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   http://functions.wolfram.com/01.22.03.0047.01
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    Coth[(4 Pi I)/7] == (7 + 7 I Sqrt[3] + ((7/2) (1 - 3 I Sqrt[3]))^(2/3) - 
   I Sqrt[3] (7/2 - (21 I Sqrt[3])/2)^(2/3) + 
   2^(2/3) (7 - 21 I Sqrt[3])^(1/3))/
  ((-I) 2^(2/3) 7^(5/6) (1 - 3 I Sqrt[3])^(1/3) + 
   Sqrt[7] (I + Sqrt[3]) (7 - (I Sqrt[7])/2 - (3 Sqrt[21])/2)^(1/3) - 
   (1/2) I (4 Sqrt[7] (7 + (I Sqrt[7])/2 + (3 Sqrt[21])/2)^(1/3) + 
     (14 - 42 I Sqrt[3])^(1/3) (-2 I (14 - I Sqrt[7] - 3 Sqrt[21])^(1/3) + 
       (-I + Sqrt[3]) (14 + I Sqrt[7] + 3 Sqrt[21])^(1/3)))) 
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   Cell[BoxData[RowBox[List[RowBox[List["Coth", "[", FractionBox[RowBox[List["4", "\[Pi]", " ", "\[ImaginaryI]"]], "7"], "]"]], "\[Equal]", RowBox[List[RowBox[List["(", RowBox[List["7", "+", RowBox[List["7", " ", "\[ImaginaryI]", " ", SqrtBox["3"]]], "+", SuperscriptBox[RowBox[List["(", RowBox[List[FractionBox["7", "2"], " ", RowBox[List["(", RowBox[List["1", "-", RowBox[List["3", " ", "\[ImaginaryI]", " ", SqrtBox["3"]]]]], ")"]]]], ")"]], RowBox[List["2", "/", "3"]]], "-", RowBox[List["\[ImaginaryI]", " ", SqrtBox["3"], " ", SuperscriptBox[RowBox[List["(", RowBox[List[FractionBox["7", "2"], "-", FractionBox[RowBox[List["21", " ", "\[ImaginaryI]", " ", SqrtBox["3"]]], "2"]]], ")"]], RowBox[List["2", "/", "3"]]]]], "+", RowBox[List[SuperscriptBox["2", RowBox[List["2", "/", "3"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["7", "-", RowBox[List["21", " ", "\[ImaginaryI]", " ", SqrtBox["3"]]]]], ")"]], RowBox[List["1", "/", "3"]]]]]]], ")"]], "/", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", SuperscriptBox["2", RowBox[List["2", "/", "3"]]], " ", SuperscriptBox["7", RowBox[List["5", "/", "6"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", RowBox[List["3", " ", "\[ImaginaryI]", " ", SqrtBox["3"]]]]], ")"]], RowBox[List["1", "/", "3"]]]]], "+", RowBox[List[SqrtBox["7"], " ", RowBox[List["(", RowBox[List["\[ImaginaryI]", "+", SqrtBox["3"]]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["7", "-", FractionBox[RowBox[List["\[ImaginaryI]", " ", SqrtBox["7"]]], "2"], "-", FractionBox[RowBox[List["3", " ", SqrtBox["21"]]], "2"]]], ")"]], RowBox[List["1", "/", "3"]]]]], "-", RowBox[List[FractionBox["1", "2"], " ", "\[ImaginaryI]", " ", RowBox[List["(", RowBox[List[RowBox[List["4", " ", SqrtBox["7"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["7", "+", FractionBox[RowBox[List["\[ImaginaryI]", " ", SqrtBox["7"]]], "2"], "+", FractionBox[RowBox[List["3", " ", SqrtBox["21"]]], "2"]]], ")"]], RowBox[List["1", "/", "3"]]]]], "+", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["14", "-", RowBox[List["42", " ", "\[ImaginaryI]", " ", SqrtBox["3"]]]]], ")"]], RowBox[List["1", "/", "3"]]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "\[ImaginaryI]", " ", SuperscriptBox[RowBox[List["(", RowBox[List["14", "-", RowBox[List["\[ImaginaryI]", " ", SqrtBox["7"]]], "-", RowBox[List["3", " ", SqrtBox["21"]]]]], ")"]], RowBox[List["1", "/", "3"]]]]], "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], "+", SqrtBox["3"]]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["14", "+", RowBox[List["\[ImaginaryI]", " ", SqrtBox["7"]]], "+", RowBox[List["3", " ", SqrtBox["21"]]]]], ")"]], RowBox[List["1", "/", "3"]]]]]]], ")"]]]]]], ")"]]]]]], ")"]]]]]]]] 
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   <math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'>  <semantics>  <mrow>  <mrow>  <mi> coth </mi>  <mo> ⁡ </mo>  <mo> ( </mo>  <mfrac>  <mrow>  <mn> 4 </mn>  <mo> ⁢ </mo>  <mi> π </mi>  <mo> ⁢ </mo>  <mi> ⅈ </mi>  </mrow>  <mn> 7 </mn>  </mfrac>  <mo> ) </mo>  </mrow>  <mo> ⩵ </mo>  <mrow>  <mrow>  <mo> ( </mo>  <mrow>  <mn> 7 </mn>  <mo> + </mo>  <mrow>  <mn> 7 </mn>  <mo> ⁢ </mo>  <mi> ⅈ </mi>  <mo> ⁢ </mo>  <msqrt>  <mn> 3 </mn>  </msqrt>  </mrow>  <mo> + </mo>  <msup>  <mrow>  <mo> ( </mo>  <mrow>  <mfrac>  <mn> 7 </mn>  <mn> 2 </mn>  </mfrac>  <mo> ⁢ </mo>  <mrow>  <mo> ( </mo>  <mrow>  <mn> 1 </mn>  <mo> - </mo>  <mrow>  <mn> 3 </mn>  <mo> ⁢ </mo>  <mi> ⅈ </mi>  <mo> ⁢ </mo>  <msqrt>  <mn> 3 </mn>  </msqrt>  </mrow>  </mrow>  <mo> ) </mo>  </mrow>  </mrow>  <mo> ) </mo>  </mrow>  <mrow>  <mn> 2 </mn>  <mo> / </mo>  <mn> 3 </mn>  </mrow>  </msup>  <mo> - </mo>  <mrow>  <mi> ⅈ </mi>  <mo> ⁢ </mo>  <msqrt>  <mn> 3 </mn>  </msqrt>  <mo> ⁢ </mo>  <msup>  <mrow>  <mo> ( </mo>  <mrow>  <mfrac>  <mn> 7 </mn>  <mn> 2 </mn>  </mfrac>  <mo> - </mo>  <mfrac>  <mrow>  <mn> 21 </mn>  <mo> ⁢ </mo>  <mi> ⅈ </mi>  <mo> ⁢ </mo>  <msqrt>  <mn> 3 </mn>  </msqrt>  </mrow>  <mn> 2 </mn>  </mfrac>  </mrow>  <mo> ) </mo>  </mrow>  <mrow>  <mn> 2 </mn>  <mo> / </mo>  <mn> 3 </mn>  </mrow>  </msup>  </mrow>  <mo> + </mo>  <mrow>  <msup>  <mn> 2 </mn>  <mrow>  <mn> 2 </mn>  <mo> / </mo>  <mn> 3 </mn>  </mrow>  </msup>  <mo> ⁢ </mo>  <mroot>  <mrow>  <mn> 7 </mn>  <mo> - </mo>  <mrow>  <mn> 21 </mn>  <mo> ⁢ </mo>  <mi> ⅈ </mi>  <mo> ⁢ </mo>  <msqrt>  <mn> 3 </mn>  </msqrt>  </mrow>  </mrow>  <mn> 3 </mn>  </mroot>  </mrow>  </mrow>  <mo> ) </mo>  </mrow>  <mo> / </mo>  <mrow>  <mo> ( </mo>  <mrow>  <mrow>  <mrow>  <mo> - </mo>  <mi> ⅈ </mi>  </mrow>  <mo> ⁢ </mo>  <msup>  <mn> 2 </mn>  <mrow>  <mn> 2 </mn>  <mo> / </mo>  <mn> 3 </mn>  </mrow>  </msup>  <mo> ⁢ </mo>  <msup>  <mn> 7 </mn>  <mrow>  <mn> 5 </mn>  <mo> / </mo>  <mn> 6 </mn>  </mrow>  </msup>  <mo> ⁢ </mo>  <mroot>  <mrow>  <mn> 1 </mn>  <mo> - </mo>  <mrow>  <mn> 3 </mn>  <mo> ⁢ </mo>  <mi> ⅈ </mi>  <mo> ⁢ </mo>  <msqrt>  <mn> 3 </mn>  </msqrt>  </mrow>  </mrow>  <mn> 3 </mn>  </mroot>  </mrow>  <mo> + </mo>  <mrow>  <msqrt>  <mn> 7 </mn>  </msqrt>  <mo> ⁢ </mo>  <mrow>  <mo> ( </mo>  <mrow>  <mi> ⅈ </mi>  <mo> + </mo>  <msqrt>  <mn> 3 </mn>  </msqrt>  </mrow>  <mo> ) </mo>  </mrow>  <mo> ⁢ </mo>  <mroot>  <mrow>  <mn> 7 </mn>  <mo> - </mo>  <mfrac>  <mrow>  <mi> ⅈ </mi>  <mo> ⁢ </mo>  <msqrt>  <mn> 7 </mn>  </msqrt>  </mrow>  <mn> 2 </mn>  </mfrac>  <mo> - </mo>  <mfrac>  <mrow>  <mn> 3 </mn>  <mo> ⁢ </mo>  <msqrt>  <mn> 21 </mn>  </msqrt>  </mrow>  <mn> 2 </mn>  </mfrac>  </mrow>  <mn> 3 </mn>  </mroot>  </mrow>  <mo> - </mo>  <mrow>  <mfrac>  <mn> 1 </mn>  <mn> 2 </mn>  </mfrac>  <mo> ⁢ </mo>  <mi> ⅈ </mi>  <mo> ⁢ </mo>  <mrow>  <mo> ( </mo>  <mrow>  <mrow>  <mn> 4 </mn>  <mo> ⁢ </mo>  <msqrt>  <mn> 7 </mn>  </msqrt>  <mo> ⁢ </mo>  <mroot>  <mrow>  <mn> 7 </mn>  <mo> + </mo>  <mfrac>  <mrow>  <mi> ⅈ </mi>  <mo> ⁢ </mo>  <msqrt>  <mn> 7 </mn>  </msqrt>  </mrow>  <mn> 2 </mn>  </mfrac>  <mo> + </mo>  <mfrac>  <mrow>  <mn> 3 </mn>  <mo> ⁢ </mo>  <msqrt>  <mn> 21 </mn>  </msqrt>  </mrow>  <mn> 2 </mn>  </mfrac>  </mrow>  <mn> 3 </mn>  </mroot>  </mrow>  <mo> + </mo>  <mrow>  <mroot>  <mrow>  <mn> 14 </mn>  <mo> - </mo>  <mrow>  <mn> 42 </mn>  <mo> ⁢ </mo>  <mi> ⅈ </mi>  <mo> ⁢ </mo>  <msqrt>  <mn> 3 </mn>  </msqrt>  </mrow>  </mrow>  <mn> 3 </mn>  </mroot>  <mo> ⁢ </mo>  <mrow>  <mo> ( </mo>  <mrow>  <mrow>  <mrow>  <mo> - </mo>  <mn> 2 </mn>  </mrow>  <mo> ⁢ </mo>  <mi> ⅈ </mi>  <mo> ⁢ </mo>  <mroot>  <mrow>  <mn> 14 </mn>  <mo> - </mo>  <mrow>  <mi> ⅈ </mi>  <mo> ⁢ </mo>  <msqrt>  <mn> 7 </mn>  </msqrt>  </mrow>  <mo> - </mo>  <mrow>  <mn> 3 </mn>  <mo> ⁢ </mo>  <msqrt>  <mn> 21 </mn>  </msqrt>  </mrow>  </mrow>  <mn> 3 </mn>  </mroot>  </mrow>  <mo> + </mo>  <mrow>  <mrow>  <mo> ( </mo>  <mrow>  <mrow>  <mo> - </mo>  <mi> ⅈ </mi>  </mrow>  <mo> + </mo>  <msqrt>  <mn> 3 </mn>  </msqrt>  </mrow>  <mo> ) </mo>  </mrow>  <mo> ⁢ </mo>  <mroot>  <mrow>  <mn> 14 </mn>  <mo> + </mo>  <mrow>  <mi> ⅈ </mi>  <mo> ⁢ </mo>  <msqrt>  <mn> 7 </mn>  </msqrt>  </mrow>  <mo> + </mo>  <mrow>  <mn> 3 </mn>  <mo> ⁢ </mo>  <msqrt>  <mn> 21 </mn>  </msqrt>  </mrow>  </mrow>  <mn> 3 </mn>  </mroot>  </mrow>  </mrow>  <mo> ) </mo>  </mrow>  </mrow>  </mrow>  <mo> ) </mo>  </mrow>  </mrow>  </mrow>  <mo> ) </mo>  </mrow>  </mrow>  </mrow>  <annotation-xml encoding='MathML-Content'>  <apply>  <eq />  <apply>  <coth />  <apply>  <times />  <cn type='integer'> 4 </cn>  <pi />  <imaginaryi />  <apply>  <power />  <cn type='integer'> 7 </cn>  <cn type='integer'> -1 </cn>  </apply>  </apply>  </apply>  <apply>  <times />  <apply>  <plus />  <cn type='integer'> 7 </cn>  <apply>  <times />  <cn type='integer'> 7 </cn>  <imaginaryi />  <apply>  <power />  <cn type='integer'> 3 </cn>  <cn type='rational'> 1 <sep /> 2 </cn>  </apply>  </apply>  <apply>  <power />  <apply>  <times />  <cn type='rational'> 7 <sep /> 2 </cn>  <apply>  <plus />  <cn type='integer'> 1 </cn>  <apply>  <times />  <cn type='integer'> -1 </cn>  <apply>  <times />  <cn type='integer'> 3 </cn>  <imaginaryi />  <apply>  <power />  <cn type='integer'> 3 </cn>  <cn type='rational'> 