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 Cosh

 http://functions.wolfram.com/01.20.21.4413.01

 Input Form

 Integrate[(a Sinh[e z]^2 + b Cosh[e z]^2)^\[Beta], z] == (-(1/(e \[Beta]))) ((2^(-1 - 2 \[Beta]) ((a (-1 + E^(2 e z))^2 + b (1 + E^(2 e z))^2)/E^(2 e z))^\[Beta] AppellF1[-\[Beta], -\[Beta], -\[Beta], 1 - \[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])])/ ((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[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[RowBox[List["-", FractionBox["1", RowBox[List["e", " ", "\[Beta]"]]]]], RowBox[List["(", RowBox[List[SuperscriptBox["2", RowBox[List[RowBox[List["-", "1"]], "-", RowBox[List["2", " ", "\[Beta]"]]]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["a", "+", "b"]], ")"]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "e", " ", "z"]]]]], RowBox[List["a", "-", "b", "+", RowBox[List["2", " ", SqrtBox[RowBox[List[RowBox[List["-", "a"]], " ", "b"]]]]]]]]]], ")"]], RowBox[List["-", "\[Beta]"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["a", "+", "b"]], ")"]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "e", " ", "z"]]]]], RowBox[List[RowBox[List["-", "a"]], "+", "b", "+", RowBox[List["2", " ", SqrtBox[RowBox[List[RowBox[List["-", "a"]], " ", "b"]]]]]]]]]], ")"]], RowBox[List["-", "\[Beta]"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "2"]], " ", "e", " ", "z"]]], " ", RowBox[List["(", RowBox[List[RowBox[List["a", " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "e", " ", "z"]]]]], ")"]], "2"]]], "+", RowBox[List["b", " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "+", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "e", " ", "z"]]]]], ")"]], "2"]]]]], ")"]]]], ")"]], "\[Beta]"], " ", RowBox[List["AppellF1", "[", RowBox[List[RowBox[List["-", "\[Beta]"]], ",", RowBox[List["-", "\[Beta]"]], ",", RowBox[List["-", "\[Beta]"]], ",", RowBox[List["1", "-", "\[Beta]"]], ",", RowBox[List["-", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["a", "+", "b"]], ")"]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "e", " ", "z"]]]]], RowBox[List[RowBox[List["-", "a"]], "+", "b", "+", RowBox[List["2", " ", SqrtBox[RowBox[List[RowBox[List["-", "a"]], " ", "b"]]]]]]]]]], ",", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["a", "+", "b"]], ")"]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "e", " ", "z"]]]]], RowBox[List["a", "-", "b", "+", RowBox[List["2", " ", SqrtBox[RowBox[List[RowBox[List["-", "a"]], " ", "b"]]]]]]]]]], "]"]]]], ")"]]]]]]]]

 MathML Form

 ( a sinh 2 ( e z ) + b cosh 2 ( e z ) ) β z - 1 e β ( 2 - 2 β - 1 ( 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 ( - β ; - β , - β ; 1 - β ; - ( a + b ) 2 e z - a + b + 2 - a b , ( a + b ) 2 e z a - b + 2 - a b ) ) z a e z 2 b e z 2 β -1 1 e β -1 2 -2 β -1 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 -1 β -1 β -1 β 1 -1 β -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

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["\[Integral]", RowBox[List[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_"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List["-", FractionBox[RowBox[List[SuperscriptBox["2", RowBox[List[RowBox[List["-", "1"]], "-", RowBox[List["2", " ", "\[Beta]"]]]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["a", "+", "b"]], ")"]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "e", " ", "z"]]]]], RowBox[List["a", "-", "b", "+", RowBox[List["2", " ", SqrtBox[RowBox[List[RowBox[List["-", "a"]], " ", "b"]]]]]]]]]], ")"]], RowBox[List["-", "\[Beta]"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["a", "+", "b"]], ")"]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "e", " ", "z"]]]]], RowBox[List[RowBox[List["-", "a"]], "+", "b", "+", RowBox[List["2", " ", SqrtBox[RowBox[List[RowBox[List["-", "a"]], " ", "b"]]]]]]]]]], ")"]], RowBox[List["-", "\[Beta]"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "2"]], " ", "e", " ", "z"]]], " ", RowBox[List["(", RowBox[List[RowBox[List["a", " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "e", " ", "z"]]]]], ")"]], "2"]]], "+", RowBox[List["b", " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "+", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "e", " ", "z"]]]]], ")"]], "2"]]]]], ")"]]]], ")"]], "\[Beta]"], " ", RowBox[List["AppellF1", "[", RowBox[List[RowBox[List["-", "\[Beta]"]], ",", RowBox[List["-", "\[Beta]"]], ",", RowBox[List["-", "\[Beta]"]], ",", RowBox[List["1", "-", "\[Beta]"]], ",", RowBox[List["-", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["a", "+", "b"]], ")"]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "e", " ", "z"]]]]], RowBox[List[RowBox[List["-", "a"]], "+", "b", "+", RowBox[List["2", " ", SqrtBox[RowBox[List[RowBox[List["-", "a"]], " ", "b"]]]]]]]]]], ",", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["a", "+", "b"]], ")"]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "e", " ", "z"]]]]], RowBox[List["a", "-", "b", "+", RowBox[List["2", " ", SqrtBox[RowBox[List[RowBox[List["-", "a"]], " ", "b"]]]]]]]]]], "]"]]]], RowBox[List["e", " ", "\[Beta]"]]]]]]]]]

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

 2002-12-18