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

View Related Information In
The Documentation Center
MathWorld

Download All Formulas For This Function
Mathematica Notebook
PDF File

Download All Introductions For This Function
Mathematica Notebook
PDF File

 

Developed with Mathematica -- Download a Free Trial Version
 











Coth




Mathematica Notation

Traditional Notation









Elementary Functions > Coth[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 sinh > Involving (a sinh(c z)+b coth(c z))-n





http://functions.wolfram.com/01.22.21.0386.01









  


  








Input Form





Integrate[1/(a Sinh[c z] + b Coth[c z])^2, z] == (Csch[c z]^2 (b Cosh[c z] + a Sinh[c z]^2) (-((2 (b + 2 a Cosh[c z]) Sinh[c z])/(4 a^2 + b^2)) + (2 Sqrt[2] (4 a^2 + b (-b + Sqrt[4 a^2 + b^2])) ArcTan[((2 a - b + Sqrt[4 a^2 + b^2]) Tanh[(c z)/2])/ (Sqrt[2] Sqrt[b] Sqrt[-b + Sqrt[4 a^2 + b^2]])] (b Cosh[c z] + a Sinh[c z]^2))/(Sqrt[b] (4 a^2 + b^2)^(3/2) Sqrt[-b + Sqrt[4 a^2 + b^2]]) - (2 Sqrt[2] (-4 a^2 + b (b + Sqrt[4 a^2 + b^2])) ArcTan[((-2 a + b + Sqrt[4 a^2 + b^2]) Tanh[(c z)/2])/ (Sqrt[2] Sqrt[(-b) (b + Sqrt[4 a^2 + b^2])])] (b Cosh[c z] + a Sinh[c z]^2))/((4 a^2 + b^2)^(3/2) Sqrt[(-b) (b + Sqrt[4 a^2 + b^2])])))/ (2 c (b Coth[c z] + a Sinh[c z])^2)