1 <sep /> 2 </cn>  </apply>  </apply>  </apply>  </apply>  </apply>  <cn type='rational'> 2 <sep /> 3 </cn>  </apply>  <apply>  <times />  <cn type='integer'> -1 </cn>  <apply>  <times />  <imaginaryi />  <apply>  <power />  <cn type='integer'> 3 </cn>  <cn type='rational'> 1 <sep /> 2 </cn>  </apply>  <apply>  <power />  <apply>  <plus />  <cn type='rational'> 7 <sep /> 2 </cn>  <apply>  <times />  <cn type='integer'> -1 </cn>  <apply>  <times />  <cn type='integer'> 21 </cn>  <imaginaryi />  <apply>  <power />  <cn type='integer'> 3 </cn>  <cn type='rational'> 1 <sep /> 2 </cn>  </apply>  <apply>  <power />  <cn type='integer'> 2 </cn>  <cn type='integer'> -1 </cn>  </apply>  </apply>  </apply>  </apply>  <cn type='rational'> 2 <sep /> 3 </cn>  </apply>  </apply>  </apply>  <apply>  <times />  <apply>  <power />  <cn type='integer'> 2 </cn>  <cn type='rational'> 2 <sep /> 3 </cn>  </apply>  <apply>  <power />  <apply>  <plus />  <cn type='integer'> 7 </cn>  <apply>  <times />  <cn type='integer'> -1 </cn>  <apply>  <times />  <cn type='integer'> 21 </cn>  <imaginaryi />  <apply>  <power />  <cn type='integer'> 3 </cn>  <cn type='rational'> 1 <sep /> 2 </cn>  </apply>  </apply>  </apply>  </apply>  <cn type='rational'> 1 <sep /> 3 </cn>  </apply>  </apply>  </apply>  <apply>  <power />  <apply>  <plus />  <apply>  <times />  <apply>  <times />  <cn type='integer'> -1 </cn>  <imaginaryi />  </apply>  <apply>  <power />  <cn type='integer'> 2 </cn>  <cn type='rational'> 2 <sep /> 3 </cn>  </apply>  <apply>  <power />  <cn type='integer'> 7 </cn>  <cn type='rational'> 5 <sep /> 6 </cn>  </apply>  <apply>  <power />  <apply>  <plus />  <cn type='integer'> 1 </cn>  <apply>  <times />  <cn type='integer'> -1 </cn>  <apply>  <times />  <cn type='integer'> 3 </cn>  <imaginaryi />  <apply>  <power />  <cn type='integer'> 3 </cn>  <cn type='rational'> 1 <sep /> 2 </cn>  </apply>  </apply>  </apply>  </apply>  <cn type='rational'> 1 <sep /> 3 </cn>  </apply>  </apply>  <apply>  <times />  <apply>  <power />  <cn type='integer'> 7 </cn>  <cn type='rational'> 1 <sep /> 2 </cn>  </apply>  <apply>  <plus />  <imaginaryi />  <apply>  <power />  <cn type='integer'> 3 </cn>  <cn type='rational'> 1 <sep /> 2 </cn>  </apply>  </apply>  <apply>  <power />  <apply>  <plus />  <cn type='integer'> 7 </cn>  <apply>  <times />  <cn type='integer'> -1 </cn>  <apply>  <times />  <imaginaryi />  <apply>  <power />  <cn type='integer'> 7 </cn>  <cn type='rational'> 1 <sep /> 2 </cn>  </apply>  <apply>  <power />  <cn type='integer'> 2 </cn>  <cn type='integer'> -1 </cn>  </apply>  </apply>  </apply>  <apply>  <times />  <cn type='integer'> -1 </cn>  <apply>  <times />  <cn type='integer'> 3 </cn>  <apply>  <power />  <cn type='integer'> 21 </cn>  <cn type='rational'> 1 <sep /> 2 </cn>  </apply>  <apply>  <power />  <cn type='integer'> 2 </cn>  <cn type='integer'> -1 </cn>  </apply>  </apply>  </apply>  </apply>  <cn type='rational'> 1 <sep /> 3 </cn>  </apply>  </apply>  <apply>  <times />  <cn type='integer'> -1 </cn>  <apply>  <times />  <cn type='rational'> 1 <sep /> 2 </cn>  <imaginaryi />  <apply>  <plus />  <apply>  <times />  <cn type='integer'> 4 </cn>  <apply>  <power />  <cn type='integer'> 7 </cn>  <cn type='rational'> 1 <sep /> 2 </cn>  </apply>  <apply>  <power />  <apply>  <plus />  <cn type='integer'> 7 </cn>  <apply>  <times />  <imaginaryi />  <apply>  <power />  <cn type='integer'> 7 </cn>  <cn type='rational'> 1 <sep /> 2 </cn>  </apply>  <apply>  <power />  <cn type='integer'> 2 </cn>  <cn type='integer'> -1 </cn>  </apply>  </apply>  <apply>  <times />  <cn type='integer'> 3 </cn>  <apply>  <power />  <cn type='integer'> 21 </cn>  <cn type='rational'> 1 <sep /> 2 </cn>  </apply>  <apply>  <power />  <cn type='integer'> 2 </cn>  <cn type='integer'> -1 </cn>  </apply>  </apply>  </apply>  <cn type='rational'> 1 <sep /> 3 </cn>  </apply>  </apply>  <apply>  <times />  <apply>  <power />  <apply>  <plus />  <cn type='integer'> 14 </cn>  <apply>  <times />  <cn type='integer'> -1 </cn>  <apply>  <times />  <cn type='integer'> 42 </cn>  <imaginaryi />  <apply>  <power />  <cn type='integer'> 3 </cn>  <cn type='rational'> 1 <sep /> 2 </cn>  </apply>  </apply>  </apply>  </apply>  <cn type='rational'> 1 <sep /> 3 </cn>  </apply>  <apply>  <plus />  <apply>  <times />  <cn type='integer'> -2 </cn>  <imaginaryi />  <apply>  <power />  <apply>  <plus />  <cn type='integer'> 14 </cn>  <apply>  <times />  <cn type='integer'> -1 </cn>  <apply>  <times />  <imaginaryi />  <apply>  <power />  <cn type='integer'> 7 </cn>  <cn type='rational'> 1 <sep /> 2 </cn>  </apply>  </apply>  </apply>  <apply>  <times />  <cn type='integer'> -1 </cn>  <apply>  <times />  <cn type='integer'> 3 </cn>  <apply>  <power />  <cn type='integer'> 21 </cn>  <cn type='rational'> 1 <sep /> 2 </cn>  </apply>  </apply>  </apply>  </apply>  <cn type='rational'> 1 <sep /> 3 </cn>  </apply>  </apply>  <apply>  <times />  <apply>  <plus />  <apply>  <times />  <cn type='integer'> -1 </cn>  <imaginaryi />  </apply>  <apply>  <power />  <cn type='integer'> 3 </cn>  <cn type='rational'> 1 <sep /> 2 </cn>  </apply>  </apply>  <apply>  <power />  <apply>  <plus />  <cn type='integer'> 14 </cn>  <apply>  <times />  <imaginaryi />  <apply>  <power />  <cn type='integer'> 7 </cn>  <cn type='rational'> 1 <sep /> 2 </cn>  </apply>  </apply>  <apply>  <times />  <cn type='integer'> 3 </cn>  <apply>  <power />  <cn type='integer'> 21 </cn>  <cn type='rational'> 1 <sep /> 2 </cn>  </apply>  </apply>  </apply>  <cn type='rational'> 1 <sep /> 3 </cn>  </apply>  </apply>  </apply>  </apply>  </apply>  </apply>  </apply>  </apply>  <cn type='integer'> -1 </cn>  </apply>  </apply>  </apply>  </annotation-xml>  </semantics>  </math> 
   
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