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 Cosh

 http://functions.wolfram.com/01.20.21.3800.01

 Input Form

 Integrate[E^(p z) Cos[d + c z] Cosh[b + a z]^\[Nu], z] == ((1/2) Cosh[b + a z]^\[Nu] (-((1/(I c - p - a \[Nu])) (E^(((-I) c + p) z) Hypergeometric2F1[ -(((-I) c + p + a \[Nu])/(2 a)), -\[Nu], (1/2) (2 + (I c - p)/a - \[Nu]), -E^(-2 (b + a z))])) + (1/(I c + p + a \[Nu])) (E^(2 I d + (I c + p) z) Hypergeometric2F1[-((I c + p + a \[Nu])/(2 a)), -\[Nu], -((I c + p + a (-2 + \[Nu]))/(2 a)), -E^(-2 (b + a z))])))/ (E^(I d) (1 + E^(-2 (b + a z)))^\[Nu])

 Standard Form

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 MathML Form

 p z cos ( d + c z ) cosh ν ( b + a z ) z 1 2 - d ( 1 + - 2 ( b + a z ) ) - ν cosh ν ( b + a z ) ( 2 d + ( c + p ) z c + p + a ν 2 F 1 ( - c + p + a ν 2 a , - ν ; - c + p + a ( ν - 2 ) 2 a ; - - 2 ( b + a z ) ) 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[RowBox[List["-", FractionBox[RowBox[List[RowBox[List["\[ImaginaryI]", " ", "c"]], "+", "p", "+", RowBox[List["a", " ", "\[Nu]"]]]], RowBox[List["2", " ", "a"]]]]], Hypergeometric2F1], ",", TagBox[RowBox[List["-", "\[Nu]"]], Hypergeometric2F1]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1], ";", TagBox[TagBox[TagBox[RowBox[List["-", FractionBox[RowBox[List[RowBox[List["\[ImaginaryI]", " ", "c"]], "+", "p", "+", RowBox[List["a", " ", RowBox[List["(", RowBox[List["\[Nu]", "-", "2"]], ")"]]]]]], RowBox[List["2", " ", "a"]]]]], Hypergeometric2F1], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1], ";", TagBox[RowBox[List["-", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "2"]], " ", RowBox[List["(", RowBox[List["b", "+", RowBox[List["a", " ", "z"]]]], ")"]]]]]]], Hypergeometric2F1]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]]], Hypergeometric2F1] - ( - c + p ) z c - p - a ν 2 F 1 ( - - c + p + a ν 2 a , - ν ; 1 2 ( c - p a - ν + 2 ) ; - - 2 ( b + a z ) ) 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[RowBox[List["-", FractionBox[RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "c"]], "+", "p", "+", RowBox[List["a", " ", "\[Nu]"]]]], RowBox[List["2", " ", "a"]]]]], Hypergeometric2F1], ",", TagBox[RowBox[List["-", "\[Nu]"]], Hypergeometric2F1]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1], ";", TagBox[TagBox[TagBox[RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[FractionBox[RowBox[List[RowBox[List["\[ImaginaryI]", " ", "c"]], "-", "p"]], "a"], "-", "\[Nu]", "+", "2"]], ")"]]]], Hypergeometric2F1], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1], ";", TagBox[RowBox[List["-", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "2"]], " ", RowBox[List["(", RowBox[List["b", "+", RowBox[List["a", " ", "z"]]]], ")"]]]]]]], Hypergeometric2F1]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]]], Hypergeometric2F1] ) z p z d c z b a z ν 1 2 -1 d 1 -2 b a z -1 ν b a z ν 2 d c p z c p a ν -1 Hypergeometric2F1 -1 c p a ν 2 a -1 -1 ν -1 c p a ν -2 2 a -1 -1 -2 b a z -1 -1 c p z c -1 p -1 a ν -1 Hypergeometric2F1 -1 -1 c p a ν 2 a -1 -1 ν 1 2 c -1 p a -1 -1 ν 2 -1 -2 b a z [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["\[Integral]", RowBox[List[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["p_", " ", "z_"]]], " ", RowBox[List["Cos", "[", RowBox[List["d_", "+", RowBox[List["c_", " ", "z_"]]]], "]"]], " ", SuperscriptBox[RowBox[List["Cosh", "[", RowBox[List["b_", "+", RowBox[List["a_", " ", "z_"]]]], "]"]], "\[Nu]_"]]], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox["1", "2"], " ", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "d"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "+", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "2"]], " ", RowBox[List["(", RowBox[List["b", "+", RowBox[List["a", " ", "z"]]]], ")"]]]]]]], ")"]], RowBox[List["-", "\[Nu]"]]], " ", SuperscriptBox[RowBox[List["Cosh", "[", RowBox[List["b", "+", RowBox[List["a", " ", "z"]]]], "]"]], "\[Nu]"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", FractionBox[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "c"]], "+", "p"]], ")"]], " ", "z"]]], " ", RowBox[List["Hypergeometric2F1", "[", RowBox[List[RowBox[List["-", FractionBox[RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "c"]], "+", "p", "+", RowBox[List["a", " ", "\[Nu]"]]]], RowBox[List["2", " ", "a"]]]]], ",", RowBox[List["-", "\[Nu]"]], ",", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["2", "+", FractionBox[RowBox[List[RowBox[List["\[ImaginaryI]", " ", "c"]], "-", "p"]], "a"], "-", "\[Nu]"]], ")"]]]], ",", RowBox[List["-", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "2"]], " ", RowBox[List["(", RowBox[List["b", "+", RowBox[List["a", " ", "z"]]]], ")"]]]]]]]]], "]"]]]], RowBox[List[RowBox[List["\[ImaginaryI]", " ", "c"]], "-", "p", "-", RowBox[List["a", " ", "\[Nu]"]]]]]]], "+", FractionBox[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["2", " ", "\[ImaginaryI]", " ", "d"]], "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "c"]], "+", "p"]], ")"]], " ", "z"]]]]], " ", RowBox[List["Hypergeometric2F1", "[", RowBox[List[RowBox[List["-", FractionBox[RowBox[List[RowBox[List["\[ImaginaryI]", " ", "c"]], "+", "p", "+", RowBox[List["a", " ", "\[Nu]"]]]], RowBox[List["2", " ", "a"]]]]], ",", RowBox[List["-", "\[Nu]"]], ",", RowBox[List["-", FractionBox[RowBox[List[RowBox[List["\[ImaginaryI]", " ", "c"]], "+", "p", "+", RowBox[List["a", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "2"]], "+", "\[Nu]"]], ")"]]]]]], RowBox[List["2", " ", "a"]]]]], ",", RowBox[List["-", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "2"]], " ", RowBox[List["(", RowBox[List["b", "+", RowBox[List["a", " ", "z"]]]], ")"]]]]]]]]], "]"]]]], RowBox[List[RowBox[List["\[ImaginaryI]", " ", "c"]], "+", "p", "+", RowBox[List["a", " ", "\[Nu]"]]]]]]], ")"]]]]]]]]

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

 2002-12-18