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Cos






Mathematica Notation

Traditional Notation









Elementary Functions > Cos[z] > Integration > Indefinite integration > Involving functions of the direct function and exponential function > Involving products of powers of two direct functions and exponential function > Involving product of powers of two direct functions and rational functions of exp > Involving ep zcosm(e z)cosv(c z)(a+b ed z)-n





http://functions.wolfram.com/01.07.21.1843.01









  


  










Input Form





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










Standard Form





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<imaginaryi /> <ci> c </ci> <ci> s </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <imaginaryi /> <ci> c </ci> <ci> v </ci> </apply> </apply> </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 /> <cn type='integer'> -2 </cn> <ci> e </ci> <imaginaryi /> <ci> k </ci> </apply> <apply> <times /> <imaginaryi /> <ci> e </ci> <ci> m </ci> </apply> <ci> p </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <imaginaryi /> <ci> c </ci> <ci> s </ci> </apply> </apply> <apply> <times /> <imaginaryi /> <ci> c </ci> <ci> v </ci> </apply> </apply> <ci> z </ci> </apply> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <imaginaryi /> <ci> e </ci> <ci> k </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <imaginaryi /> <ci> e </ci> <ci> m </ci> </apply> </apply> <ci> p </ci> <apply> <times /> <cn type='integer'> 2 </cn> <imaginaryi /> <ci> c </ci> <ci> s </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <imaginaryi /> <ci> c </ci> <ci> v </ci> </apply> </apply> </apply> <apply> <ci> Hypergeometric2F1 </ci> <apply> <times /> <apply> <plus /> <apply> <times /> <cn type='integer'> -2 </cn> <ci> e </ci> <imaginaryi /> <ci> k </ci> </apply> <apply> <times /> <imaginaryi /> <ci> e </ci> <ci> m </ci> </apply> <ci> p </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <imaginaryi /> <ci> c </ci> <ci> s </ci> </apply> </apply> 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</apply> </apply> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <times /> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <imaginaryi /> <ci> e </ci> <ci> k </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <imaginaryi /> <ci> e </ci> <ci> m </ci> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> p </ci> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <imaginaryi /> <ci> c </ci> <ci> s </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <imaginaryi /> <ci> c </ci> <ci> v </ci> </apply> </apply> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <imaginaryi /> <ci> e </ci> <ci> k </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <imaginaryi /> <ci> e </ci> <ci> m </ci> </apply> </apply> <ci> p </ci> <apply> <times /> <cn type='integer'> 2 </cn> <imaginaryi /> <ci> c </ci> <ci> s </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <imaginaryi /> <ci> c </ci> <ci> v </ci> </apply> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <apply> <plus /> <apply> <times /> <apply> <power /> <exponentiale /> <apply> <times /> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <imaginaryi /> <ci> e </ci> <ci> k </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <imaginaryi /> <ci> e </ci> <ci> m </ci> </apply> </apply> <ci> p </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <imaginaryi /> <ci> c </ci> <ci> s </ci> </apply> </apply> <apply> <times /> <imaginaryi /> <ci> c </ci> <ci> v </ci> </apply> </apply> <ci> z </ci> </apply> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <imaginaryi /> <ci> e </ci> <ci> k </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <imaginaryi /> <ci> e </ci> <ci> m </ci> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> p </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <imaginaryi /> <ci> c </ci> <ci> s </ci> </apply> </apply> <apply> <times /> <imaginaryi /> <ci> c </ci> <ci> v </ci> </apply> </apply> <apply> <ci> Hypergeometric2F1 </ci> <apply> <times /> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <imaginaryi /> <ci> e </ci> <ci> k </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <imaginaryi /> <ci> e </ci> <ci> m </ci> </apply> </apply> <ci> p </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <imaginaryi /> <ci> c </ci> <ci> s </ci> </apply> </apply> <apply> <times /> <imaginaryi /> <ci> c </ci> <ci> v </ci> </apply> </apply> <apply> <power /> <ci> d </ci> <cn type='integer'> -1 </cn> </apply> </apply> <ci> n </ci> <apply> <times /> <apply> <plus /> <ci> d </ci> 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</cn> <ci> e </ci> <imaginaryi /> <ci> k </ci> </apply> <apply> <times /> <imaginaryi /> <ci> e </ci> <ci> m </ci> </apply> <ci> p </ci> <apply> <times /> <cn type='integer'> 2 </cn> <imaginaryi /> <ci> c </ci> <ci> s </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <imaginaryi /> <ci> c </ci> <ci> v </ci> </apply> </apply> </apply> <ci> z </ci> </apply> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <imaginaryi /> <ci> e </ci> <ci> k </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <imaginaryi /> <ci> e </ci> <ci> m </ci> </apply> </apply> <ci> p </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <imaginaryi /> <ci> c </ci> <ci> s </ci> </apply> </apply> <apply> <times /> <imaginaryi /> <ci> c </ci> <ci> v </ci> </apply> </apply> <apply> <ci> Hypergeometric2F1 </ci> <apply> <times /> <apply> <plus /> <apply> <times /> <cn type='integer'> -2 </cn> <ci> 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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> <power /> <apply> <times /> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <imaginaryi /> <ci> e </ci> <ci> k </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <imaginaryi /> <ci> e </ci> <ci> m </ci> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> p </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <imaginaryi /> <ci> c </ci> <ci> s </ci> </apply> </apply> <apply> <times /> <imaginaryi /> <ci> c </ci> <ci> v </ci> </apply> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <imaginaryi /> <ci> e </ci> <ci> k </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <imaginaryi /> <ci> e </ci> <ci> m </ci> </apply> </apply> <ci> p </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <imaginaryi /> <ci> c </ci> <ci> s </ci> </apply> </apply> <apply> <times /> <imaginaryi /> <ci> c </ci> <ci> v </ci> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <times /> <apply> <times /> <apply> <power /> <cn type='integer'> 2 </cn> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> m </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> v </ci> </apply> </apply> </apply> <apply> <power /> <ci> a </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> n </ci> </apply> </apply> <apply> <power /> <exponentiale /> <apply> <times /> <ci> p </ci> <ci> z </ci> </apply> </apply> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <rem /> <ci> $CellContext`m </ci> <cn type='integer'> 2 </cn> </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> <power /> <ci> p </ci> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <ci> Binomial </ci> <ci> m </ci> <apply> <times /> <ci> m </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </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> <ci> Hypergeometric2F1 </ci> <apply> <times /> <ci> p </ci> <apply> <power /> <ci> d </ci> <cn type='integer'> -1 </cn> </apply> </apply> <ci> n </ci> <apply> <times /> <apply> <plus /> <ci> d </ci> <ci> p 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Date Added to functions.wolfram.com (modification date)





2002-12-18