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Sinh






Mathematica Notation

Traditional Notation









Elementary Functions > Sinh[z] > Integration > Indefinite integration > Involving one direct function and elementary functions > Involving trigonometric and exponential functions > Involving powers of sin and algebraic functions of exp > Involving (a+b ed z)beta sinm(e z)sinh(c z)





http://functions.wolfram.com/01.19.21.1238.01









  


  










Input Form





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










Standard Form





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MathML Form







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Rule Form





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





2002-12-18