frac{9}{10}} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{3} - \frac{3^{\frac{9}{10}}}{12} - \frac{3^{\frac{9}{10}} \sqrt{5} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{12} + \frac{3^{\frac{9}{10}} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{12} + \frac{3^{\frac{9}{10}} i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{12} + \frac{3^{\frac{9}{10}} \sqrt{5} i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{12} \right)}\right)} - i \operatorname{im}{\left(\operatorname{atan}{\left(- \frac{3^{\frac{9}{10}} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{3} - \frac{3^{\frac{9}{10}}}{12} - \frac{3^{\frac{9}{10}} \sqrt{5} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{12} + \frac{3^{\frac{9}{10}} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{12} + \frac{3^{\frac{9}{10}} i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{12} + \frac{3^{\frac{9}{10}} \sqrt{5} i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{12} \right)}\right)} 
         /    /                   ___________      ___________                ___________                ___________                      ___________                      ___________\\     /    /                   ___________      ___________                ___________                ___________                      ___________                      ___________\\
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         |    | 9/10   3    *  /   - - ----- *  /   - + -----    I*3    *  /   - - -----    I*3    *  /   - + -----    I*3    *\/ 5 *  /   - - -----    I*3    *\/ 5 *  /   - + ----- ||     |    | 9/10   3    *  /   - - ----- *  /   - + -----    I*3    *  /   - - -----    I*3    *  /   - + -----    I*3    *\/ 5 *  /   - - -----    I*3    *\/ 5 *  /   - + ----- ||
         |    |3             \/    8     8    \/    8     8              \/    8     8              \/    8     8                    \/    8     8                    \/    8     8   ||     |    |3             \/    8     8    \/    8     8              \/    8     8              \/    8     8                    \/    8     8                    \/    8     8   ||
x4 = I*im|atan|----- - --------------------------------------- - ------------------------ + ------------------------ + ------------------------------ + ------------------------------|| + re|atan|----- - --------------------------------------- - ------------------------ + ------------------------ + ------------------------------ + ------------------------------||
         \    \  12                       3                                 12                         12                            12                               12              //     \    \  12                       3                                 12                         12                            12                               12              //
x4=re⁡(atan⁡(−391058−5858+583+391012−3910i58−5812+3910i58+5812+39105i58−5812+39105i58+5812))+iim⁡(atan⁡(−391058−5858+583+391012−3910i58−5812+3910i58+5812+39105i58−5812+39105i58+5812))x_{4} = \operatorname{re}{\left(\operatorname{atan}{\left(- \frac{3^{\frac{9}{10}} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{3} + \frac{3^{\frac{9}{10}}}{12} - \frac{3^{\frac{9}{10}} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{12} + \frac{3^{\frac{9}{10}} i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{12} + \frac{3^{\frac{9}{10}} \sqrt{5} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{12} + \frac{3^{\frac{9}{10}} \sqrt{5} i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{12} \right)}\right)} + i \operatorname{im}{\left(\operatorname{atan}{\left(- \frac{3^{\frac{9}{10}} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{3} + \frac{3^{\frac{9}{10}}}{12} - \frac{3^{\frac{9}{10}} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{12} + \frac{3^{\frac{9}{10}} i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{12} + \frac{3^{\frac{9}{10}} \sqrt{5} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{12} + \frac{3^{\frac{9}{10}} \sqrt{5} i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{12} \right)}\right)} 
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         |    | 9/10    9/10   ___   I*3    *  /   - - ----- ||     |    | 9/10    9/10   ___   I*3    *  /   - - ----- ||
         |    |3       3    *\/ 5            \/    8     8   ||     |    |3       3    *\/ 5            \/    8     8   ||
x5 = I*im|atan|----- + ----------- + ------------------------|| + re|atan|----- + ----------- + ------------------------||
         \    \  12         12                  3            //     \    \  12         12                  3            //
x5=re⁡(atan⁡(391012+3910512+3910i58−583))+iim⁡(atan⁡(391012+3910512+3910i58−583))x_{5} = \operatorname{re}{\left(\operatorname{atan}{\left(\frac{3^{\frac{9}{10}}}{12} + \frac{3^{\frac{9}{10}} \sqrt{5}}{12} + \frac{3^{\frac{9}{10}} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{3} \right)}\right)} + i \operatorname{im}{\left(\operatorname{atan}{\left(\frac{3^{\frac{9}{10}}}{12} + \frac{3^{\frac{9}{10}} \sqrt{5}}{12} + \frac{3^{\frac{9}{10}} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{3} \right)}\right)} 
x5 = re(atan(3^(9/10)/12 + 3^(9/10)*sqrt(5)/12 + 3^(9/10)*i*sqrt(5/8 - sqrt(5)/8)/3)) + i*im(atan(3^(9/10)/12 + 3^(9/10)*sqrt(5)/12 + 3^(9/10)*i*sqrt(5/8 - sqrt(5)/8)/3))
Sum and product of roots [src] 
sum
