Wolfram Researchfunctions.wolfram.comOther Wolfram Sites
Search Site
Function CategoriesGraphics GalleryNotationsGeneral IdentitiesAbout This Site ContributeEmail Comments

View Related Information In
The Documentation Center
MathWorld

Download All Formulas For This Function
Mathematica Notebook
PDF File

 

Developed with Mathematica -- Download a Free Trial Version
 











variants of this functions
Log






Mathematica Notation

Traditional Notation









Elementary Functions > Log[z] > Integration > Indefinite integration > Involving the direct function





http://functions.wolfram.com/01.04.21.0028.01









  


  










Input Form





Integrate[Log[Cot[z]] Cot[z], z] == (1/2) ((-Log[1 - I Cot[z]]) Log[Cot[z]] - Log[1 + I Cot[z]] Log[Cot[z]] - PolyLog[2, (-I) Cot[z]] - PolyLog[2, I Cot[z]])










Standard Form





Cell[BoxData[RowBox[List[RowBox[List["\[Integral]", RowBox[List[RowBox[List["Log", "[", RowBox[List["Cot", "[", "z", "]"]], "]"]], RowBox[List["Cot", "[", "z", "]"]], RowBox[List["\[DifferentialD]", " ", "z"]]]]]], "\[Equal]", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", RowBox[List["Log", "[", RowBox[List["1", "-", RowBox[List["\[ImaginaryI]", " ", RowBox[List["Cot", "[", "z", "]"]]]]]], "]"]]]], " ", RowBox[List["Log", "[", RowBox[List["Cot", "[", "z", "]"]], "]"]]]], "-", RowBox[List[RowBox[List["Log", "[", RowBox[List["1", "+", RowBox[List["\[ImaginaryI]", " ", RowBox[List["Cot", "[", "z", "]"]]]]]], "]"]], " ", RowBox[List["Log", "[", RowBox[List["Cot", "[", "z", "]"]], "]"]]]], "-", RowBox[List["PolyLog", "[", RowBox[List["2", ",", RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", RowBox[List["Cot", "[", "z", "]"]]]]]], "]"]], "-", RowBox[List["PolyLog", "[", RowBox[List["2", ",", RowBox[List["\[ImaginaryI]", " ", RowBox[List["Cot", "[", "z", "]"]]]]]], "]"]]]], ")"]]]]]]]]










MathML Form







<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mo> &#8747; </mo> <mrow> <mrow> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> cot </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> cot </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> <mo> &#8290; </mo> <mrow> <mo> &#8518; </mo> <mi> z </mi> </mrow> </mrow> </mrow> <mo> &#10869; </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mo> - </mo> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> cot </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> &#8290; </mo> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mrow> <mi> cot </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> cot </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mrow> <mi> cot </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </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> <mrow> <mo> - </mo> <mi> &#8520; </mi> </mrow> <mo> &#8290; </mo> <mrow> <mi> cot </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> - </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> <mi> &#8520; </mi> <mo> &#8290; </mo> <mrow> <mi> cot </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <int /> <bvar> <ci> z </ci> </bvar> <apply> <times /> <apply> <ln /> <apply> <cot /> <ci> z </ci> </apply> </apply> <apply> <cot /> <ci> z </ci> </apply> </apply> </apply> <apply> <times /> <cn type='rational'> 1 <sep /> 2 </cn> <apply> <plus /> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ln /> <apply> <cot /> <ci> z </ci> </apply> </apply> </apply> <apply> <ln /> <apply> <plus /> <apply> <times /> <imaginaryi /> <apply> <cot /> <ci> z </ci> </apply> </apply> <cn type='integer'> 1 </cn> </apply> </apply> </apply> <apply> <times /> <apply> <ln /> <apply> <cot /> <ci> z </ci> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ln /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <imaginaryi /> <apply> <cot /> <ci> z </ci> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> PolyLog </ci> <cn type='integer'> 2 </cn> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <imaginaryi /> </apply> <apply> <cot /> <ci> z </ci> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> PolyLog </ci> <cn type='integer'> 2 </cn> <apply> <times /> <imaginaryi /> <apply> <cot /> <ci> z </ci> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["\[Integral]", RowBox[List[RowBox[List[RowBox[List["Log", "[", RowBox[List["Cot", "[", "z_", "]"]], "]"]], " ", RowBox[List["Cot", "[", "z_", "]"]]]], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", RowBox[List["Log", "[", RowBox[List["1", "-", RowBox[List["\[ImaginaryI]", " ", RowBox[List["Cot", "[", "z", "]"]]]]]], "]"]]]], " ", RowBox[List["Log", "[", RowBox[List["Cot", "[", "z", "]"]], "]"]]]], "-", RowBox[List[RowBox[List["Log", "[", RowBox[List["1", "+", RowBox[List["\[ImaginaryI]", " ", RowBox[List["Cot", "[", "z", "]"]]]]]], "]"]], " ", RowBox[List["Log", "[", RowBox[List["Cot", "[", "z", "]"]], "]"]]]], "-", RowBox[List["PolyLog", "[", RowBox[List["2", ",", RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", RowBox[List["Cot", "[", "z", "]"]]]]]], "]"]], "-", RowBox[List["PolyLog", "[", RowBox[List["2", ",", RowBox[List["\[ImaginaryI]", " ", RowBox[List["Cot", "[", "z", "]"]]]]]], "]"]]]], ")"]]]]]]]]










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





2001-10-29