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 Sech

 http://functions.wolfram.com/01.24.21.0444.01

 Input Form

 Integrate[((-Cosh[2 z] + 2 Tanh[z]^2) Sech[z]^2)/Sqrt[Tanh[z]^3 Tanh[2 z]^3], z] == (Cosh[z]^4 (Cosh[2 z] - 2 Tanh[z]^2) (-24 AppellF1[1/2, 1, -(1/2), 3/2, Coth[z]^2, -Coth[z]^2] Cosh[2 z] Csch[z] Sech[z]^3 + Sqrt[1 + Coth[z]^2] (3 Coth[z] + 28 Cosh[z] Sinh[z] - Tanh[z] (6 + Tanh[z]^2 - 8 Tanh[z]^4 + 4 Tanh[z]^6 - 15 Log[Tanh[z]] Sech[z]^2 Sqrt[1 + Tanh[z]^2] + 15 Log[1 + Sqrt[1 + Tanh[z]^2]] Sech[z]^2 Sqrt[1 + Tanh[z]^2] + Sinh[z]^2 (26 + 32 Tanh[z]^2 - 30 Tanh[z]^4)))) Tanh[2 z]^2)/ (12 Sqrt[2] (5 - 2 Cosh[2 z] + Cosh[4 z]) Sqrt[1 + Coth[z]^2] Sqrt[Sech[2 z]^3 Sinh[z]^6])