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      / 9/10\       |    |   9/10    9/10   ___   I*3    *  /   - - -----    I*3    *\/ 5 *  /   - - ----- ||       |    |   9/10    9/10   ___   I*3    *  /   - - -----    I*3    *\/ 5 *  /   - - ----- ||       |    |   9/10   3    *  /   - - ----- *  /   - + -----    I*3    *  /   - - -----    I*3    *  /   - + -----    I*3    *\/ 5 *  /   - - -----    I*3    *\/ 5 *  /   - + ----- ||       |    |   9/10   3    *  /   - - ----- *  /   - + -----    I*3    *  /   - - -----    I*3    *  /   - + -----    I*3    *\/ 5 *  /   - - -----    I*3    *\/ 5 *  /   - + ----- ||       |    | 9/10   3    *  /   - - ----- *  /   - + -----    I*3    *  /   - - -----    I*3    *  /   - + -----    I*3    *\/ 5 *  /   - - -----    I*3    *\/ 5 *  /   - + ----- ||     |    | 9/10   3    *  /   - - ----- *  /   - + -----    I*3    *  /   - - -----    I*3    *  /   - + -----    I*3    *\/ 5 *  /   - - -----    I*3    *\/ 5 *  /   - + ----- ||       |    | 9/10    9/10   ___   I*3    *  /   - - ----- ||     |    | 9/10    9/10   ___   I*3    *  /   - - ----- ||
      |3    |       |    |  3       3    *\/ 5            \/    8     8                    \/    8     8   ||       |    |  3       3    *\/ 5            \/    8     8                    \/    8     8   ||       |    |  3             \/    8     8    \/    8     8              \/    8     8              \/    8     8                    \/    8     8                    \/    8     8   ||       |    |  3             \/    8     8    \/    8     8              \/    8     8              \/    8     8                    \/    8     8                    \/    8     8   ||       |    |3             \/    8     8    \/    8     8              \/    8     8              \/    8     8                    \/    8     8                    \/    8     8   ||     |    |3             \/    8     8    \/    8     8              \/    8     8              \/    8     8                    \/    8     8                    \/    8     8   ||       |    |3       3    *\/ 5            \/    8     8   ||     |    |3       3    *\/ 5            \/    8     8   ||
- atan|-----| + - re|atan|- ----- + ----------- + ------------------------ + ------------------------------|| - I*im|atan|- ----- + ----------- + ------------------------ + ------------------------------|| + - re|atan|- ----- - --------------------------------------- + ------------------------ + ------------------------ - ------------------------------ + ------------------------------|| - I*im|atan|- ----- - --------------------------------------- + ------------------------ + ------------------------ - ------------------------------ + ------------------------------|| + I*im|atan|----- - --------------------------------------- - ------------------------ + ------------------------ + ------------------------------ + ------------------------------|| + re|atan|----- - --------------------------------------- - ------------------------ + ------------------------ + ------------------------------ + ------------------------------|| + I*im|atan|----- + ----------- + ------------------------|| + re|atan|----- + ----------- + ------------------------||
      \  3  /       \    \    12         12                  6                             6               //       \    \    12         12                  6                             6               //       \    \    12                       3                                 12                         12                            12                               12              //       \    \    12                       3                                 12                         12                            12                               12              //       \    \  12                       3                                 12                         12                            12                               12              //     \    \  12                       3                                 12                         12                            12                               12              //       \    \  12         12                  3            //     \    \  12         12                  3            //
(((−atan⁡(39103)+(−re⁡(atan⁡(−391012+3910512+3910i58−586+39105i58−586))−iim⁡(atan⁡(−391012+3910512+3910i58−586+39105i58−586))))+(−re⁡(atan⁡(−391058−5858+583−391012−39105i58−5812+3910i58−5812+3910i58+5812+39105i58+5812))−iim⁡(atan⁡(−391058−5858+583−391012−39105i58−5812+3910i58−5812+3910i58+5812+39105i58+5812))))+(re⁡(atan⁡(−391058−5858+583+391012−3910i58−5812+3910i58+5812+39105i58−5812+39105i58+5812))+iim⁡(atan⁡(−391058−5858+583+391012−3910i58−5812+3910i58+5812+39105i58−5812+39105i58+5812))))+(re⁡(atan⁡(391012+3910512+3910i58−583))+iim⁡(atan⁡(391012+3910512+3910i58−583)))\left(\left(\left(- \operatorname{atan}{\left(\frac{3^{\frac{9}{10}}}{3} \right)} + \left(- \operatorname{re}{\left(\operatorname{atan}{\left(- \frac{3^{\frac{9}{10}}}{12} + \frac{3^{\frac{9}{10}} \sqrt{5}}{12} + \frac{3^{\frac{9}{10}} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{6} + \frac{3^{\frac{9}{10}} \sqrt{5} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{6} \right)}\right)} - i \operatorname{im}{\left(\operatorname{atan}{\left(- \frac{3^{\frac{9}{10}}}{12} + \frac{3^{\frac{9}{10}} \sqrt{5}}{12} + \frac{3^{\frac{9}{10}} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{6} + \frac{3^{\frac{9}{10}} \sqrt{5} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{6} \right)}\right)}\right)\right) + \left(- \operatorname{re}{\left(\operatorname{atan}{\left(- \frac{3^{\frac{9}{10}} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{3} - \frac{3^{\frac{9}{10}}}{12} - \frac{3^{\frac{9}{10}} \sqrt{5} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{12} + \frac{3^{\frac{9}{10}} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{12} + \frac{3^{\frac{9}{10}} i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{12} + \frac{3^{\frac{9}{10}} \sqrt{5} i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{12} \right)}\right)} - i \operatorname{im}{\left(\operatorname{atan}{\left(- \frac{3^{\frac{9}{10}} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{3} - \frac{3^{\frac{9}{10}}}{12} - \frac{3^{\frac{9}{10}} \sqrt{5} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{12} + \frac{3^{\frac{9}{10}} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{12} + \frac{3^{\frac{9}{10}} i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{12} + \frac{3^{\frac{9}{10}} \sqrt{5} i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{12} \right)}\right)}\right)\right) + \left(\operatorname{re}{\left(\operatorname{atan}{\left(- \frac{3^{\frac{9}{10}} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{3} + \frac{3^{\frac{9}{10}}}{12} - \frac{3^{\frac{9}{10}} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{12} + \frac{3^{\frac{9}{10}} i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{12} + \frac{3^{\frac{9}{10}} \sqrt{5} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{12} + \frac{3^{\frac{9}{10}} \sqrt{5} i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{12} \right)}\right)} + i \operatorname{im}{\left(\operatorname{atan}{\left(- \frac{3^{\frac{9}{10}} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{3} + \frac{3^{\frac{9}{10}}}{12} - \frac{3^{\frac{9}{10}} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{12} + \frac{3^{\frac{9}{10}} i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{12} + \frac{3^{\frac{9}{10}} \sqrt{5} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{12} + \frac{3^{\frac{9}{10}} \sqrt{5} i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{12} \right)}\right)}\right)\right) + \left(\operatorname{re}{\left(\operatorname{atan}{\left(\frac{3^{\frac{9}{10}}}{12} + \frac{3^{\frac{9}{10}} \sqrt{5}}{12} + \frac{3^{\frac{9}{10}} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{3} \right)}\right)} + i \operatorname{im}{\left(\operatorname{atan}{\left(\frac{3^{\frac{9}{10}}}{12} + \frac{3^{\frac{9}{10}} \sqrt{5}}{12} + \frac{3^{\frac{9}{10}} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{3} \right)}\right)}\right) 
=
                  /    /                                     ___________                      ___________\\     /    /                     ___________      ___________                ___________                ___________                      ___________                      ___________\\       /    /                                   ___________\\       /    /                   ___________      ___________                ___________                ___________                      ___________                      ___________\\       /    /                                     ___________                      ___________\\       /    /                     ___________      ___________                ___________                ___________                      ___________                      ___________\\     /    /                                   ___________\\     /    /                   ___________      ___________                ___________                ___________                      ___________                      ___________\\
                  |    |                                    /       ___                      /       ___ ||     |    |                    /       ___      /       ___                /       ___                /       ___                      /       ___                      /       ___ ||       |    |                                  /       ___ ||       |    |                  /       ___      /       ___                /       ___                /       ___                      /       ___                      /       ___ ||       |    |                                    /       ___                      /       ___ ||       |    |                    /       ___      /       ___                /       ___                /       ___                      /       ___                      /       ___ ||     |    |                                  /       ___ ||     |    |                  /       ___      /       ___                /       ___                /       ___                      /       ___                      /       ___ ||
                  |    |                           9/10    /  5   \/ 5        9/10   ___    /  5   \/ 5  ||     |    |           9/10    /  5   \/ 5      /  5   \/ 5        9/10    /  5   \/ 5        9/10    /  5   \/ 5        9/10   ___    /  5   \/ 5        9/10   ___    /  5   \/ 5  ||       |    |                         9/10    /  5   \/ 5  ||       |    |         9/10    /  5   \/ 5      /  5   \/ 5        9/10    /  5   \/ 5        9/10    /  5   \/ 5        9/10   ___    /  5   \/ 5        9/10   ___    /  5   \/ 5  ||       |    |                           9/10    /  5   \/ 5        9/10   ___    /  5   \/ 5  ||       |    |           9/10    /  5   \/ 5      /  5   \/ 5        9/10    /  5   \/ 5        9/10    /  5   \/ 5        9/10   ___    /  5   \/ 5        9/10   ___    /  5   \/ 5  ||     |    |                         9/10    /  5   \/ 5  ||     |    |         9/10    /  5   \/ 5      /  5   \/ 5        9/10    /  5   \/ 5        9/10    /  5   \/ 5        9/10   ___    /  5   \/ 5        9/10   ___    /  5   \/ 5  ||
