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 Cosh

 http://functions.wolfram.com/01.20.21.3901.01

 Input Form

 Integrate[E^(p z) Cos[d z] (a + b Cosh[c z]^2)^\[Beta], z] == (1/2) ((1/((-I) d + p - 2 c \[Beta])) ((E^(((-I) d + p) z) (a + ((1/4) b (1 + E^(2 c z))^2)/E^(2 c z))^\[Beta] AppellF1[((-I) d + p)/(2 c) - \[Beta], -\[Beta], -\[Beta], 1 + ((-I) d + p)/(2 c) - \[Beta], -((b E^(2 c z))/(2 a + b + 2 Sqrt[a (a + b)])), -((b E^(2 c z))/(2 a + b - 2 Sqrt[a (a + b)]))])/ ((1 + (b E^(2 c z))/(2 a + b - 2 Sqrt[a (a + b)]))^\[Beta] (1 + (b E^(2 c z))/(2 a + b + 2 Sqrt[a (a + b)]))^\[Beta])) + (1/(I d + p - 2 c \[Beta])) ((E^((I d + p) z) (a + ((1/4) b (1 + E^(2 c z))^2)/E^(2 c z))^\[Beta] AppellF1[(I d + p)/(2 c) - \[Beta], -\[Beta], -\[Beta], 1 + (I d + p)/(2 c) - \[Beta], -((b E^(2 c z))/(2 a + b + 2 Sqrt[a (a + b)])), -((b E^(2 c z))/(2 a + b - 2 Sqrt[a (a + b)]))])/ ((1 + (b E^(2 c z))/(2 a + b - 2 Sqrt[a (a + b)]))^\[Beta] (1 + (b E^(2 c z))/(2 a + b + 2 Sqrt[a (a + b)]))^\[Beta])))

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List["\[Integral]", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["p", " ", "z"]]], RowBox[List["Cos", "[", RowBox[List["d", " ", "z"]], "]"]], SuperscriptBox[RowBox[List["(", RowBox[List["a", "+", RowBox[List["b", " ", SuperscriptBox[RowBox[List["Cosh", "[", RowBox[List["c", " ", "z"]], "]"]], "2"]]]]], ")"]], "\[Beta]"], RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List[FractionBox["1", RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "d"]], "+", "p", "-", RowBox[List["2", " ", "c", " ", "\[Beta]"]]]]], RowBox[List["(", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "d"]], "+", "p"]], ")"]], " ", "z"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "+", FractionBox[RowBox[List["b", " ", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " 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SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "c", " ", "z"]]]]], RowBox[List[RowBox[List["2", " ", "a"]], "+", "b", "+", RowBox[List["2", " ", SqrtBox[RowBox[List["a", " ", RowBox[List["(", RowBox[List["a", "+", "b"]], ")"]]]]]]]]]]]], ",", RowBox[List["-", FractionBox[RowBox[List["b", " ", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "c", " ", "z"]]]]], RowBox[List[RowBox[List["2", " ", "a"]], "+", "b", "-", RowBox[List["2", " ", SqrtBox[RowBox[List["a", " ", RowBox[List["(", RowBox[List["a", "+", "b"]], ")"]]]]]]]]]]]]]], "]"]]]], ")"]]]]]], ")"]]]]]]]]

 MathML Form

 p z cos ( d z ) ( a + b cosh 2 ( c z ) ) β z 1 2 ( 1 - d + p - 2 c β ( ( - d + p ) z ( 2 c z b 2 a + b - 2 a ( a + b ) + 1 ) - β ( 2 c z b 2 a + b + 2 a ( a + b ) + 1 ) - β ( 1 4 b - 2 c z ( 1 + 2 c z ) 2 + a ) β F 1 AppellF1 ( - d + p 2 c - β ; - β , - β ; - d + p 2 c - β + 1 ; - b 2 c z 2 a + b + 2 a ( a + b ) , - b 2 c z 2 a + b - 2 a ( a + b ) ) ) + 1 d + p - 2 c β ( ( d + p ) z ( 2 c z b 2 a + b - 2 a ( a + b ) + 1 ) - β ( 2 c z b 2 a + b + 2 a ( a + b ) + 1 ) - β ( 1 4 b - 2 c z ( 1 + 2 c z ) 2 + a ) β F 1 AppellF1 ( d + p 2 c - β ; - β , - β ; d + p 2 c - β + 1 ; - b 2 c z 2 a + b + 2 a ( a + b ) , - b 2 c z 2 a + b - 2 a ( a + b ) ) ) ) z p z d z a b c z 2 β 1 2 1 -1 d p -1 2 c β -1 -1 d p z 2 c z b 2 a b -1 2 a a b 1 2 -1 1 -1 β 2 c z b 2 a b 2 a a b 1 2 -1 1 -1 β 1 4 b -2 c z 1 2 c z 2 a β AppellF1 -1 d p 2 c -1 -1 β -1 β -1 β -1 d p 2 c -1 -1 β 1 -1 b 2 c z 2 a b 2 a a b 1 2 -1 -1 b 2 c z 2 a b -1 2 a a b 1 2 -1 1 d p -1 2 c β -1 d p z 2 c z b 2 a b -1 2 a a b 1 2 -1 1 -1 β 2 c z b 2 a b 2 a a b 1 2 -1 1 -1 β 1 4 b -2 c z 1 2 c z 2 a β AppellF1 d p 2 c -1 -1 β -1 β -1 β d p 2 c -1 -1 β 1 -1 b 2 c z 2 a b 2 a a b 1 2 -1 -1 b 2 c z 2 a b -1 2 a a b 1 2 -1 [/itex]

 Rule Form

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

 2002-12-18