 Standard Form

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 MathML Form

 ( 2 tanh 2 ( z ) - cosh ( 2 z ) ) sech 2 ( z ) tanh 3 ( z ) tanh 3 ( 2 z ) z ( cosh 4 ( z ) ( cosh ( 2 z ) - 2 tanh 2 ( z ) ) ( coth 2 ( z ) + 1 ( 3 coth ( z ) + 28 cosh ( z ) sinh ( z ) - tanh ( z ) ( 4 tanh 6 ( z ) - 8 tanh 4 ( z ) + tanh 2 ( z ) + sinh 2 ( z ) ( - 30 tanh 4 ( z ) + 32 tanh 2 ( z ) + 26 ) - 15 log ( tanh ( z ) ) sech 2 ( z ) tanh 2 ( z ) + 1 + 15 log ( tanh 2 ( z ) + 1 + 1 ) sech 2 ( z ) tanh 2 ( z ) + 1 + 6 ) ) - 24 F 1 AppellF1 ( 1 2 ; 1 , - 1 2 ; 3 2 ; coth 2 ( z ) , - coth 2 ( z ) ) cosh ( 2 z ) csch ( z ) sech 3 ( z ) ) tanh 2 ( 2 z ) ) / ( 12 2 ( - 2 cosh ( 2 z ) + cosh ( 4 z ) + 5 ) coth 2 ( z ) + 1 sech 3 ( 2 z ) sinh 6 ( z ) ) z 2 z 2 -1 2 z z 2 z 3 2 z 3 1 2 -1 z 4 2 z -1 2 z 2 z 2 1 1 2 3 z 28 z z -1 z 4 z 6 -1 8 z 4 z 2 z 2 -30 z 4 32 z 2 26 -1 15 z z 2 z 2 1 1 2 15 z 2 1 1 2 1 z 2 z 2 1 1 2 6 -1 24 AppellF1 1 2 1 -1 1 2 3 2 z 2 -1 z 2 2 z z z 3 2 z 2 12 2 1 2 -2 2 z 4 z 5 z 2 1 1 2 2 z 3 z 6 1 2 -1 [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["\[Integral]", RowBox[List[FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", RowBox[List["Cosh", "[", RowBox[List["2", " ", "z_"]], "]"]]]], "+", RowBox[List["2", " ", SuperscriptBox[RowBox[List["Tanh", "[", "z_", "]"]], "2"]]]]], ")"]], " ", SuperscriptBox[RowBox[List["Sech", "[", "z_", "]"]], "2"]]], SqrtBox[RowBox[List[SuperscriptBox[RowBox[List["Tanh", "[", "z_", "]"]], "3"], " ", SuperscriptBox[RowBox[List["Tanh", "[", RowBox[List["2", " ", "z_"]], "]"]], "3"]]]]], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", FractionBox[RowBox[List[SuperscriptBox[RowBox[List["Cosh", "[", "z", "]"]], "4"], " ", RowBox[List["(", RowBox[List[RowBox[List["Cosh", "[", RowBox[List["2", " ", "z"]], "]"]], "-", RowBox[List["2", " ", SuperscriptBox[RowBox[List["Tanh", "[", "z", "]"]], "2"]]]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "24"]], " ", RowBox[List["AppellF1", "[", RowBox[List[FractionBox["1", "2"], ",", "1", ",", RowBox[List["-", FractionBox["1", "2"]]], ",", FractionBox["3", "2"], ",", SuperscriptBox[RowBox[List["Coth", "[", "z", "]"]], "2"], ",", RowBox[List["-", SuperscriptBox[RowBox[List["Coth", "[", "z", "]"]], "2"]]]]], "]"]], " ", RowBox[List["Cosh", "[", RowBox[List["2", " ", "z"]], "]"]], " ", RowBox[List["Csch", "[", "z", "]"]], " ", SuperscriptBox[RowBox[List["Sech", "[", "z", "]"]], "3"]]], "+", RowBox[List[SqrtBox[RowBox[List["1", "+", SuperscriptBox[RowBox[List["Coth", "[", "z", "]"]], "2"]]]], " ", RowBox[List["(", RowBox[List[RowBox[List["3", " ", RowBox[List["Coth", "[", "z", "]"]]]], "+", RowBox[List["28", " ", RowBox[List["Cosh", "[", "z", "]"]], " ", RowBox[List["Sinh", "[", "z", "]"]]]], "-", RowBox[List[RowBox[List["Tanh", "[", "z", "]"]], " ", RowBox[List["(", RowBox[List["6", "+", SuperscriptBox[RowBox[List["Tanh", "[", "z", "]"]], "2"], "-", RowBox[List["8", " ", SuperscriptBox[RowBox[List["Tanh", "[", "z", "]"]], "4"]]], "+", RowBox[List["4", " ", SuperscriptBox[RowBox[List["Tanh", "[", "z", "]"]], "6"]]], "-", RowBox[List["15", " ", RowBox[List["Log", "[", RowBox[List["Tanh", "[", "z", "]"]], "]"]], " ", SuperscriptBox[RowBox[List["Sech", "[", "z", "]"]], "2"], " ", SqrtBox[RowBox[List["1", "+", SuperscriptBox[RowBox[List["Tanh", "[", "z", "]"]], "2"]]]]]], "+", RowBox[List["15", " ", RowBox[List["Log", "[", RowBox[List["1", "+", SqrtBox[RowBox[List["1", "+", SuperscriptBox[RowBox[List["Tanh", "[", "z", "]"]], "2"]]]]]], "]"]], " ", SuperscriptBox[RowBox[List["Sech", "[", "z", "]"]], "2"], " ", SqrtBox[RowBox[List["1", "+", SuperscriptBox[RowBox[List["Tanh", "[", "z", "]"]], "2"]]]]]], "+", RowBox[List[SuperscriptBox[RowBox[List["Sinh", "[", "z", "]"]], "2"], " ", RowBox[List["(", RowBox[List["26", "+", RowBox[List["32", " ", SuperscriptBox[RowBox[List["Tanh", "[", "z", "]"]], "2"]]], "-", RowBox[List["30", " ", SuperscriptBox[RowBox[List["Tanh", "[", "z", "]"]], "4"]]]]], ")"]]]]]], ")"]]]]]], ")"]]]]]], ")"]], " ", SuperscriptBox[RowBox[List["Tanh", "[", RowBox[List["2", " ", "z"]], "]"]], "2"]]], RowBox[List["12", " ", SqrtBox["2"], " ", RowBox[List["(", RowBox[List["5", "-", RowBox[List["2", " ", RowBox[List["Cosh", "[", RowBox[List["2", " ", "z"]], "]"]]]], "+", RowBox[List["Cosh", "[", RowBox[List["4", " ", "z"]], "]"]]]], ")"]], " ", SqrtBox[RowBox[List["1", "+", SuperscriptBox[RowBox[List["Coth", "[", "z", "]"]], "2"]]]], " ", SqrtBox[RowBox[List[SuperscriptBox[RowBox[List["Sech", "[", RowBox[List["2", " ", "z"]], "]"]], "3"], " ", SuperscriptBox[RowBox[List["Sinh", "[", "z", "]"]], "6"]]]]]]]]]]]

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

 2002-12-18