      / 9/10\     |    |   9/10    9/10   ___   I*3    *  /   - - -----    I*3    *\/ 5 *  /   - - ----- ||     |    |   9/10   3    *  /   - - ----- *  /   - + -----    I*3    *  /   - - -----    I*3    *  /   - + -----    I*3    *\/ 5 *  /   - - -----    I*3    *\/ 5 *  /   - + ----- ||       |    | 9/10    9/10   ___   I*3    *  /   - - ----- ||       |    | 9/10   3    *  /   - - ----- *  /   - + -----    I*3    *  /   - - -----    I*3    *  /   - + -----    I*3    *\/ 5 *  /   - - -----    I*3    *\/ 5 *  /   - + ----- ||       |    |   9/10    9/10   ___   I*3    *  /   - - -----    I*3    *\/ 5 *  /   - - ----- ||       |    |   9/10   3    *  /   - - ----- *  /   - + -----    I*3    *  /   - - -----    I*3    *  /   - + -----    I*3    *\/ 5 *  /   - - -----    I*3    *\/ 5 *  /   - + ----- ||     |    | 9/10    9/10   ___   I*3    *  /   - - ----- ||     |    | 9/10   3    *  /   - - ----- *  /   - + -----    I*3    *  /   - - -----    I*3    *  /   - + -----    I*3    *\/ 5 *  /   - - -----    I*3    *\/ 5 *  /   - + ----- ||
      |3    |     |    |  3       3    *\/ 5            \/    8     8                    \/    8     8   ||     |    |  3             \/    8     8    \/    8     8              \/    8     8              \/    8     8                    \/    8     8                    \/    8     8   ||       |    |3       3    *\/ 5            \/    8     8   ||       |    |3             \/    8     8    \/    8     8              \/    8     8              \/    8     8                    \/    8     8                    \/    8     8   ||       |    |  3       3    *\/ 5            \/    8     8                    \/    8     8   ||       |    |  3             \/    8     8    \/    8     8              \/    8     8              \/    8     8                    \/    8     8                    \/    8     8   ||     |    |3       3    *\/ 5            \/    8     8   ||     |    |3             \/    8     8    \/    8     8              \/    8     8              \/    8     8                    \/    8     8                    \/    8     8   ||
- atan|-----| - re|atan|- ----- + ----------- + ------------------------ + ------------------------------|| - re|atan|- ----- - --------------------------------------- + ------------------------ + ------------------------ - ------------------------------ + ------------------------------|| + I*im|atan|----- + ----------- + ------------------------|| + I*im|atan|----- - --------------------------------------- - ------------------------ + ------------------------ + ------------------------------ + ------------------------------|| - I*im|atan|- ----- + ----------- + ------------------------ + ------------------------------|| - I*im|atan|- ----- - --------------------------------------- + ------------------------ + ------------------------ - ------------------------------ + ------------------------------|| + re|atan|----- + ----------- + ------------------------|| + re|atan|----- - --------------------------------------- - ------------------------ + ------------------------ + ------------------------------ + ------------------------------||
      \  3  /     \    \    12         12                  6                             6               //     \    \    12                       3                                 12                         12                            12                               12              //       \    \  12         12                  3            //       \    \  12                       3                                 12                         12                            12                               12              //       \    \    12         12                  6                             6               //       \    \    12                       3                                 12                         12                            12                               12              //     \    \  12         12                  3            //     \    \  12                       3                                 12                         12                            12                               12              //
−atan⁡(39103)−re⁡(atan⁡(−391012+3910512+3910i58−586+39105i58−586))+re⁡(atan⁡(−391058−5858+583+391012−3910i58−5812+3910i58+5812+39105i58−5812+39105i58+5812))−re⁡(atan⁡(−391058−5858+583−391012−39105i58−5812+3910i58−5812+3910i58+5812+39105i58+5812))+re⁡(atan⁡(391012+3910512+3910i58−583))−iim⁡(atan⁡(−391012+3910512+3910i58−586+39105i58−586))−iim⁡(atan⁡(−391058−5858+583−391012−39105i58−5812+3910i58−5812+3910i58+5812+39105i58+5812))+iim⁡(atan⁡(391012+3910512+3910i58−583))+iim⁡(atan⁡(−391058−5858+583+391012−3910i58−5812+3910i58+5812+39105i58−5812+39105i58+5812))- \operatorname{atan}{\left(\frac{3^{\frac{9}{10}}}{3} \right)} - \operatorname{re}{\left(\operatorname{atan}{\left(- \frac{3^{\frac{9}{10}}}{12} + \frac{3^{\frac{9}{10}} \sqrt{5}}{12} + \frac{3^{\frac{9}{10}} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{6} + \frac{3^{\frac{9}{10}} \sqrt{5} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{6} \right)}\right)} + \operatorname{re}{\left(\operatorname{atan}{\left(- \frac{3^{\frac{9}{10}} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{3} + \frac{3^{\frac{9}{10}}}{12} - \frac{3^{\frac{9}{10}} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{12} + \frac{3^{\frac{9}{10}} i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{12} + \frac{3^{\frac{9}{10}} \sqrt{5} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{12} + \frac{3^{\frac{9}{10}} \sqrt{5} i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{12} \right)}\right)} - \operatorname{re}{\left(\operatorname{atan}{\left(- \frac{3^{\frac{9}{10}} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{3} - \frac{3^{\frac{9}{10}}}{12} - \frac{3^{\frac{9}{10}} \sqrt{5} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{12} + \frac{3^{\frac{9}{10}} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{12} + \frac{3^{\frac{9}{10}} i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{12} + \frac{3^{\frac{9}{10}} \sqrt{5} i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{12} \right)}\right)} + \operatorname{re}{\left(\operatorname{atan}{\left(\frac{3^{\frac{9}{10}}}{12} + \frac{3^{\frac{9}{10}} \sqrt{5}}{12} + \frac{3^{\frac{9}{10}} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{3} \right)}\right)} - i \operatorname{im}{\left(\operatorname{atan}{\left(- \frac{3^{\frac{9}{10}}}{12} + \frac{3^{\frac{9}{10}} \sqrt{5}}{12} + \frac{3^{\frac{9}{10}} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{6} + \frac{3^{\frac{9}{10}} \sqrt{5} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{6} \right)}\right)} - i \operatorname{im}{\left(\operatorname{atan}{\left(- \frac{3^{\frac{9}{10}} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{3} - \frac{3^{\frac{9}{10}}}{12} - \frac{3^{\frac{9}{10}} \sqrt{5} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{12} + \frac{3^{\frac{9}{10}} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{12} + \frac{3^{\frac{9}{10}} i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{12} + \frac{3^{\frac{9}{10}} \sqrt{5} i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{12} \right)}\right)} + i \operatorname{im}{\left(\operatorname{atan}{\left(\frac{3^{\frac{9}{10}}}{12} + \frac{3^{\frac{9}{10}} \sqrt{5}}{12} + \frac{3^{\frac{9}{10}} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{3} \right)}\right)} + i \operatorname{im}{\left(\operatorname{atan}{\left(- \frac{3^{\frac{9}{10}} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{3} + \frac{3^{\frac{9}{10}}}{12} - \frac{3^{\frac{9}{10}} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{12} + \frac{3^{\frac{9}{10}} i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{12} + \frac{3^{\frac{9}{10}} \sqrt{5} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{12} + \frac{3^{\frac{9}{10}} \sqrt{5} i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{12} \right)}\right)} 
product
             /    /    /                                     ___________                      ___________\\       /    /                                     ___________                      ___________\\\ /    /    /                     ___________      ___________                ___________                ___________                      ___________                      ___________\\       /    /                     ___________      ___________                ___________                ___________                      ___________                      ___________\\\ /    /    /                   ___________      ___________                ___________                ___________                      ___________                      ___________\\     /    /                   ___________      ___________                ___________                ___________                      ___________                      ___________\\\ /    /    /                                   ___________\\     /    /                                   ___________\\\
             |    |    |                                    /       ___                      /       ___ ||       |    |                                    /       ___                      /       ___ ||| |    |    |                    /       ___      /       ___                /       ___                /       ___                      /       ___                      /       ___ ||       |    |                    /       ___      /       ___                /       ___                /       ___                      /       ___                      /       ___ ||| |    |    |                  /       ___      /       ___                /       ___                /       ___                      /       ___                      /       ___ ||     |    |                  /       ___      /       ___                /       ___                /       ___                      /       ___                      /       ___ ||| |    |    |                                  /       ___ ||     |    |                                  /       ___ |||
             |    |    |                           9/10    /  5   \/ 5        9/10   ___    /  5   \/ 5  ||       |    |                           9/10    /  5   \/ 5        9/10   ___    /  5   \/ 5  ||| |    |    |           9/10    /  5   \/ 5      /  5   \/ 5        9/10    /  5   \/ 5        9/10    /  5   \/ 5        9/10   ___    /  5   \/ 5        9/10   ___    /  5   \/ 5  ||       |    |           9/10    /  5   \/ 5      /  5   \/ 5        9/10    /  5   \/ 5        9/10    /  5   \/ 5        9/10   ___    /  5   \/ 5        9/10   ___    /  5   \/ 5  ||| |    |    |         9/10    /  5   \/ 5      /  5   \/ 5        9/10    /  5   \/ 5        9/10    /  5   \/ 5        9/10   ___    /  5   \/ 5        9/10   ___    /  5   \/ 5  ||     |    |         9/10    /  5   \/ 5      /  5   \/ 5        9/10    /  5   \/ 5        9/10    /  5   \/ 5        9/10   ___    /  5   \/ 5        9/10   ___    /  5   \/ 5  ||| |    |    |                         9/10    /  5   \/ 5  ||     |    |                         9/10    /  5   \/ 5  |||