Standard Form





Cell[BoxData[RowBox[List[RowBox[List["\[Integral]", RowBox[List[FractionBox["1", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["a", " ", RowBox[List["Sinh", "[", RowBox[List["c", " ", "z"]], "]"]]]], "+", RowBox[List["b", " ", RowBox[List["Coth", "[", RowBox[List["c", " ", "z"]], "]"]]]]]], ")"]], "2"]], RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["Csch", "[", RowBox[List["c", " ", "z"]], "]"]], "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["b", " ", RowBox[List["Cosh", "[", RowBox[List["c", " ", "z"]], "]"]]]], "+", RowBox[List["a", " ", SuperscriptBox[RowBox[List["Sinh", "[", RowBox[List["c", " ", "z"]], "]"]], "2"]]]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", FractionBox[RowBox[List["2", " ", RowBox[List["(", RowBox[List["b", "+", RowBox[List["2", " ", "a", " ", RowBox[List["Cosh", "[", RowBox[List["c", " ", "z"]], "]"]]]]]], ")"]], " ", RowBox[List["Sinh", "[", RowBox[List["c", " ", "z"]], "]"]]]], RowBox[List[RowBox[List["4", " ", SuperscriptBox["a", "2"]]], "+", SuperscriptBox["b", "2"]]]]]], "+", RowBox[List[RowBox[List["(", RowBox[List["2", " ", SqrtBox["2"], " ", RowBox[List["(", RowBox[List[RowBox[List["4", " ", SuperscriptBox["a", "2"]]], "+", RowBox[List["b", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "b"]], "+", SqrtBox[RowBox[List[RowBox[List["4", " ", SuperscriptBox["a", "2"]]], "+", SuperscriptBox["b", "2"]]]]]], ")"]]]]]], ")"]], " ", RowBox[List["ArcTan", "[", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["2", " ", "a"]], "-", "b", "+", SqrtBox[RowBox[List[RowBox[List["4", " ", SuperscriptBox["a", "2"]]], "+", SuperscriptBox["b", "2"]]]]]], ")"]], " ", RowBox[List["Tanh", "[", FractionBox[RowBox[List["c", " ", "z"]], "2"], "]"]]]], RowBox[List[SqrtBox["2"], " ", SqrtBox["b"], " ", SqrtBox[RowBox[List[RowBox[List["-", "b"]], "+", SqrtBox[RowBox[List[RowBox[List["4", " ", SuperscriptBox["a", "2"]]], "+", SuperscriptBox["b", "2"]]]]]]]]]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["b", " ", RowBox[List["Cosh", "[", RowBox[List["c", " ", "z"]], "]"]]]], "+", RowBox[List["a", " ", SuperscriptBox[RowBox[List["Sinh", "[", RowBox[List["c", " ", "z"]], "]"]], "2"]]]]], ")"]]]], ")"]], "/", RowBox[List["(", RowBox[List[SqrtBox["b"], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["4", " ", SuperscriptBox["a", "2"]]], "+", SuperscriptBox["b", "2"]]], ")"]], RowBox[List["3", "/", "2"]]], " ", SqrtBox[RowBox[List[RowBox[List["-", "b"]], "+", SqrtBox[RowBox[List[RowBox[List["4", " ", SuperscriptBox["a", "2"]]], "+", SuperscriptBox["b", "2"]]]]]]]]], ")"]]]], "-", RowBox[List[RowBox[List["(", RowBox[List["2", " ", SqrtBox["2"], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "4"]], " ", SuperscriptBox["a", "2"]]], "+", RowBox[List["b", " ", RowBox[List["(", RowBox[List["b", "+", SqrtBox[RowBox[List[RowBox[List["4", " ", SuperscriptBox["a", "2"]]], "+", SuperscriptBox["b", "2"]]]]]], ")"]]]]]], ")"]], " ", RowBox[List["ArcTan", "[", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "a"]], "+", "b", "+", SqrtBox[RowBox[List[RowBox[List["4", " ", SuperscriptBox["a", "2"]]], "+", SuperscriptBox["b", "2"]]]]]], ")"]], " ", RowBox[List["Tanh", "[", FractionBox[RowBox[List["c", " ", "z"]], "2"], "]"]]]], RowBox[List[SqrtBox["2"], " ", SqrtBox[RowBox[List[RowBox[List["-", "b"]], " ", RowBox[List["(", RowBox[List["b", "+", SqrtBox[RowBox[List[RowBox[List["4", " ", SuperscriptBox["a", "2"]]], "+", SuperscriptBox["b", "2"]]]]]], ")"]]]]]]]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["b", " ", RowBox[List["Cosh", "[", RowBox[List["c", " ", "z"]], "]"]]]], "+", RowBox[List["a", " ", SuperscriptBox[RowBox[List["Sinh", "[", RowBox[List["c", " ", "z"]], "]"]], "2"]]]]], ")"]]]], ")"]], "/", RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["4", " ", SuperscriptBox["a", "2"]]], "+", SuperscriptBox["b", "2"]]], ")"]], RowBox[List["3", "/", "2"]]], " ", SqrtBox[RowBox[List[RowBox[List["-", "b"]], " ", RowBox[List["(", RowBox[List["b", "+", SqrtBox[RowBox[List[RowBox[List["4", " ", SuperscriptBox["a", "2"]]], "+", SuperscriptBox["b", "2"]]]]]], ")"]]]]]]], ")"]]]]]], ")"]]]], ")"]], "/", RowBox[List["(", RowBox[List["2", " ", "c", " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["b", " ", RowBox[List["Coth", "[", RowBox[List["c", " ", "z"]], "]"]]]], "+", RowBox[List["a", " ", RowBox[List["Sinh", "[", RowBox[List["c", " ", "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> <mfrac> <mn> 1 </mn> <msup> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> a </mi> <mo> &#8290; </mo> <mrow> <mi> sinh </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> c </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mi> b </mi> <mo> &#8290; </mo> <mrow> <mi> coth </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> c </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> </mfrac> <mo> &#8290; </mo> <mrow> <mo> &#8518; </mo> <mi> z </mi> </mrow> </mrow> </mrow> <mo> &#10869; </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <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> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> a </mi> <mo> &#8290; </mo> <mrow> <msup> <mi> sinh </mi> <mn> 2 </mn> </msup> <mo> ( </mo> <mrow> <mi> c </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mi> b </mi> <mo> &#8290; </mo> <mrow> <mi> cosh </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> c </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <mfrac> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> b </mi> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> a </mi> <mo> &#8290; </mo> <mrow> <mi> cosh </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> c </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> sinh </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> c </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mrow> <mrow> <mn> 4 </mn> <mo> &#8290; </mo> <msup> <mi> a </mi> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <msup> <mi> b </mi> <mn> 2 </mn> </msup> </mrow> </mfrac> </mrow> <mo> + </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <msqrt> <mn> 2 </mn> </msqrt> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 4 </mn> <mo> &#8290; </mo> <msup> <mi> a </mi> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mi> b </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <msqrt> <mrow> <mrow> <mn> 4 </mn> <mo> &#8290; </mo> <msup> <mi> a </mi> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <msup> <mi> b </mi> <mn> 2 </mn> </msup> </mrow> </msqrt> <mo> - </mo> <mi> b </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <msup> <mi> tan </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mfrac> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> a </mi> </mrow> <mo> - </mo> <mi> b </mi> <mo> + </mo> <msqrt> <mrow> <mrow> <mn> 4 </mn> <mo> &#8290; </mo> <msup> <mi> a </mi> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <msup> <mi> b </mi> <mn> 2 </mn> </msup> </mrow> </msqrt> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> tanh </mi> <mo> &#8289; </mo> <mo> ( </mo> <mfrac> <mrow> <mi> c </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mn> 2 </mn> </mfrac> <mo> ) </mo> </mrow> </mrow> <mrow> <msqrt> <mn> 2 </mn> </msqrt> <mo> &#8290; </mo> <msqrt> <mi> b </mi> </msqrt> <mo> &#8290; </mo> <msqrt> <mrow> <msqrt> <mrow> <mrow> <mn> 4 </mn> <mo> &#8290; </mo> <msup> <mi> a </mi> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <msup> <mi> b </mi> <mn> 2 </mn> </msup> </mrow> </msqrt> <mo> - </mo> <mi> b </mi> </mrow> </msqrt> </mrow> </mfrac> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> a </mi> <mo> &#8290; </mo> <mrow> <msup> <mi> sinh </mi> <mn> 2 </mn> </msup> <mo> ( </mo> <mrow> <mi> c </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mi> b </mi> <mo> &#8290; </mo> <mrow> <mi> cosh </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> c </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> / </mo> <mrow> <mo> ( </mo> <mrow> <msqrt> <mi> b </mi> </msqrt> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 4 </mn> <mo> &#8290; </mo> <msup> <mi> a </mi> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <msup> <mi> b </mi> <mn> 2 </mn> </msup> </mrow> <mo> ) </mo> </mrow> <mrow> <mn> 3 </mn> <mo> / </mo> <mn> 2 </mn> </mrow> </msup> <mo> &#8290; </mo> <msqrt> <mrow> <msqrt> <mrow> <mrow> <mn> 4 </mn> <mo> &#8290; </mo> <msup> <mi> a </mi> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <msup> <mi> b </mi> <mn> 2 </mn> </msup> </mrow> </msqrt> <mo> - </mo> <mi> b </mi> </mrow> </msqrt> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <msqrt> <mn> 2 </mn> </msqrt> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> b </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> b </mi> <mo> + </mo> <msqrt> <mrow> <mrow> <mn> 4 </mn> <mo> &#8290; </mo> <msup> <mi> a </mi> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <msup> <mi> b </mi> <mn> 2 </mn> </msup> </mrow> </msqrt> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <mn> 