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Coth






Mathematica Notation

Traditional Notation









Elementary Functions > Coth[z] > Integration > Definite integration > Involving related functions





http://functions.wolfram.com/01.22.21.0012.01









  


  










Input Form





Integrate[Log[Coth[t]], {t, 0, Pi/4}] == (1/12) (6 Log[Coth[Pi/4] + 1] Log[Coth[Pi/4]] + Pi^2 - 3 Log[Coth[Pi/4]]^2 + 6 PolyLog[2, 1 - Coth[Pi/4]] - 6 PolyLog[2, -Tanh[Pi/4]])










Standard Form





Cell[BoxData[RowBox[List[RowBox[List[SubsuperscriptBox["\[Integral]", "0", FractionBox["\[Pi]", "4"]], RowBox[List[RowBox[List["Log", "[", RowBox[List["Coth", "[", "t", "]"]], "]"]], RowBox[List["\[DifferentialD]", "t"]]]]]], "\[Equal]", RowBox[List[FractionBox["1", "12"], " ", RowBox[List["(", RowBox[List[RowBox[List["6", " ", RowBox[List["Log", "[", RowBox[List[RowBox[List["Coth", "[", FractionBox["\[Pi]", "4"], "]"]], "+", "1"]], "]"]], " ", RowBox[List["Log", "[", RowBox[List["Coth", "[", FractionBox["\[Pi]", "4"], "]"]], "]"]]]], "+", SuperscriptBox["\[Pi]", "2"], "-", RowBox[List["3", " ", SuperscriptBox[RowBox[List["Log", "[", RowBox[List["Coth", "[", FractionBox["\[Pi]", "4"], "]"]], "]"]], "2"]]], "+", RowBox[List["6", " ", RowBox[List["PolyLog", "[", RowBox[List["2", ",", RowBox[List["1", "-", RowBox[List["Coth", "[", FractionBox["\[Pi]", "4"], "]"]]]]]], "]"]]]], "-", RowBox[List["6", " ", RowBox[List["PolyLog", "[", RowBox[List["2", ",", RowBox[List["-", RowBox[List["Tanh", "[", FractionBox["\[Pi]", "4"], "]"]]]]]], "]"]]]]]], ")"]]]]]]]]










MathML Form







<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <msubsup> <mo> &#8747; </mo> <mn> 0 </mn> <mfrac> <mi> &#960; </mi> <mn> 4 </mn> </mfrac> </msubsup> <mrow> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> coth </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> t </mi> <mo> ) </mo> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> &#8518; </mo> <mi> t </mi> </mrow> </mrow> </mrow> <mo> &#10869; </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 12 </mn> </mfrac> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mo> - </mo> <mn> 3 </mn> </mrow> <mo> &#8290; </mo> <mrow> <msup> <mi> log </mi> <mn> 2 </mn> </msup> <mo> ( </mo> <mrow> <mi> coth </mi> <mo> &#8289; </mo> <mo> ( </mo> <mfrac> <mi> &#960; </mi> <mn> 4 </mn> </mfrac> <mo> ) </mo> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mn> 6 </mn> <mo> &#8290; </mo> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mrow> <mi> coth </mi> <mo> &#8289; </mo> <mo> ( </mo> <mfrac> <mi> &#960; </mi> <mn> 4 </mn> </mfrac> <mo> ) </mo> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> coth </mi> <mo> &#8289; </mo> <mo> ( </mo> <mfrac> <mi> &#960; </mi> <mn> 4 </mn> </mfrac> <mo> ) </mo> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <msup> <mi> &#960; </mi> <mn> 2 </mn> </msup> <mo> + </mo> <mrow> <mn> 6 </mn> <mo> &#8290; </mo> <mrow> <msub> <semantics> <mi> Li </mi> <annotation-xml encoding='MathML-Content'> <ci> PolyLog </ci> </annotation-xml> </semantics> <mn> 2 </mn> </msub> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mrow> <mi> coth </mi> <mo> &#8289; </mo> <mo> ( </mo> <mfrac> <mi> &#960; </mi> <mn> 4 </mn> </mfrac> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <mn> 6 </mn> <mo> &#8290; </mo> <mrow> <msub> <semantics> <mi> Li </mi> <annotation-xml encoding='MathML-Content'> <ci> PolyLog </ci> </annotation-xml> </semantics> <mn> 2 </mn> </msub> <mo> ( </mo> <mrow> <mo> - </mo> <mrow> <mi> tanh </mi> <mo> &#8289; </mo> <mo> ( </mo> <mfrac> <mi> &#960; </mi> <mn> 4 </mn> </mfrac> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <int /> <bvar> <ci> t </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <apply> <times /> <pi /> <apply> <power /> <cn type='integer'> 4 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </uplimit> <apply> <ln /> <apply> <coth /> <ci> t </ci> </apply> </apply> </apply> <apply> <times /> <cn type='rational'> 1 <sep /> 12 </cn> <apply> <plus /> <apply> <times /> <cn type='integer'> -3 </cn> <apply> <power /> <apply> <ln /> <apply> <coth /> <apply> <times /> <pi /> <apply> <power /> <cn type='integer'> 4 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 6 </cn> <apply> <ln /> <apply> <plus /> <apply> <coth /> <apply> <times /> <pi /> <apply> <power /> <cn type='integer'> 4 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <ln /> <apply> <coth /> <apply> <times /> <pi /> <apply> <power /> <cn type='integer'> 4 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <power /> <pi /> <cn type='integer'> 2 </cn> </apply> <apply> <times /> <cn type='integer'> 6 </cn> <apply> <ci> PolyLog </ci> <cn type='integer'> 2 </cn> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <coth /> <apply> <times /> <pi /> <apply> <power /> <cn type='integer'> 4 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 6 </cn> <apply> <ci> PolyLog </ci> <cn type='integer'> 2 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <tanh /> <apply> <times /> <pi /> <apply> <power /> <cn type='integer'> 4 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[SubsuperscriptBox["\[Integral]", "0", FractionBox["\[Pi]", "4"]], RowBox[List[RowBox[List["Log", "[", RowBox[List["Coth", "[", "t_", "]"]], "]"]], RowBox[List["\[DifferentialD]", "t_"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox["1", "12"], " ", RowBox[List["(", RowBox[List[RowBox[List["6", " ", RowBox[List["Log", "[", RowBox[List[RowBox[List["Coth", "[", FractionBox["\[Pi]", "4"], "]"]], "+", "1"]], "]"]], " ", RowBox[List["Log", "[", RowBox[List["Coth", "[", FractionBox["\[Pi]", "4"], "]"]], "]"]]]], "+", SuperscriptBox["\[Pi]", "2"], "-", RowBox[List["3", " ", SuperscriptBox[RowBox[List["Log", "[", RowBox[List["Coth", "[", FractionBox["\[Pi]", "4"], "]"]], "]"]], "2"]]], "+", RowBox[List["6", " ", RowBox[List["PolyLog", "[", RowBox[List["2", ",", RowBox[List["1", "-", RowBox[List["Coth", "[", FractionBox["\[Pi]", "4"], "]"]]]]]], "]"]]]], "-", RowBox[List["6", " ", RowBox[List["PolyLog", "[", RowBox[List["2", ",", RowBox[List["-", RowBox[List["Tanh", "[", FractionBox["\[Pi]", "4"], "]"]]]]]], "]"]]]]]], ")"]]]]]]]]










Date Added to functions.wolfram.com (modification date)





2001-10-29