     / 9/10\ |    |    |   9/10    9/10   ___   I*3    *  /   - - -----    I*3    *\/ 5 *  /   - - ----- ||       |    |   9/10    9/10   ___   I*3    *  /   - - -----    I*3    *\/ 5 *  /   - - ----- ||| |    |    |   9/10   3    *  /   - - ----- *  /   - + -----    I*3    *  /   - - -----    I*3    *  /   - + -----    I*3    *\/ 5 *  /   - - -----    I*3    *\/ 5 *  /   - + ----- ||       |    |   9/10   3    *  /   - - ----- *  /   - + -----    I*3    *  /   - - -----    I*3    *  /   - + -----    I*3    *\/ 5 *  /   - - -----    I*3    *\/ 5 *  /   - + ----- ||| |    |    | 9/10   3    *  /   - - ----- *  /   - + -----    I*3    *  /   - - -----    I*3    *  /   - + -----    I*3    *\/ 5 *  /   - - -----    I*3    *\/ 5 *  /   - + ----- ||     |    | 9/10   3    *  /   - - ----- *  /   - + -----    I*3    *  /   - - -----    I*3    *  /   - + -----    I*3    *\/ 5 *  /   - - -----    I*3    *\/ 5 *  /   - + ----- ||| |    |    | 9/10    9/10   ___   I*3    *  /   - - ----- ||     |    | 9/10    9/10   ___   I*3    *  /   - - ----- |||
     |3    | |    |    |  3       3    *\/ 5            \/    8     8                    \/    8     8   ||       |    |  3       3    *\/ 5            \/    8     8                    \/    8     8   ||| |    |    |  3             \/    8     8    \/    8     8              \/    8     8              \/    8     8                    \/    8     8                    \/    8     8   ||       |    |  3             \/    8     8    \/    8     8              \/    8     8              \/    8     8                    \/    8     8                    \/    8     8   ||| |    |    |3             \/    8     8    \/    8     8              \/    8     8              \/    8     8                    \/    8     8                    \/    8     8   ||     |    |3             \/    8     8    \/    8     8              \/    8     8              \/    8     8                    \/    8     8                    \/    8     8   ||| |    |    |3       3    *\/ 5            \/    8     8   ||     |    |3       3    *\/ 5            \/    8     8   |||
-atan|-----|*|- re|atan|- ----- + ----------- + ------------------------ + ------------------------------|| - I*im|atan|- ----- + ----------- + ------------------------ + ------------------------------|||*|- re|atan|- ----- - --------------------------------------- + ------------------------ + ------------------------ - ------------------------------ + ------------------------------|| - I*im|atan|- ----- - --------------------------------------- + ------------------------ + ------------------------ - ------------------------------ + ------------------------------|||*|I*im|atan|----- - --------------------------------------- - ------------------------ + ------------------------ + ------------------------------ + ------------------------------|| + re|atan|----- - --------------------------------------- - ------------------------ + ------------------------ + ------------------------------ + ------------------------------|||*|I*im|atan|----- + ----------- + ------------------------|| + re|atan|----- + ----------- + ------------------------|||
     \  3  / \    \    \    12         12                  6                             6               //       \    \    12         12                  6                             6               /// \    \    \    12                       3                                 12                         12                            12                               12              //       \    \    12                       3                                 12                         12                            12                               12              /// \    \    \  12                       3                                 12                         12                            12                               12              //     \    \  12                       3                                 12                         12                            12                               12              /// \    \    \  12         12                  3            //     \    \  12         12                  3            ///
(−re⁡(atan⁡(−391012+3910512+3910i58−586+39105i58−586))−iim⁡(atan⁡(−391012+3910512+3910i58−586+39105i58−586)))(−atan⁡(39103))(−re⁡(atan⁡(−391058−5858+583−391012−39105i58−5812+3910i58−5812+3910i58+5812+39105i58+5812))−iim⁡(atan⁡(−391058−5858+583−391012−39105i58−5812+3910i58−5812+3910i58+5812+39105i58+5812)))(re⁡(atan⁡(−391058−5858+583+391012−3910i58−5812+3910i58+5812+39105i58−5812+39105i58+5812))+iim⁡(atan⁡(−391058−5858+583+391012−3910i58−5812+3910i58+5812+39105i58−5812+39105i58+5812)))(re⁡(atan⁡(391012+3910512+3910i58−583))+iim⁡(atan⁡(391012+3910512+3910i58−583)))\left(- \operatorname{re}{\left(\operatorname{atan}{\left(- \frac{3^{\frac{9}{10}}}{12} + \frac{3^{\frac{9}{10}} \sqrt{5}}{12} + \frac{3^{\frac{9}{10}} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{6} + \frac{3^{\frac{9}{10}} \sqrt{5} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{6} \right)}\right)} - i \operatorname{im}{\left(\operatorname{atan}{\left(- \frac{3^{\frac{9}{10}}}{12} + \frac{3^{\frac{9}{10}} \sqrt{5}}{12} + \frac{3^{\frac{9}{10}} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{6} + \frac{3^{\frac{9}{10}} \sqrt{5} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{6} \right)}\right)}\right) \left(- \operatorname{atan}{\left(\frac{3^{\frac{9}{10}}}{3} \right)}\right) \left(- \operatorname{re}{\left(\operatorname{atan}{\left(- \frac{3^{\frac{9}{10}} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{3} - \frac{3^{\frac{9}{10}}}{12} - \frac{3^{\frac{9}{10}} \sqrt{5} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{12} + \frac{3^{\frac{9}{10}} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{12} + \frac{3^{\frac{9}{10}} i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{12} + \frac{3^{\frac{9}{10}} \sqrt{5} i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{12} \right)}\right)} - i \operatorname{im}{\left(\operatorname{atan}{\left(- \frac{3^{\frac{9}{10}} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{3} - \frac{3^{\frac{9}{10}}}{12} - \frac{3^{\frac{9}{10}} \sqrt{5} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{12} + \frac{3^{\frac{9}{10}} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{12} + \frac{3^{\frac{9}{10}} i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{12} + \frac{3^{\frac{9}{10}} \sqrt{5} i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{12} \right)}\right)}\right) \left(\operatorname{re}{\left(\operatorname{atan}{\left(- \frac{3^{\frac{9}{10}} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{3} + \frac{3^{\frac{9}{10}}}{12} - \frac{3^{\frac{9}{10}} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{12} + \frac{3^{\frac{9}{10}} i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{12} + \frac{3^{\frac{9}{10}} \sqrt{5} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{12} + \frac{3^{\frac{9}{10}} \sqrt{5} i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{12} \right)}\right)} + i \operatorname{im}{\left(\operatorname{atan}{\left(- \frac{3^{\frac{9}{10}} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{3} + \frac{3^{\frac{9}{10}}}{12} - \frac{3^{\frac{9}{10}} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{12} + \frac{3^{\frac{9}{10}} i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{12} + \frac{3^{\frac{9}{10}} \sqrt{5} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{12} + \frac{3^{\frac{9}{10}} \sqrt{5} i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{12} \right)}\right)}\right) \left(\operatorname{re}{\left(\operatorname{atan}{\left(\frac{3^{\frac{9}{10}}}{12} + \frac{3^{\frac{9}{10}} \sqrt{5}}{12} + \frac{3^{\frac{9}{10}} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{3} \right)}\right)} + i \operatorname{im}{\left(\operatorname{atan}{\left(\frac{3^{\frac{9}{10}}}{12} + \frac{3^{\frac{9}{10}} \sqrt{5}}{12} + \frac{3^{\frac{9}{10}} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{3} \right)}\right)}\right) 
=
 /    /    /      /                       ___________\\\     /    /      /                       ___________\\\\ /    /    /      /                          ___________               ___________\\\     /    /      /                          ___________               ___________\\\\ /    /    /      /                          ___________              ___________               ___________               ___________\\\     /    /      /                          ___________              ___________               ___________               ___________\\\\ /    /    /      /                         ___________               ___________               ___________              ___________\\\     /    /      /                         ___________               ___________               ___________              ___________\\\\            
 |    |    | 9/10 |      ___       ___   /       ___ |||     |    | 9/10 |      ___       ___   /       ___ |||| |    |    | 9/10 |         ___       ___   /       ___        ____   /       ___ |||     |    | 9/10 |         ___       ___   /       ___        ____   /       ___ |||| |    |    | 9/10 |         ___       ___   /       ___        ___   /       ___        ____   /       ___        ____   /       ___ |||     |    | 9/10 |         ___       ___   /       ___        ___   /       ___        ____   /       ___        ____   /       ___ |||| |    |    | 9/10 |        ___       ___   /       ___        ____   /       ___        ____   /       ___        ___   /       ___ |||     |    | 9/10 |        ___       ___   /       ___        ____   /       ___        ____   /       ___        ___   /       ___ ||||     / 9/10\
 |    |    |3    *\1 + \/ 5  + I*\/ 2 *\/  5 - \/ 5  /||     |    |3    *\1 + \/ 5  + I*\/ 2 *\/  5 - \/ 5  /||| |    |    |3    *\-2 + 2*\/ 5  + I*\/ 2 *\/  5 - \/ 5   + I*\/ 10 *\/  5 - \/ 5  /||     |    |3    *\-2 + 2*\/ 5  + I*\/ 2 *\/  5 - \/ 5   + I*\/ 10 *\/  5 - \/ 5  /||| |    |    |3    *\-4 - 4*\/ 5  + I*\/ 2 *\/  5 + \/ 5   + I*\/ 2 *\/  5 - \/ 5   + I*\/ 10 *\/  5 + \/ 5   - I*\/ 10 *\/  5 - \/ 5  /||     |    |3    *\-4 - 4*\/ 5  + I*\/ 2 *\/  5 + \/ 5   + I*\/ 2 *\/  5 - \/ 5   + I*\/ 10 *\/  5 + \/ 5   - I*\/ 10 *\/  5 - \/ 5  /||| |    |    |3    *\4 - 4*\/ 5  + I*\/ 2 *\/  5 + \/ 5   + I*\/ 10 *\/  5 + \/ 5   + I*\/ 10 *\/  5 - \/ 5   - I*\/ 2 *\/  5 - \/ 5  /||     |    |3    *\4 - 4*\/ 5  + I*\/ 2 *\/  5 + \/ 5   + I*\/ 10 *\/  5 + \/ 5   + I*\/ 10 *\/  5 - \/ 5   - I*\/ 2 *\/  5 - \/ 5  /|||     |3    |
-|I*im|atan|------------------------------------------|| + re|atan|------------------------------------------|||*|I*im|atan|-----------------------------------------------------------------------|| + re|atan|-----------------------------------------------------------------------|||*|I*im|atan|--------------------------------------------------------------------------------------------------------------------------|| + re|atan|--------------------------------------------------------------------------------------------------------------------------|||*|I*im|atan|-------------------------------------------------------------------------------------------------------------------------|| + re|atan|-------------------------------------------------------------------------------------------------------------------------|||*atan|-----|
