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 Cosh

 http://functions.wolfram.com/01.20.21.1280.01

 Input Form

 Integrate[E^(p z) (a + b E^(d z))^\[Beta] Cos[e z] Cosh[c z], z] == ((1/4) (a + b E^(d z))^\[Beta] ((E^((c - I e + p) z) (c - I e - p) Hypergeometric2F1[(c - I e + p)/d, -\[Beta], (c + d - I e + p)/d, -((b E^(d z))/a)] - E^((-c + I e + p) z) (c - I e + p) Hypergeometric2F1[(-c + I e + p)/d, -\[Beta], (-c + d + I e + p)/d, -((b E^(d z))/a)])/ ((c - I e - p) (c - I e + p)) + ((-E^((-c - I e + p) z)) (c + I e + p) Hypergeometric2F1[ (-c - I e + p)/d, -\[Beta], (-c + d - I e + p)/d, -((b E^(d z))/a)] + E^((c + I e + p) z) (c + I e - p) Hypergeometric2F1[(c + I e + p)/d, -\[Beta], (c + d + I e + p)/d, -((b E^(d z))/a)])/ ((c + I e - p) (c + I e + p))))/(1 + (b E^(d z))/a)^\[Beta]

 Standard Form

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 MathML Form

 p z ( a + b d z ) β cos ( e z ) cosh ( c z ) z 1 4 ( a + b d z ) β ( d z b a + 1 ) - β ( 1 ( c - e - p ) ( c - e + p ) ( ( c - e + p ) z ( c - e - p ) 2 F 1 ( c - e + p d , - β ; c + d - e + p 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["c", "-", RowBox[List["\[ImaginaryI]", " ", "e"]], "+", "p"]], "d"], Hypergeometric2F1], ",", TagBox[RowBox[List["-", "\[Beta]"]], Hypergeometric2F1]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1], ";", TagBox[TagBox[TagBox[FractionBox[RowBox[List["c", "+", "d", "-", RowBox[List["\[ImaginaryI]", " ", "e"]], "+", "p"]], "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] - ( - c + e + p ) z ( c - e + p ) 2 F 1 ( - c + e + p d , - β ; - c + d + e + p 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["-", "c"]], "+", RowBox[List["\[ImaginaryI]", " ", "e"]], " ", "+", "p"]], "d"], Hypergeometric2F1], ",", TagBox[RowBox[List["-", "\[Beta]"]], Hypergeometric2F1]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1], ";", TagBox[TagBox[TagBox[FractionBox[RowBox[List[RowBox[List["-", "c"]], "+", "d", "+", RowBox[List["\[ImaginaryI]", " ", "e"]], " ", "+", "p"]], "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] ) + 1 ( c + e - p ) ( c + e + p ) ( ( c + e + p ) z ( c + e - p ) 2 F 1 ( c + e + p d , - β ; c + d + e + p 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["c", "+", RowBox[List["\[ImaginaryI]", " ", "e"]], " ", "+", "p"]], "d"], Hypergeometric2F1], ",", TagBox[RowBox[List["-", "\[Beta]"]], Hypergeometric2F1]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1], ";", TagBox[TagBox[TagBox[FractionBox[RowBox[List["c", "+", "d", "+", RowBox[List["\[ImaginaryI]", " ", "e"]], " ", "+", "p"]], "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] - ( - c - e + p ) z ( c + e + p ) 2 F 1 ( - c - e + p d , - β ; - c + d - e + p 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["-", "c"]], "-", RowBox[List["\[ImaginaryI]", " ", "e"]], "+", "p"]], "d"], Hypergeometric2F1], ",", TagBox[RowBox[List["-", "\[Beta]"]], Hypergeometric2F1]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1], ";", TagBox[TagBox[TagBox[FractionBox[RowBox[List[RowBox[List["-", "c"]], "+", "d", "-", RowBox[List["\[ImaginaryI]", " ", "e"]], "+", "p"]], "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] ) ) z p z a b d z β e z c z 1 4 a b d z β d z b a -1 1 -1 β 1 c -1 e -1 p c -1 e p -1 c -1 e p z c -1 e -1 p Hypergeometric2F1 c -1 e p d -1 -1 β c d -1 e p d -1 -1 b d z a -1 -1 -1 c e p z c -1 e p Hypergeometric2F1 -1 c e p d -1 -1 β -1 c d e p d -1 -1 b d z a -1 1 c e -1 p c e p -1 c e p z c e -1 p Hypergeometric2F1 c e p d -1 -1 β c d e p d -1 -1 b d z a -1 -1 -1 c -1 e p z c e p Hypergeometric2F1 -1 c -1 e p d -1 -1 β -1 c d -1 e p d -1 -1 b d z a -1 [/itex]

 Rule Form

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

 2002-12-18