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Cosh






Mathematica Notation

Traditional Notation









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





http://functions.wolfram.com/01.20.21.4783.01









  


  










Input Form





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










Standard Form





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MathML Form







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<apply> <times /> <apply> <plus /> <ci> a </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> b </ci> </apply> </apply> <apply> <plus /> <ci> a </ci> <ci> c </ci> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <power /> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> a </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> b </ci> </apply> <ci> c </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <apply> <times /> <apply> <plus /> <ci> a </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> b </ci> </apply> </apply> <apply> <plus /> <ci> a </ci> <ci> c </ci> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <apply> <times /> <apply> <plus /> <ci> a </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> b </ci> </apply> </apply> <apply> <plus /> <ci> a </ci> <ci> c </ci> </apply> </apply> <cn type='rational'> 3 <sep /> 2 </cn> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> e </ci> </apply> <apply> <times /> <imaginaryi /> <ci> d </ci> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </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