 \    \    \                    12                    //     \    \                    12                    /// \    \    \                                   24                                  //     \    \                                   24                                  /// \    \    \                                                            48                                                            //     \    \                                                            48                                                            /// \    \    \                                                            48                                                           //     \    \                                                            48                                                           ///     \  3  /
−(re⁡(atan⁡(3910(1+5+2i5−5)12))+iim⁡(atan⁡(3910(1+5+2i5−5)12)))(re⁡(atan⁡(3910(−2+25+2i5−5+10i5−5)24))+iim⁡(atan⁡(3910(−2+25+2i5−5+10i5−5)24)))(re⁡(atan⁡(3910(−45−4−10i5−5+2i5−5+2i5+5+10i5+5)48))+iim⁡(atan⁡(3910(−45−4−10i5−5+2i5−5+2i5+5+10i5+5)48)))(re⁡(atan⁡(3910(−45+4−2i5−5+2i5+5+10i5−5+10i5+5)48))+iim⁡(atan⁡(3910(−45+4−2i5−5+2i5+5+10i5−5+10i5+5)48)))atan⁡(39103)- \left(\operatorname{re}{\left(\operatorname{atan}{\left(\frac{3^{\frac{9}{10}} \left(1 + \sqrt{5} + \sqrt{2} i \sqrt{5 - \sqrt{5}}\right)}{12} \right)}\right)} + i \operatorname{im}{\left(\operatorname{atan}{\left(\frac{3^{\frac{9}{10}} \left(1 + \sqrt{5} + \sqrt{2} i \sqrt{5 - \sqrt{5}}\right)}{12} \right)}\right)}\right) \left(\operatorname{re}{\left(\operatorname{atan}{\left(\frac{3^{\frac{9}{10}} \left(-2 + 2 \sqrt{5} + \sqrt{2} i \sqrt{5 - \sqrt{5}} + \sqrt{10} i \sqrt{5 - \sqrt{5}}\right)}{24} \right)}\right)} + i \operatorname{im}{\left(\operatorname{atan}{\left(\frac{3^{\frac{9}{10}} \left(-2 + 2 \sqrt{5} + \sqrt{2} i \sqrt{5 - \sqrt{5}} + \sqrt{10} i \sqrt{5 - \sqrt{5}}\right)}{24} \right)}\right)}\right) \left(\operatorname{re}{\left(\operatorname{atan}{\left(\frac{3^{\frac{9}{10}} \left(- 4 \sqrt{5} - 4 - \sqrt{10} i \sqrt{5 - \sqrt{5}} + \sqrt{2} i \sqrt{5 - \sqrt{5}} + \sqrt{2} i \sqrt{\sqrt{5} + 5} + \sqrt{10} i \sqrt{\sqrt{5} + 5}\right)}{48} \right)}\right)} + i \operatorname{im}{\left(\operatorname{atan}{\left(\frac{3^{\frac{9}{10}} \left(- 4 \sqrt{5} - 4 - \sqrt{10} i \sqrt{5 - \sqrt{5}} + \sqrt{2} i \sqrt{5 - \sqrt{5}} + \sqrt{2} i \sqrt{\sqrt{5} + 5} + \sqrt{10} i \sqrt{\sqrt{5} + 5}\right)}{48} \right)}\right)}\right) \left(\operatorname{re}{\left(\operatorname{atan}{\left(\frac{3^{\frac{9}{10}} \left(- 4 \sqrt{5} + 4 - \sqrt{2} i \sqrt{5 - \sqrt{5}} + \sqrt{2} i \sqrt{\sqrt{5} + 5} + \sqrt{10} i \sqrt{5 - \sqrt{5}} + \sqrt{10} i \sqrt{\sqrt{5} + 5}\right)}{48} \right)}\right)} + i \operatorname{im}{\left(\operatorname{atan}{\left(\frac{3^{\frac{9}{10}} \left(- 4 \sqrt{5} + 4 - \sqrt{2} i \sqrt{5 - \sqrt{5}} + \sqrt{2} i \sqrt{\sqrt{5} + 5} + \sqrt{10} i \sqrt{5 - \sqrt{5}} + \sqrt{10} i \sqrt{\sqrt{5} + 5}\right)}{48} \right)}\right)}\right) \operatorname{atan}{\left(\frac{3^{\frac{9}{10}}}{3} \right)} 
-(i*im(atan(3^(9/10)*(1 + sqrt(5) + i*sqrt(2)*sqrt(5 - sqrt(5)))/12)) + re(atan(3^(9/10)*(1 + sqrt(5) + i*sqrt(2)*sqrt(5 - sqrt(5)))/12)))*(i*im(atan(3^(9/10)*(-2 + 2*sqrt(5) + i*sqrt(2)*sqrt(5 - sqrt(5)) + i*sqrt(10)*sqrt(5 - sqrt(5)))/24)) + re(atan(3^(9/10)*(-2 + 2*sqrt(5) + i*sqrt(2)*sqrt(5 - sqrt(5)) + i*sqrt(10)*sqrt(5 - sqrt(5)))/24)))*(i*im(atan(3^(9/10)*(-4 - 4*sqrt(5) + i*sqrt(2)*sqrt(5 + sqrt(5)) + i*sqrt(2)*sqrt(5 - sqrt(5)) + i*sqrt(10)*sqrt(5 + sqrt(5)) - i*sqrt(10)*sqrt(5 - sqrt(5)))/48)) + re(atan(3^(9/10)*(-4 - 4*sqrt(5) + i*sqrt(2)*sqrt(5 + sqrt(5)) + i*sqrt(2)*sqrt(5 - sqrt(5)) + i*sqrt(10)*sqrt(5 + sqrt(5)) - i*sqrt(10)*sqrt(5 - sqrt(5)))/48)))*(i*im(atan(3^(9/10)*(4 - 4*sqrt(5) + i*sqrt(2)*sqrt(5 + sqrt(5)) + i*sqrt(10)*sqrt(5 + sqrt(5)) + i*sqrt(10)*sqrt(5 - sqrt(5)) - i*sqrt(2)*sqrt(5 - sqrt(5)))/48)) + re(atan(3^(9/10)*(4 - 4*sqrt(5) + i*sqrt(2)*sqrt(5 + sqrt(5)) + i*sqrt(10)*sqrt(5 + sqrt(5)) + i*sqrt(10)*sqrt(5 - sqrt(5)) - i*sqrt(2)*sqrt(5 - sqrt(5)))/48)))*atan(3^(9/10)/3)
Numerical answer [src] 
x1 = 178322.35162535
x2 = -123.252691204132
x3 = 14.9773855538189
x4 = -35.2880969036177
x5 = -101.261542629003
x6 = 18.1189782074087
x7 = -79.2703940538749
x8 = -25.8633189428484
x9 = 46.3933120897169
x10 = 2.41101493945977
x11 = -11447607863.8933
x12 = 84.0924239327944
x13 = -63.5624307859259
x14 = -44.7128748643871
x15 = -899.226076640811
x16 = 96.6587945471536
x17 = -33625.1967490857
x18 = -66.7040234395157
x19 = 58.9596827040761
x20 = -468.827883099009
x21 = -41.5712822107973
x22 = 62.1012753576658
x23 = -10848.6500105597
x24 = -7.01376302130961
x25 = 17931.4802889764
x26 = -57.2792454787463
x27 = -47.8544675179769
x28 = 52.6764973968965
x29 = -22.7217262892586
x30 = -69.8456160931055
x31 = 36.9685341289475
x32 = 40.1101267825373
x33 = -91.836764668234
x34 = 74.667645972025
x35 = -19.5801336356688
x36 = 68.3844606648454
x37 = 80.9508312792046
x38 = 43.2517194361271
x39 = -955.774744405427
x40 = 3480.15408246336
x41 = -305.46506511234
x42 = -3396.7922362447
x43 = 30.6853488217679
x44 = 8.69420024663936
x45 = -85.5535793610544
x46 = 24.4021635145883
x47 = 39187.4961831649
x48 = 90.375609239974
x49 = -1569405.33182707
x50 = -3.87217036771981
x51 = -13.2969483284892
x52 = -5297621.29843638
x53 = -32.146504250028
x53 = -32.146504250028
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