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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 rational functions of the direct function and hyperbolic functions > Involving rational functions of cosh > Involving (a cosh(z)+b csch(z))-n





http://functions.wolfram.com/01.23.21.0440.01









  


  










Input Form





Integrate[1/(a Cosh[z] + b Csch[z])^2, z] == ((Csch[z]^2 (2 b + a Sinh[2 z]))/(4 (a^2 + 4 b^2) (a Cosh[z] + b Csch[z])^2)) (-a - (4 b^2)/a + a Cosh[2 z] - ((4 b (2 b + a Sinh[2 z]))/Sqrt[-a^2 - 4 b^2]) ArcTan[(a - 2 b Tanh[z])/Sqrt[-a^2 - 4 b^2]])










Standard Form





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MathML Form







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</mo> <msup> <mi> b </mi> <mn> 2 </mn> </msup> </mrow> </mrow> </msqrt> </mfrac> <mo> ) </mo> </mrow> </mrow> <mtext> </mtext> <mo> - </mo> <mi> a </mi> <mo> + </mo> <mrow> <mi> a </mi> <mo> &#8290; </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> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <int /> <bvar> <ci> z </ci> </bvar> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <power /> <apply> <plus /> <apply> <times /> <ci> a </ci> <apply> <cosh /> <ci> z </ci> </apply> </apply> <apply> <times /> <ci> b </ci> <apply> <csch /> <ci> z </ci> </apply> </apply> </apply> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <times /> <apply> <times /> <apply> <power /> <apply> <csch /> <ci> z </ci> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> b </ci> </apply> <apply> <times /> <ci> a </ci> <apply> <sinh /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> z </ci> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 4 </cn> <apply> <plus /> <apply> <power /> <ci> a </ci> <cn type='integer'> 2 </cn> </apply> <apply> <times /> <cn type='integer'> 4 </cn> <apply> <power /> <ci> b </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <apply> <power /> <apply> <plus /> <apply> <times /> <ci> a </ci> <apply> <cosh /> <ci> z </ci> </apply> </apply> <apply> <times /> <ci> b </ci> <apply> <csch /> <ci> z </ci> </apply> </apply> </apply> <cn type='integer'> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 4 </cn> <apply> <power /> <ci> b </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <ci> a </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <times /> <cn type='integer'> 4 </cn> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> b </ci> </apply> <apply> <times /> <ci> a </ci> <apply> <sinh /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> z </ci> </apply> </apply> </apply> </apply> <ci> b </ci> <apply> <power /> <apply> <power /> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <ci> a </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 4 </cn> <apply> <power /> <ci> b </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <arctan /> <apply> <times /> <apply> <plus /> <ci> a </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <ci> b </ci> <apply> <tanh /> <ci> z </ci> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <power /> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <ci> a </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 4 </cn> <apply> <power /> <ci> b </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> a </ci> </apply> <apply> <times /> <ci> a </ci> <apply> <cosh /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> z </ci> </apply> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["\[Integral]", RowBox[List[FractionBox["1", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["a_", " ", RowBox[List["Cosh", "[", "z_", "]"]]]], "+", RowBox[List["b_", " ", RowBox[List["Csch", "[", "z_", "]"]]]]]], ")"]], "2"]], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["Csch", "[", "z", "]"]], "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "b"]], "+", RowBox[List["a", " ", RowBox[List["Sinh", "[", RowBox[List["2", " ", "z"]], "]"]]]]]], ")"]]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "a"]], "-", FractionBox[RowBox[List["4", " ", SuperscriptBox["b", "2"]]], "a"], "+", RowBox[List["a", " ", RowBox[List["Cosh", "[", RowBox[List["2", " ", "z"]], "]"]]]], "-", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["4", " ", "b", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "b"]], "+", RowBox[List["a", " ", RowBox[List["Sinh", "[", RowBox[List["2", " ", "z"]], "]"]]]]]], ")"]]]], ")"]], " ", RowBox[List["ArcTan", "[", FractionBox[RowBox[List["a", "-", RowBox[List["2", " ", "b", " ", RowBox[List["Tanh", "[", "z", "]"]]]]]], SqrtBox[RowBox[List[RowBox[List["-", SuperscriptBox["a", "2"]]], "-", RowBox[List["4", " ", SuperscriptBox["b", "2"]]]]]]], "]"]]]], SqrtBox[RowBox[List[RowBox[List["-", SuperscriptBox["a", "2"]]], "-", RowBox[List["4", " ", SuperscriptBox["b", "2"]]]]]]]]], ")"]]]], RowBox[List["4", " ", RowBox[List["(", RowBox[List[SuperscriptBox["a", "2"], "+", RowBox[List["4", " ", SuperscriptBox["b", "2"]]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["a", " ", RowBox[List["Cosh", "[", "z", "]"]]]], "+", RowBox[List["b", " ", RowBox[List["Csch", "[", "z", "]"]]]]]], ")"]], "2"]]]]]]]]










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





2002-12-18





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