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 Cosh

 http://functions.wolfram.com/01.20.21.3825.01

 Input Form

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

 Standard Form

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 MathML Form

 p z ( a + b d z ) β cos ( e z ) cosh v ( c z ) z 2 - v - 1 ( a + b d z ) β ( d z b a + 1 ) - β s = 0 v - 1 2 ( v s ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["v", Identity]], List[TagBox["s", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( ( ( - e + p + c ( 2 s - v ) ) z ( - e - p + c ( 2 s - v ) ) 2 F 1 ( - e + p + c ( 2 s - v ) d , - β ; d - e + p + c ( 2 s - v ) d ; - b d z a ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["2", TraditionalForm]], SubscriptBox["F", FormBox["1", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox[FractionBox[RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "e"]], "+", "p", "+", RowBox[List["c", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "s"]], "-", "v"]], ")"]]]]]], "d"], Hypergeometric2F1], ",", TagBox[RowBox[List["-", "\[Beta]"]], Hypergeometric2F1]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1], ";", TagBox[TagBox[TagBox[FractionBox[RowBox[List["d", "-", RowBox[List["\[ImaginaryI]", " ", "e"]], "+", "p", "+", RowBox[List["c", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "s"]], "-", "v"]], ")"]]]]]], "d"], Hypergeometric2F1], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1], ";", TagBox[RowBox[List["-", FractionBox[RowBox[List["b", " ", SuperscriptBox["\[ExponentialE]", RowBox[List["d", " ", "z"]]]]], "a"]]], Hypergeometric2F1]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]]], Hypergeometric2F1] - ( e + p - c ( 2 s - v ) ) z ( - e + p + c ( 2 s - v ) ) 2 F 1 ( e + p - c ( 2 s - v ) d , - β ; 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v + Condition z p z a b d z β e z c z v 2 -1 v -1 a b d z β d z b a -1 1 -1 β s 0 v -1 2 -1 Binomial v s -1 e p c 2 s -1 v z -1 e -1 p c 2 s -1 v Hypergeometric2F1 -1 e p c 2 s -1 v d -1 -1 β d -1 e p c 2 s -1 v d -1 -1 b d z a -1 -1 e p -1 c 2 s -1 v z -1 e p c 2 s -1 v Hypergeometric2F1 e p -1 c 2 s -1 v d -1 -1 β d e p -1 c 2 s -1 v d -1 -1 b d z a -1 -1 e -1 p c 2 s -1 v -1 e p c 2 s -1 v -1 -1 e p -1 c 2 s -1 v z -1 e -1 p -1 c 2 s -1 v Hypergeometric2F1 -1 e p -1 c 2 s -1 v d -1 -1 β d -1 e p -1 c 2 s -1 v d -1 -1 b d z a -1 -1 e p c 2 s -1 v z -1 e p -1 c 2 s -1 v Hypergeometric2F1 e p c 2 s -1 v d -1 -1 β d e p c 2 s -1 v d -1 -1 b d z a -1 -1 e -1 p -1 c 2 s -1 v -1 e p -1 c 2 s -1 v -1 1 e -1 p e p -1 2 -1 v -1 a b d z β d z b a -1 1 -1 β Binomial v v 2 -1 e p z e -1 p Hypergeometric2F1 e p d -1 -1 β d e p d -1 -1 b d z a -1 -1 -1 e p z e p Hypergeometric2F1 -1 e p d -1 -1 β d -1 e p d -1 -1 b d z a -1 1 -1 \$CellContext`v 2 v SuperPlus [/itex]

 Rule Form

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

 2002-12-18