4 </mn> <mo> &#8290; </mo> <msup> <mi> a </mi> <mn> 2 </mn> </msup> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <msup> <mi> tan </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mfrac> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mo> - </mo> <mn> 2 </mn> </mrow> <mo> &#8290; </mo> <mi> a </mi> </mrow> <mo> + </mo> <mi> b </mi> <mo> + </mo> <msqrt> <mrow> <mrow> <mn> 4 </mn> <mo> &#8290; </mo> <msup> <mi> a </mi> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <msup> <mi> b </mi> <mn> 2 </mn> </msup> </mrow> </msqrt> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> tanh </mi> <mo> &#8289; </mo> <mo> ( </mo> <mfrac> <mrow> <mi> c </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mn> 2 </mn> </mfrac> <mo> ) </mo> </mrow> </mrow> <mrow> <msqrt> <mn> 2 </mn> </msqrt> <mo> &#8290; </mo> <msqrt> <mrow> <mrow> <mo> - </mo> <mi> b </mi> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> b </mi> <mo> + </mo> <msqrt> <mrow> <mrow> <mn> 4 </mn> <mo> &#8290; </mo> <msup> <mi> a </mi> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <msup> <mi> b </mi> <mn> 2 </mn> </msup> </mrow> </msqrt> </mrow> <mo> ) </mo> </mrow> </mrow> </msqrt> </mrow> </mfrac> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> a </mi> <mo> &#8290; </mo> <mrow> <msup> <mi> sinh </mi> <mn> 2 </mn> </msup> <mo> ( </mo> <mrow> <mi> c </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mi> b </mi> <mo> &#8290; </mo> <mrow> <mi> cosh </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> c </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> / </mo> <mrow> <mo> ( </mo> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 4 </mn> <mo> &#8290; </mo> <msup> <mi> a </mi> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <msup> <mi> b </mi> <mn> 2 </mn> </msup> </mrow> <mo> ) </mo> </mrow> <mrow> <mn> 3 </mn> <mo> / </mo> <mn> 2 </mn> </mrow> </msup> <mo> &#8290; </mo> <msqrt> <mrow> <mrow> <mo> - </mo> <mi> b </mi> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> b </mi> <mo> + </mo> <msqrt> <mrow> <mrow> <mn> 4 </mn> <mo> &#8290; </mo> <msup> <mi> a </mi> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <msup> <mi> b </mi> <mn> 2 </mn> </msup> </mrow> </msqrt> </mrow> <mo> ) </mo> </mrow> </mrow> </msqrt> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> / </mo> <mrow> <mo> ( </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> c </mi> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> b </mi> <mo> &#8290; </mo> <mrow> <mi> coth </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> c </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mi> a </mi> <mo> &#8290; </mo> <mrow> <mi> sinh </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> c </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> </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> <sinh /> <apply> <times /> <ci> c </ci> <ci> z </ci> </apply> </apply> </apply> <apply> <times /> <ci> b </ci> <apply> <coth /> <apply> <times /> <ci> c </ci> <ci> z </ci> </apply> </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 /> <apply> <times /> <ci> c </ci> <ci> z </ci> </apply> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <plus /> <apply> <times /> <ci> a </ci> <apply> <power /> <apply> <sinh /> <apply> <times /> <ci> c </ci> <ci> z </ci> </apply> </apply> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <ci> b </ci> <apply> <cosh /> <apply> <times /> <ci> c </ci> <ci> z </ci> </apply> </apply> </apply> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <plus /> <ci> b </ci> <apply> <times /> <cn type='integer'> 2 </cn> <ci> a </ci> <apply> <cosh /> <apply> <times /> <ci> c </ci> <ci> z </ci> </apply> </apply> </apply> </apply> <apply> <sinh /> <apply> <times /> <ci> c </ci> <ci> z </ci> </apply> </apply> <apply> <power /> <apply> <plus /> <apply> <times /> <cn type='integer'> 4 </cn> <apply> <power /> <ci> a </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <power /> <ci> b </ci> <cn type='integer'> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <times /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 4 </cn> <apply> <power /> <ci> a </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <ci> b </ci> <apply> <plus /> <apply> <power /> <apply> <plus /> <apply> <times /> <cn type='integer'> 4 </cn> <apply> <power /> <ci> a </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <power /> <ci> b </ci> <cn type='integer'> 2 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> b </ci> </apply> </apply> </apply> </apply> <apply> <arctan /> <apply> <times /> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> a </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> b </ci> </apply> <apply> <power /> <apply> <plus /> <apply> <times /> <cn type='integer'> 4 </cn> <apply> <power /> <ci> a </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <power /> <ci> b </ci> <cn type='integer'> 2 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <apply> <tanh /> <apply> <times /> <ci> c </ci> <ci> z </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <power /> <apply> <times /> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <ci> b </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <plus /> <apply> <power /> <apply> <plus /> <apply> <times /> <cn type='integer'> 4 </cn> <apply> <power /> <ci> a </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <power /> <ci> b </ci> <cn type='integer'> 2 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> b </ci> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <plus /> <apply> <times /> <ci> a </ci> <apply> <power /> <apply> <sinh /> <apply> <times /> <ci> c </ci> <ci> z </ci> </apply> </apply> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <ci> b </ci> <apply> <cosh /> <apply> <times /> <ci> c </ci> <ci> z </ci> </apply> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <times /> <apply> <power /> <ci> b </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <plus /> <apply> <times /> <cn type='integer'> 4 </cn> <apply> <power /> <ci> a </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <power /> <ci> b </ci> <cn type='integer'> 2 </cn> </apply> </apply> <cn type='rational'> 3 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <plus /> <apply> <power /> <apply> <plus /> <apply> <times /> <cn type='integer'> 4 </cn> <apply> <power /> <ci> a </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <power /> <ci> b </ci> <cn type='integer'> 2 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> b </ci> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <plus /> <apply> <times /> <ci> b </ci> <apply> <plus /> <ci> b </ci> <apply> <power /> <apply> <plus /> <apply> <times /> <cn type='integer'> 4 </cn> <apply> <power /> <ci> a </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <power /> <ci> b </ci> <cn type='integer'> 2 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 4 </cn> <apply> <power /> <ci> a </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> </apply> <apply> <arctan /> <apply> <times /> <apply> <plus /> <apply> <times /> <cn type='integer'> -2 </cn> <ci> a </ci> </apply> <ci> b </ci> <apply> <power /> <apply> <plus /> <apply> <times /> <cn type='integer'> 4 </cn> <apply> <power /> <ci> a </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <power /> <ci> b </ci> <cn type='integer'> 2 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <apply> <tanh /> <apply> <times /> <ci> c </ci> <ci> z </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <power /> <apply> <times /> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> b </ci> </apply> <apply> <plus /> <ci> b </ci> <apply> <power /> <apply> <plus /> <apply> <times /> <cn type='integer'> 4 </cn> <apply> <power /> <ci> a </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <power /> <ci> b </ci> <cn type='integer'> 2 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <plus /> <apply> <times /> <ci> a </ci> <apply> <power /> <apply> <sinh /> <apply> <times /> <ci> c </ci> <ci> z </ci> </apply> </apply> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <ci> b </ci> <apply> <cosh /> <apply> <times /> <ci> c </ci> <ci> z </ci> </apply> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <times /> <apply> <power /> <apply> <plus /> <apply> <times /> <cn type='integer'> 4 </cn> <apply> <power /> <ci> a </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <power /> <ci> b </ci> <cn type='integer'> 2 </cn> </apply> </apply> <cn type='rational'> 3 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> b </ci> </apply> <apply> <plus /> <ci> b </ci> <apply> <power /> <apply> <plus /> <apply> <times /> <cn type='integer'> 4 </cn> <apply> <power /> <ci> a </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <power /> <ci> b </ci> <cn type='integer'> 2 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> c </ci> <apply> <power /> <apply> <plus /> <apply> <times /> <ci> b </ci> <apply> <coth /> <apply> <times /> <ci> c </ci> <ci> z </ci> </apply> </apply> </apply> <apply> <times /> <ci> a </ci> <apply> <sinh /> <apply> <times /> <ci> c </ci> <ci> z </ci> </apply> </apply> </apply> </apply> <cn type='integer'> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </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["Sinh", "[", RowBox[List["c_", " ", "z_"]], "]"]]]], "+", RowBox[List["b_", " ", RowBox[List["Coth", "[", RowBox[List["c_", " ", "z_"]], "]"]]]]]], ")"]], "2"]], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", FractionBox[RowBox[List[SuperscriptBox[RowBox[List["Csch", "[", RowBox[List["c", " ", "z"]], "]"]], "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["b", " ", RowBox[List["Cosh", "[", RowBox[List["c", " ", "z"]], "]"]]]], "+", RowBox[List["a", " ", SuperscriptBox[RowBox[List["Sinh", "[", RowBox[List["c", " ", "z"]], "]"]], "2"]]]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", FractionBox[RowBox[List["2", " ", RowBox[List["(", RowBox[List["b", "+", RowBox[List["2", " ", "a", " ", RowBox[List["Cosh", "[", RowBox[List["c", " ", "z"]], "]"]]]]]], ")"]], " ", RowBox[List["Sinh", "[", RowBox[List["c", " ", "z"]], "]"]]]], RowBox[List[RowBox[List["4", " ", SuperscriptBox["a", "2"]]], "+", SuperscriptBox["b", "2"]]]]]], "+", FractionBox[RowBox[List["2", " ", SqrtBox["2"], " ", RowBox[List["(", RowBox[List[RowBox[List["4", " ", SuperscriptBox["a", "2"]]], "+", RowBox[List["b", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "b"]], "+", SqrtBox[RowBox[List[RowBox[List["4", " ", SuperscriptBox["a", "2"]]], "+", SuperscriptBox["b", "2"]]]]]], ")"]]]]]], ")"]], " ", RowBox[List["ArcTan", "[", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["2", " ", "a"]], "-", "b", "+", SqrtBox[RowBox[List[RowBox[List["4", " ", SuperscriptBox["a", "2"]]], "+", SuperscriptBox["b", "2"]]]]]], ")"]], " ", RowBox[List["Tanh", "[", FractionBox[RowBox[List["c", " ", "z"]], "2"], "]"]]]], RowBox[List[SqrtBox["2"], " ", SqrtBox["b"], " ", SqrtBox[RowBox[List[RowBox[List["-", "b"]], "+", SqrtBox[RowBox[List[RowBox[List["4", " ", SuperscriptBox["a", "2"]]], "+", SuperscriptBox["b", "2"]]]]]]]]]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["b", " ", RowBox[List["Cosh", "[", RowBox[List["c", " ", "z"]], "]"]]]], "+", RowBox[List["a", " ", SuperscriptBox[RowBox[List["Sinh", "[", RowBox[List["c", " ", "z"]], "]"]], "2"]]]]], ")"]]]], RowBox[List[SqrtBox["b"], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["4", " ", SuperscriptBox["a", "2"]]], "+", SuperscriptBox["b", "2"]]], ")"]], RowBox[List["3", "/", "2"]]], " ", SqrtBox[RowBox[List[RowBox[List["-", "b"]], "+", SqrtBox[RowBox[List[RowBox[List["4", " ", SuperscriptBox["a", "2"]]], "+", SuperscriptBox["b", "2"]]]]]]]]]], "-", FractionBox[RowBox[List["2", " ", SqrtBox["2"], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "4"]], " ", SuperscriptBox["a", "2"]]], "+", RowBox[List["b", " ", RowBox[List["(", RowBox[List["b", "+", SqrtBox[RowBox[List[RowBox[List["4", " ", SuperscriptBox["a", "2"]]], "+", SuperscriptBox["b", "2"]]]]]], ")"]]]]]], ")"]], " ", RowBox[List["ArcTan", "[", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "a"]], "+", "b", "+", SqrtBox[RowBox[List[RowBox[List["4", " ", SuperscriptBox["a", "2"]]], "+", SuperscriptBox["b", "2"]]]]]], ")"]], " ", RowBox[List["Tanh", "[", FractionBox[RowBox[List["c", " ", "z"]], "2"], "]"]]]], RowBox[List[SqrtBox["2"], " ", SqrtBox[RowBox[List[RowBox[List["-", "b"]], " ", RowBox[List["(", RowBox[List["b", "+", SqrtBox[RowBox[List[RowBox[List["4", " ", SuperscriptBox["a", "2"]]], "+", SuperscriptBox["b", "2"]]]]]], ")"]]]]]]]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["b", " ", RowBox[List["Cosh", "[", RowBox[List["c", " ", "z"]], "]"]]]], "+", RowBox[List["a", " ", SuperscriptBox[RowBox[List["Sinh", "[", RowBox[List["c", " ", "z"]], "]"]], "2"]]]]], ")"]]]], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["4", " ", SuperscriptBox["a", "2"]]], "+", SuperscriptBox["b", "2"]]], ")"]], RowBox[List["3", "/", "2"]]], " ", SqrtBox[RowBox[List[RowBox[List["-", "b"]], " ", RowBox[List["(", RowBox[List["b", "+", SqrtBox[RowBox[List[RowBox[List["4", " ", SuperscriptBox["a", "2"]]], "+", SuperscriptBox["b", "2"]]]]]], ")"]]]]]]]]]], ")"]]]], RowBox[List["2", " ", "c", " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["b", " ", RowBox[List["Coth", "[", RowBox[List["c", " ", "z"]], "]"]]]], "+", RowBox[List["a", " ", RowBox[List["Sinh", "[", RowBox[List["c", " ", "z"]], "]"]]]]]], ")"]], "2"]]]]]]]]


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





2002-12-18