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Cosh




Mathematica Notation

Traditional Notation









Elementary Functions > Cosh[z] > Integration > Indefinite integration > Involving functions of the direct function, hyperbolic and exponential functions > Involving rational functions of the direct function, hyperbolic and exponential functions > Involving rational functions of sinh and exp > Involving ep zsinh(d z)(a sinh2(e z)+b cosh2(e z))-n





http://functions.wolfram.com/01.20.21.4710.01









  


  








Input Form





Integrate[(E^(p z) Sinh[d z])/(a Sinh[e z]^2 + b Cosh[e z]^2)^2, z] == (1/2) ((I E^((-d + 2 e + p) z) ((Sqrt[a] + I Sqrt[b])^2 (a - b) Hypergeometric2F1[1 + (-d + p)/(2 e), 1, 2 + (-d + p)/(2 e), ((a + b) E^(2 e z))/(Sqrt[a] - I Sqrt[b])^2] - (Sqrt[a] - I Sqrt[b])^2 (a - b) Hypergeometric2F1[1 + (-d + p)/(2 e), 1, 2 + (-d + p)/(2 e), ((a + b) E^(2 e z))/(Sqrt[a] + I Sqrt[b])^2] - 2 I Sqrt[a] Sqrt[b] ((Sqrt[a] + I Sqrt[b])^2 Hypergeometric2F1[ 1 + (-d + p)/(2 e), 2, 2 + (-d + p)/(2 e), ((a + b) E^(2 e z))/ (Sqrt[a] - I Sqrt[b])^2] + (Sqrt[a] - I Sqrt[b])^2 Hypergeometric2F1[1 + (-d + p)/(2 e), 2, 2 + (-d + p)/(2 e), ((a + b) E^(2 e z))/(Sqrt[a] + I Sqrt[b])^2])))/ (2 a^(3/2) b^(3/2) (a + b) (-d + 2 e + p)) - (I E^((d + 2 e + p) z) ((Sqrt[a] + I Sqrt[b])^2 (a - b) Hypergeometric2F1[1 + (d + p)/(2 e), 1, 2 + (d + p)/(2 e), ((a + b) E^(2 e z))/(Sqrt[a] - I Sqrt[b])^2] - (Sqrt[a] - I Sqrt[b])^2 (a - b) Hypergeometric2F1[1 + (d + p)/(2 e), 1, 2 + (d + p)/(2 e), ((a + b) E^(2 e z))/(Sqrt[a] + I Sqrt[b])^2] - 2 I Sqrt[a] Sqrt[b] ((Sqrt[a] + I Sqrt[b])^2 Hypergeometric2F1[ 1 + (d + p)/(2 e), 2, 2 + (d + p)/(2 e), ((a + b) E^(2 e z))/ (Sqrt[a] - I Sqrt[b])^2] + (Sqrt[a] - I Sqrt[b])^2 Hypergeometric2F1[1 + (d + p)/(2 e), 2, 2 + (d + p)/(2 e), ((a + b) E^(2 e z))/(Sqrt[a] + I Sqrt[b])^2])))/ (2 a^(3/2) b^(3/2) (a + b) (d + 2 e + p)))








Standard Form





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MathML Form







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<apply> <power /> <apply> <power /> <apply> <plus /> <apply> <power /> <ci> a </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <imaginaryi /> <apply> <power /> <ci> b </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> </apply> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <power /> <apply> <plus /> <apply> <power /> <ci> a </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <times /> <imaginaryi /> <apply> <power /> <ci> b </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <power /> <apply> <plus /> <apply> <power /> <ci> a </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <imaginaryi /> <apply> <power /> <ci> b </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <plus /> <ci> a </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> b </ci> </apply> </apply> <apply> <ci> Hypergeometric2F1 </ci> <apply> <plus /> <apply> <times /> <apply> <plus /> <ci> d </ci> <ci> p </ci> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> e </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='integer'> 1 </cn> </apply> <cn type='integer'> 1 </cn> <apply> <plus /> <apply> <times /> <apply> <plus /> <ci> d </ci> <ci> p </ci> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> e </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <times /> <apply> <plus /> <ci> a </ci> <ci> b </ci> </apply> <apply> <power /> <exponentiale /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> e </ci> <ci> z </ci> </apply> </apply> <apply> <power /> <apply> <power /> <apply> <plus /> <apply> <power /> <ci> a </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <times /> <imaginaryi /> <apply> <power /> <ci> b </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <imaginaryi /> <apply> <power /> <ci> a </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <ci> b </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <plus /> <apply> <times /> <apply> <ci> Hypergeometric2F1 </ci> <apply> <plus /> <apply> <times /> <apply> <plus /> <ci> d </ci> <ci> p </ci> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> e </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='integer'> 1 </cn> </apply> <cn type='integer'> 2 </cn> <apply> <plus /> <apply> <times /> <apply> <plus /> <ci> d </ci> <ci> p </ci> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> e </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <times /> <apply> <plus /> <ci> a </ci> <ci> b </ci> </apply> <apply> <power /> <exponentiale /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> e </ci> <ci> z </ci> </apply> </apply> <apply> <power /> <apply> <power /> <apply> <plus /> <apply> <power /> <ci> a </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <imaginaryi /> <apply> <power /> <ci> b </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> </apply> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <power /> <apply> <plus /> <apply> <power /> <ci> a </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <times /> <imaginaryi /> <apply> <power /> <ci> b </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <apply> <power /> <apply> <plus /> <apply> <power /> <ci> a </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <imaginaryi /> <apply> <power /> <ci> b </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <ci> Hypergeometric2F1 </ci> <apply> <plus /> <apply> <times /> <apply> <plus /> <ci> d </ci> <ci> p </ci> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> e </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='integer'> 1 </cn> </apply> <cn type='integer'> 2 </cn> <apply> <plus /> <apply> <times /> <apply> <plus /> <ci> d </ci> <ci> p </ci> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> e </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <times /> <apply> <plus /> <ci> a </ci> <ci> b </ci> </apply> <apply> <power /> <exponentiale /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> e </ci> <ci> z </ci> </apply> </apply> <apply> <power /> <apply> <power /> <apply> <plus /> <apply> <power /> <ci> a </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <times /> <imaginaryi /> <apply> <power /> <ci> b </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





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





2002-12-18