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Csch






Mathematica Notation

Traditional Notation









Elementary Functions > Csch[z] > Integration > Indefinite integration > Involving functions of the direct function > Involving powers of products of the direct function





http://functions.wolfram.com/01.23.21.0307.01









  


  










Input Form





Integrate[Sqrt[Csch[c z] Csch[2 c z]], z] == -(2^(3/4) (Coth[c z]^2)^(3/4) Sqrt[Csch[c z]/Sqrt[Cosh[c z]^2]] Sqrt[Csch[c z] Csch[2 c z]] Hypergeometric2F1[3/4, 3/4, 7/4, -Csch[c z]^2] Tanh[c z])/(3 c Sqrt[Csch[c z]/Sqrt[1 + Cosh[2 c z]]])










Standard Form





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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> <mrow> <mi> csch </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> c </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> csch </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> c </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </msqrt> <mo> &#8290; </mo> <mrow> <mo> &#8518; </mo> <mi> z </mi> </mrow> </mrow> </mrow> <mo> &#10869; </mo> <mrow> <mrow> <mo> - </mo> <mfrac> <mrow> <msup> <mn> 2 </mn> <mrow> <mn> 3 </mn> <mo> / </mo> <mn> 4 </mn> </mrow> </msup> <mo> &#8290; </mo> <msup> <mrow> <msup> <mi> coth </mi> <mn> 2 </mn> </msup> <mo> ( </mo> <mrow> <mi> c </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mrow> <mn> 3 </mn> <mo> / </mo> <mn> 4 </mn> </mrow> </msup> <mo> &#8290; </mo> <mtext> </mtext> <msqrt> <mrow> <mrow> <mi> csch </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> c </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> csch </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> c </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </msqrt> <mo> &#8290; </mo> <mrow> <mi> tanh </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> c </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mrow> <mn> 3 </mn> <mo> &#8290; </mo> <mi> c </mi> <mo> &#8290; </mo> <msqrt> <mfrac> <mrow> <mi> csch </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> c </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <msqrt> <mrow> <mrow> <mi> cosh </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> c </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> </msqrt> </mfrac> </msqrt> </mrow> </mfrac> </mrow> <mo> &#8290; </mo> <msqrt> <mfrac> <mrow> <mi> csch </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> c </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <msqrt> <mrow> <msup> <mi> cosh </mi> <mn> 2 </mn> </msup> <mo> ( </mo> <mrow> <mi> c </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </msqrt> </mfrac> </msqrt> <mo> &#8290; </mo> <semantics> <mrow> <mrow> <msub> <mo> &#8202; </mo> <mn> 2 </mn> </msub> <msub> <mi> F </mi> <mn> 1 </mn> </msub> </mrow> <mo> &#8289; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mfrac> <mn> 3 </mn> <mn> 4 </mn> </mfrac> <mo> , </mo> <mfrac> <mn> 3 </mn> <mn> 4 </mn> </mfrac> </mrow> <mo> ; </mo> <mfrac> <mn> 7 </mn> <mn> 4 </mn> </mfrac> <mo> ; </mo> <mrow> <mo> - </mo> <mrow> <msup> <mi> csch </mi> <mn> 2 </mn> </msup> <mo> ( </mo> <mrow> <mi> c </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation encoding='Mathematica'> TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox[&quot;\[InvisiblePrefixScriptBase]&quot;, FormBox[&quot;2&quot;, TraditionalForm]], SubscriptBox[&quot;F&quot;, FormBox[&quot;1&quot;, TraditionalForm]]]], &quot;\[InvisibleApplication]&quot;, RowBox[List[&quot;(&quot;, RowBox[List[TagBox[TagBox[RowBox[List[TagBox[FractionBox[&quot;3&quot;, &quot;4&quot;], Hypergeometric2F1], &quot;,&quot;, TagBox[FractionBox[&quot;3&quot;, &quot;4&quot;], Hypergeometric2F1]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1], &quot;;&quot;, TagBox[TagBox[TagBox[FractionBox[&quot;7&quot;, &quot;4&quot;], Hypergeometric2F1], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1], &quot;;&quot;, TagBox[RowBox[List[&quot;-&quot;, RowBox[List[SuperscriptBox[&quot;csch&quot;, &quot;2&quot;], &quot;(&quot;, RowBox[List[&quot;c&quot;, &quot; &quot;, &quot;z&quot;]], &quot;)&quot;]]]], Hypergeometric2F1]]], &quot;)&quot;]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]]], Hypergeometric2F1] </annotation> </semantics> <mtext> </mtext> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <int /> <bvar> <ci> z </ci> </bvar> <apply> <power /> <apply> <times /> <apply> <csch /> <apply> <times /> <ci> c </ci> <ci> z </ci> </apply> </apply> <apply> <csch /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> c </ci> <ci> z </ci> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='rational'> 3 <sep /> 4 </cn> </apply> <apply> <power /> <apply> <power /> <apply> <coth /> <apply> <times /> <ci> c </ci> <ci> z </ci> </apply> </apply> <cn type='integer'> 2 </cn> </apply> <cn type='rational'> 3 <sep /> 4 </cn> </apply> <apply> <power /> <apply> <times /> <apply> <csch /> <apply> <times /> <ci> c </ci> <ci> z </ci> </apply> </apply> <apply> <csch /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> c </ci> <ci> z </ci> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <tanh /> <apply> <times /> <ci> c </ci> <ci> z </ci> </apply> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 3 </cn> <ci> c </ci> <apply> <power /> <apply> <times /> <apply> <csch /> <apply> <times /> <ci> c </ci> <ci> z </ci> </apply> </apply> <apply> <power /> <apply> <power /> <apply> <plus /> <apply> <cosh /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> c </ci> <ci> z </ci> </apply> </apply> <cn type='integer'> 1 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <power /> <apply> <times /> <apply> <csch /> <apply> <times /> <ci> c </ci> <ci> z </ci> </apply> </apply> <apply> <power /> <apply> <power /> <apply> <power /> <apply> <cosh /> <apply> <times /> <ci> c </ci> <ci> z </ci> </apply> </apply> <cn type='integer'> 2 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <ci> Hypergeometric2F1 </ci> <cn type='rational'> 3 <sep /> 4 </cn> <cn type='rational'> 3 <sep /> 4 </cn> <cn type='rational'> 7 <sep /> 4 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <apply> <csch /> <apply> <times /> <ci> c </ci> <ci> z </ci> </apply> </apply> <cn type='integer'> 2 </cn> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["\[Integral]", RowBox[List[SqrtBox[RowBox[List[RowBox[List["Csch", "[", RowBox[List["c_", " ", "z_"]], "]"]], " ", RowBox[List["Csch", "[", RowBox[List["2", " ", "c_", " ", "z_"]], "]"]]]]], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List["-", FractionBox[RowBox[List[SuperscriptBox["2", RowBox[List["3", "/", "4"]]], " ", SuperscriptBox[RowBox[List["(", SuperscriptBox[RowBox[List["Coth", "[", RowBox[List["c", " ", "z"]], "]"]], "2"], ")"]], RowBox[List["3", "/", "4"]]], " ", SqrtBox[FractionBox[RowBox[List["Csch", "[", RowBox[List["c", " ", "z"]], "]"]], SqrtBox[SuperscriptBox[RowBox[List["Cosh", "[", RowBox[List["c", " ", "z"]], "]"]], "2"]]]], " ", SqrtBox[RowBox[List[RowBox[List["Csch", "[", RowBox[List["c", " ", "z"]], "]"]], " ", RowBox[List["Csch", "[", RowBox[List["2", " ", "c", " ", "z"]], "]"]]]]], " ", RowBox[List["Hypergeometric2F1", "[", RowBox[List[FractionBox["3", "4"], ",", FractionBox["3", "4"], ",", FractionBox["7", "4"], ",", RowBox[List["-", SuperscriptBox[RowBox[List["Csch", "[", RowBox[List["c", " ", "z"]], "]"]], "2"]]]]], "]"]], " ", RowBox[List["Tanh", "[", RowBox[List["c", " ", "z"]], "]"]]]], RowBox[List["3", " ", "c", " ", SqrtBox[FractionBox[RowBox[List["Csch", "[", RowBox[List["c", " ", "z"]], "]"]], SqrtBox[RowBox[List["1", "+", RowBox[List["Cosh", "[", RowBox[List["2", " ", "c", " ", "z"]], "]"]]]]]]]]]]]]]]]]










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





2002-12-18





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