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Cosh






Mathematica Notation

Traditional Notation









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





http://functions.wolfram.com/01.20.21.4893.01









  


  










Input Form





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










Standard Form





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<times /> <cn type='integer'> -1 </cn> <apply> <times /> <imaginaryi /> <ci> d </ci> </apply> </apply> <ci> p </ci> </apply> <ci> z </ci> </apply> </apply> <apply> <plus /> <apply> <times /> <apply> <ci> Hypergeometric2F1 </ci> <apply> <times /> <apply> <plus /> <ci> e </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <imaginaryi /> <ci> d </ci> </apply> </apply> <ci> p </ci> </apply> <apply> <power /> <ci> e </ci> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='integer'> 2 </cn> <apply> <plus /> <apply> <times /> <apply> <plus /> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <imaginaryi /> </apply> <ci> d </ci> </apply> <ci> p </ci> </apply> <apply> <power /> <ci> e </ci> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <times /> <apply> <plus /> <ci> b </ci> <ci> c </ci> </apply> <apply> <power /> <exponentiale /> <apply> <times /> <ci> e </ci> <ci> z </ci> </apply> </apply> <apply> <power /> <apply> <plus /> <apply> <power /> <apply> <plus /> <apply> <power /> <ci> a </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <ci> b </ci> <cn type='integer'> 2 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <ci> c </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> a </ci> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <power /> <ci> a </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <ci> Hypergeometric2F1 </ci> <apply> <times /> <apply> <plus /> <ci> e </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <imaginaryi /> <ci> d </ci> </apply> </apply> <ci> p </ci> </apply> <apply> <power /> <ci> e </ci> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='integer'> 2 </cn> <apply> <plus /> <apply> <times /> <apply> <plus /> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <imaginaryi /> </apply> <ci> d </ci> </apply> <ci> p </ci> </apply> <apply> <power /> <ci> e </ci> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <plus /> <ci> b </ci> <ci> c </ci> </apply> <apply> <power /> <exponentiale /> <apply> <times /> <ci> e </ci> <ci> z </ci> </apply> </apply> <apply> <power /> <apply> <plus /> <ci> a </ci> <apply> <power /> <apply> <plus /> <apply> <power /> <ci> a </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <ci> b </ci> <cn type='integer'> 2 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <ci> c </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <power /> <ci> a </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <apply> <times /> <apply> <plus /> <ci> a </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <apply> <plus /> <apply> <power /> <ci> a </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <ci> b </ci> <cn type='integer'> 2 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <ci> c </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> <apply> <ci> Hypergeometric2F1 </ci> <apply> <times /> <apply> <plus /> <ci> e </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <imaginaryi /> <ci> d </ci> </apply> </apply> <ci> p </ci> </apply> <apply> <power /> <ci> e </ci> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='integer'> 1 </cn> <apply> <plus /> <apply> <times /> <apply> <plus /> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <imaginaryi /> </apply> <ci> d </ci> 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</apply> </apply> <apply> <power /> <apply> <plus /> <apply> <power /> <apply> <plus /> <apply> <power /> <ci> a </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <ci> b </ci> <cn type='integer'> 2 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <ci> c </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> a </ci> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <ci> a </ci> </apply> <apply> <times /> <apply> <power /> <apply> <plus /> <apply> <power /> <ci> a </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <ci> b </ci> <cn type='integer'> 2 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <ci> c </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <ci> Hypergeometric2F1 </ci> <apply> <times /> <apply> <plus /> <ci> e </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <imaginaryi /> <ci> d </ci> </apply> </apply> <ci> p </ci> </apply> <apply> <power /> <ci> e </ci> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='integer'> 2 </cn> <apply> <plus /> <apply> <times /> <apply> <plus /> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <imaginaryi /> </apply> <ci> d </ci> </apply> <ci> p </ci> </apply> <apply> <power /> <ci> e </ci> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <plus /> <ci> b </ci> <ci> c </ci> </apply> <apply> <power /> <exponentiale /> <apply> <times /> <ci> e </ci> <ci> z </ci> </apply> </apply> <apply> <power /> <apply> <plus /> <ci> a </ci> <apply> <power /> <apply> <plus /> <apply> <power /> <ci> a </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <ci> b </ci> <cn type='integer'> 2 </cn> </apply> <apply> 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type='integer'> -1 </cn> </apply> </apply> <cn type='integer'> 1 </cn> <apply> <plus /> <apply> <times /> <apply> <plus /> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <imaginaryi /> </apply> <ci> d </ci> </apply> <ci> p </ci> </apply> <apply> <power /> <ci> e </ci> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <times /> <apply> <plus /> <ci> b </ci> <ci> c </ci> </apply> <apply> <power /> <exponentiale /> <apply> <times /> <ci> e </ci> <ci> z </ci> </apply> </apply> <apply> <power /> <apply> <plus /> <apply> <power /> <apply> <plus /> <apply> <power /> <ci> a </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <ci> b </ci> <cn type='integer'> 2 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <ci> c </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> a </ci> </apply> 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/> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <imaginaryi /> </apply> <ci> d </ci> </apply> <ci> p </ci> </apply> <apply> <power /> <ci> e </ci> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <times /> <apply> <plus /> <ci> b </ci> <ci> c </ci> </apply> <apply> <power /> <exponentiale /> <apply> <times /> <ci> e </ci> <ci> z </ci> </apply> </apply> <apply> <power /> <apply> <plus /> <apply> <power /> <apply> <plus /> <apply> <power /> <ci> a </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <ci> b </ci> <cn type='integer'> 2 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <ci> c </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> a </ci> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <power /> <ci> b </ci> <cn type='integer'> 2 </cn> </apply> <apply> <ci> Hypergeometric2F1 </ci> <apply> <times /> <apply> <plus /> <ci> e </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <imaginaryi /> <ci> d </ci> </apply> </apply> <ci> p </ci> </apply> <apply> <power /> <ci> e </ci> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='integer'> 2 </cn> <apply> <plus /> <apply> <times /> <apply> <plus /> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <imaginaryi /> </apply> <ci> d </ci> </apply> <ci> p </ci> </apply> <apply> <power /> <ci> e </ci> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <plus /> <ci> b </ci> <ci> c </ci> </apply> <apply> <power /> <exponentiale /> <apply> <times /> <ci> e </ci> <ci> z </ci> </apply> </apply> <apply> <power /> <apply> <plus /> <ci> a </ci> <apply> <power /> <apply> <plus /> <apply> <power /> <ci> a </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <ci> b </ci> <cn type='integer'> 2 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <ci> c </ci> <cn type='integer'> 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> <apply> <times /> <apply> <power /> <ci> c </ci> <cn type='integer'> 2 </cn> </apply> <apply> <ci> Hypergeometric2F1 </ci> <apply> <times /> <apply> <plus /> <ci> e </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <imaginaryi /> <ci> d </ci> </apply> </apply> <ci> p </ci> </apply> <apply> <power /> <ci> e </ci> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='integer'> 2 </cn> <apply> <plus /> <apply> <times /> <apply> <plus /> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <imaginaryi /> </apply> <ci> d </ci> </apply> 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Date Added to functions.wolfram.com (modification date)





2002-12-18