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 Cosh

 http://functions.wolfram.com/01.20.21.4737.01

 Input Form

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

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List["\[Integral]", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["p", " ", "z"]]], RowBox[List["Cosh", "[", RowBox[List["d", " ", "z"]], "]"]], SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["a", " ", SuperscriptBox[RowBox[List["Sinh", "[", RowBox[List["e", " ", "z"]], "]"]], "2"]]], "+", RowBox[List["b", " ", SuperscriptBox[RowBox[List["Cosh", "[", RowBox[List["e", " ", "z"]], "]"]], "2"]]]]], ")"]], "\[Beta]"], RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", RowBox[List[FractionBox["1", RowBox[List["d", "-", "p", "+", RowBox[List["2", " ", "e", " ", "\[Beta]"]]]]], RowBox[List["(", RowBox[List[SuperscriptBox["4", RowBox[List["-", "\[Beta]"]]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "d"]], "+", "p"]], ")"]], " ", "z"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", 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 MathML Form

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

 Rule Form

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

 2002-12-18