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Csch




Mathematica Notation

Traditional Notation









Elementary Functions > Csch[z] > Integration > Indefinite integration > Involving functions of the direct function and hyperbolic functions > Involving powers of the direct function and hyperbolic functions > Involving power of cosh > Involving coshu(b z) cschnu(c z)





http://functions.wolfram.com/01.23.21.0395.01









  


  








Input Form





Integrate[Sqrt[Cosh[2 z]^3] Csch[z]^3, z] == (-(1/4)) Sqrt[Cosh[2 z]^3] (2 ArcTan[Cosh[z]/Sqrt[-Cosh[2 z]]] Sqrt[-Cosh[2 z]] + 8 ArcTanh[Cosh[z]/Sqrt[Cosh[2 z]]] Sqrt[Cosh[2 z]] + Cosh[z] (3 + Coth[z]^2) + Coth[z] Csch[z] - 8 Sqrt[2] Sqrt[Cosh[2 z]] Log[Sqrt[2] Cosh[z] + Sqrt[Cosh[2 z]]]) Sech[2 z]^2








Standard Form





Cell[BoxData[RowBox[List[RowBox[List["\[Integral]", RowBox[List[SqrtBox[SuperscriptBox[RowBox[List["Cosh", "[", RowBox[List["2", " ", "z"]], "]"]], "3"]], " ", SuperscriptBox[RowBox[List["Csch", "[", "z", "]"]], "3"], RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", RowBox[List[RowBox[List["-", FractionBox["1", "4"]]], " ", SqrtBox[SuperscriptBox[RowBox[List["Cosh", "[", RowBox[List["2", " ", "z"]], "]"]], "3"]], " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", RowBox[List["ArcTan", "[", FractionBox[RowBox[List["Cosh", "[", "z", "]"]], SqrtBox[RowBox[List["-", RowBox[List["Cosh", "[", RowBox[List["2", " ", "z"]], "]"]]]]]], "]"]], " ", SqrtBox[RowBox[List["-", RowBox[List["Cosh", "[", RowBox[List["2", " ", "z"]], "]"]]]]]]], "+", RowBox[List["8", " ", RowBox[List["ArcTanh", "[", FractionBox[RowBox[List["Cosh", "[", "z", "]"]], SqrtBox[RowBox[List["Cosh", "[", RowBox[List["2", " ", "z"]], "]"]]]], "]"]], " ", SqrtBox[RowBox[List["Cosh", "[", RowBox[List["2", " ", "z"]], "]"]]]]], "+", RowBox[List[RowBox[List["Cosh", "[", "z", "]"]], " ", RowBox[List["(", RowBox[List["3", "+", SuperscriptBox[RowBox[List["Coth", "[", "z", "]"]], "2"]]], ")"]]]], "+", RowBox[List[RowBox[List["Coth", "[", "z", "]"]], " ", RowBox[List["Csch", "[", "z", "]"]]]], "-", RowBox[List["8", " ", SqrtBox["2"], " ", SqrtBox[RowBox[List["Cosh", "[", RowBox[List["2", " ", "z"]], "]"]]], " ", RowBox[List["Log", "[", RowBox[List[RowBox[List[SqrtBox["2"], " ", RowBox[List["Cosh", "[", "z", "]"]]]], "+", SqrtBox[RowBox[List["Cosh", "[", RowBox[List["2", " ", "z"]], "]"]]]]], "]"]]]]]], ")"]], " ", SuperscriptBox[RowBox[List["Sech", "[", RowBox[List["2", " ", "z"]], "]"]], "2"]]]]]]]








