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Cosh






Mathematica Notation

Traditional Notation









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





http://functions.wolfram.com/01.20.21.3823.01









  


  










Input Form





Integrate[(E^(p z) Cos[e z] Cosh[c z]^v)/(a + b E^(d z))^n, z] == (2^(-1 - v) Binomial[v, v/2] ((-E^(((-I) e + p) z)) (I e + p) Hypergeometric2F1[((-I) e + p)/d, n, (d - I e + p)/d, -((b E^(d z))/a)] + E^((I e + p) z) (I e - p) Hypergeometric2F1[(I e + p)/d, n, (d + I e + p)/d, -((b E^(d z))/a)]) (1 - Mod[v, 2]))/a^n/((I e - p) (I e + p)) + (2^(-1 - v) Sum[Binomial[v, s] (((-E^((I e + p - c (2 s - v)) z)) ((-I) e + p + c (2 s - v)) Hypergeometric2F1[(I e + p - c (2 s - v))/d, n, (d + I e + p - c (2 s - v))/d, -((b E^(d z))/a)] + E^(((-I) e + p + c (2 s - v)) z) ((-I) e - p + c (2 s - v)) Hypergeometric2F1[((-I) e + p + c (2 s - v))/d, n, (d - I e + p + c (2 s - v))/d, -((b E^(d z))/a)])/ (((-I) e - p + c (2 s - v)) ((-I) e + p + c (2 s - v))) + (E^(((-I) e + p - c (2 s - v)) z) ((-I) e - p - c (2 s - v)) Hypergeometric2F1[((-I) e + p - c (2 s - v))/d, n, (d - I e + p - c (2 s - v))/d, -((b E^(d z))/a)] - E^((I e + p + c (2 s - v)) z) ((-I) e + p - c (2 s - v)) Hypergeometric2F1[(I e + p + c (2 s - v))/d, n, (d + I e + p + c (2 s - v))/d, -((b E^(d z))/a)])/ (((-I) e - p - c (2 s - v)) ((-I) e + p - c (2 s - v)))), {s, 0, Floor[(1/2) (-1 + v)]}])/a^n /; Element[n, Integers] && n > 0 && Element[v, Integers] && v > 0










Standard Form





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MathML Form







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<apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <imaginaryi /> </apply> <ci> e </ci> </apply> <ci> p </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <ci> c </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> s </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> v </ci> </apply> </apply> </apply> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <times /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <times /> <apply> <plus /> <apply> <times /> <imaginaryi /> <ci> e </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> p </ci> </apply> </apply> <apply> <plus /> <apply> <times /> <imaginaryi /> <ci> e </ci> </apply> <ci> p </ci> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <apply> <power /> <cn type='integer'> 2 </cn> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> v </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <ci> a </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> n </ci> </apply> </apply> <apply> <ci> Binomial </ci> <ci> v </ci> <apply> <times /> <ci> v </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <plus /> <apply> <times /> <apply> <power /> <exponentiale /> <apply> <times /> <apply> <plus /> <apply> <times /> <imaginaryi /> <ci> e </ci> </apply> <ci> p </ci> </apply> <ci> z </ci> </apply> </apply> <apply> <plus /> <apply> <times /> <imaginaryi /> <ci> e </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> p </ci> </apply> </apply> <apply> <ci> Hypergeometric2F1 </ci> <apply> <times /> <apply> <plus /> <apply> <times /> <imaginaryi /> <ci> e </ci> </apply> <ci> p </ci> </apply> <apply> <power /> <ci> d </ci> <cn type='integer'> -1 </cn> </apply> </apply> <ci> n </ci> <apply> <times /> <apply> <plus /> <ci> d </ci> <apply> <times /> <imaginaryi /> <ci> e </ci> </apply> <ci> p </ci> </apply> <apply> <power /> <ci> d </ci> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <ci> b </ci> <apply> <power /> <exponentiale /> <apply> <times /> <ci> d </ci> <ci> z </ci> </apply> </apply> <apply> <power /> <ci> a </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <power /> <exponentiale /> <apply> <times /> <apply> <plus /> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <imaginaryi /> </apply> <ci> e </ci> </apply> <ci> p </ci> </apply> <ci> z </ci> </apply> </apply> <apply> <plus /> <apply> <times /> <imaginaryi /> <ci> e </ci> </apply> <ci> p </ci> </apply> <apply> <ci> Hypergeometric2F1 </ci> <apply> <times /> <apply> <plus /> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <imaginaryi /> </apply> <ci> e </ci> </apply> <ci> p </ci> </apply> <apply> <power /> <ci> d </ci> <cn type='integer'> -1 </cn> </apply> </apply> <ci> n </ci> <apply> <times /> <apply> <plus /> <ci> d </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <imaginaryi /> <ci> e </ci> </apply> </apply> <ci> p </ci> </apply> <apply> <power /> <ci> d </ci> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <ci> b </ci> <apply> <power /> <exponentiale /> <apply> <times /> <ci> d </ci> <ci> z </ci> </apply> </apply> <apply> <power /> <ci> a </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <rem /> <ci> $CellContext`v </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <and /> <apply> <in /> <ci> n </ci> <apply> <ci> SuperPlus </ci> <ci> &#8469; </ci> </apply> </apply> <apply> <in /> <ci> v </ci> <apply> <ci> SuperPlus </ci> <ci> &#8469; </ci> </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