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> <msqrt> <mrow> <msup> <mi> cosh </mi> <mn> 3 </mn> </msup> <mo> ( </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </msqrt> <mo> &#8290; </mo> <mrow> <msup> <mi> csch </mi> <mn> 3 </mn> </msup> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> &#8518; </mo> <mi> z </mi> </mrow> </mrow> </mrow> <mo> &#10869; </mo> <mrow> <mrow> <mo> - </mo> <mfrac> <mn> 1 </mn> <mn> 4 </mn> </mfrac> </mrow> <mo> &#8290; </mo> <msqrt> <mrow> <msup> <mi> cosh </mi> <mn> 3 </mn> </msup> <mo> ( </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </msqrt> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <msqrt> <mrow> <mo> - </mo> <mrow> <mi> cosh </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </msqrt> <mo> &#8290; </mo> <mrow> <msup> <mi> tan </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mfrac> <mrow> <mi> cosh </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <msqrt> <mrow> <mo> - </mo> <mrow> <mi> cosh </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </msqrt> </mfrac> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mrow> <mi> cosh </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <msup> <mi> coth </mi> <mn> 2 </mn> </msup> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> + </mo> <mn> 3 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mrow> <mi> coth </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> csch </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <mn> 8 </mn> <mo> &#8290; </mo> <msqrt> <mn> 2 </mn> </msqrt> <mo> &#8290; </mo> <mrow> <msup> <mi> cosh </mi> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </msup> <mo> ( </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mrow> <msqrt> <mn> 2 </mn> </msqrt> <mo> &#8290; </mo> <mrow> <mi> cosh </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <msup> <mi> cosh </mi> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </msup> <mo> ( </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mn> 8 </mn> <mo> &#8290; </mo> <mrow> <msup> <mi> tanh </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mfrac> <mrow> <mi> cosh </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mrow> <msup> <mi> cosh </mi> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </msup> <mo> ( </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mfrac> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <msup> <mi> cosh </mi> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </msup> <mo> ( </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <msup> <mi> sech </mi> <mn> 2 </mn> </msup> <mo> ( </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <int /> <bvar> <ci> z </ci> </bvar> <apply> <times /> <apply> <power /> <apply> <power /> <apply> <cosh /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> z </ci> </apply> </apply> <cn type='integer'> 3 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <csch /> <ci> z </ci> </apply> <cn type='integer'> 3 </cn> </apply> </apply> </apply> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <cn type='rational'> 1 <sep /> 4 </cn> </apply> <apply> <power /> <apply> <power /> <apply> <cosh /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> z </ci> </apply> </apply> <cn type='integer'> 3 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <cosh /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> z </ci> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <arctan /> <apply> <times /> <apply> <cosh /> <ci> z </ci> </apply> <apply> <power /> <apply> <power /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <cosh /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> z </ci> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <times /> <apply> <cosh /> <ci> z </ci> </apply> <apply> <plus /> <apply> <power /> <apply> <coth /> <ci> z </ci> </apply> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> 3 </cn> </apply> </apply> <apply> <times /> <apply> <coth /> <ci> z </ci> </apply> <apply> <csch /> <ci> z </ci> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 8 </cn> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <cosh /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> z </ci> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <ln /> <apply> <plus /> <apply> <times /> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <cosh /> <ci> z </ci> </apply> </apply> <apply> <power /> <apply> <cosh /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> z </ci> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 8 </cn> <apply> <arctanh /> <apply> <times /> <apply> <cosh /> <ci> z </ci> </apply> <apply> <power /> <apply> <power /> <apply> <cosh /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> z </ci> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <power /> <apply> <cosh /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> z </ci> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> <apply> <power /> <apply> <sech /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> z </ci> </apply> </apply> <cn type='integer'> 2 </cn> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["\[Integral]", RowBox[List[RowBox[List[SqrtBox[SuperscriptBox[RowBox[List["Cosh", "[", RowBox[List["2", " ", "z_"]], "]"]], "3"]], " ", SuperscriptBox[RowBox[List["Csch", "[", "z_", "]"]], "3"]]], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List["-", FractionBox["1", "4"]]], " ", SqrtBox[SuperscriptBox[RowBox[List["Cosh", "[", RowBox[List["2", " ", "z"]], "]"]], "3"]], " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", RowBox[List["ArcTan", "[", FractionBox[RowBox[List["Cosh", "[", "z", "]"]], SqrtBox[RowBox[List["-", RowBox[List["Cosh", "[", RowBox[List["2", " ", "z"]], "]"]]]]]], "]"]], " ", SqrtBox[RowBox[List["-", RowBox[List["Cosh", "[", RowBox[List["2", " ", "z"]], "]"]]]]]]], "+", RowBox[List["8", " ", RowBox[List["ArcTanh", "[", FractionBox[RowBox[List["Cosh", "[", "z", "]"]], SqrtBox[RowBox[List["Cosh", "[", RowBox[List["2", " ", "z"]], "]"]]]], "]"]], " ", SqrtBox[RowBox[List["Cosh", "[", RowBox[List["2", " ", "z"]], "]"]]]]], "+", RowBox[List[RowBox[List["Cosh", "[", "z", "]"]], " ", RowBox[List["(", RowBox[List["3", "+", SuperscriptBox[RowBox[List["Coth", "[", "z", "]"]], "2"]]], ")"]]]], "+", RowBox[List[RowBox[List["Coth", "[", "z", "]"]], " ", RowBox[List["Csch", "[", "z", "]"]]]], "-", RowBox[List["8", " ", SqrtBox["2"], " ", SqrtBox[RowBox[List["Cosh", "[", RowBox[List["2", " ", "z"]], "]"]]], " ", RowBox[List["Log", "[", RowBox[List[RowBox[List[SqrtBox["2"], " ", RowBox[List["Cosh", "[", "z", "]"]]]], "+", SqrtBox[RowBox[List["Cosh", "[", RowBox[List["2", " ", "z"]], "]"]]]]], "]"]]]]]], ")"]], " ", SuperscriptBox[RowBox[List["Sech", "[", RowBox[List["2", " ", "z"]], "]"]], "2"]]]]]]]


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





2002-12-18