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 Tanh

 http://functions.wolfram.com/01.21.21.0366.01

 Input Form

 Integrate[(Cosh[z]^3 (Cosh[2 z] - 3 Tanh[z]))/((Sinh[z]^2 - Sinh[2 z]) Sqrt[Sinh[2 z]^5]), z] == (1/(240 Sqrt[Sinh[2 z]^5])) (Cosh[z]^2 (51 Cosh[z] - 3 Cosh[3 z] - 50 Sinh[z] - 50 Sinh[3 z] - 690 (-1)^(1/4) Sqrt[1 + Coth[z/2]^2] EllipticF[I ArcSinh[(-1)^(1/4)/Sqrt[Tanh[z/2]]], -1] Sinh[z]^3 Sqrt[Tanh[z/2]] + 690 (-1)^(1/4) Sqrt[1 + Coth[z/2]^2] EllipticPi[(-1)^(1/6), I ArcSinh[(-1)^(1/4)/Sqrt[Tanh[z/2]]], -1] Sinh[z]^3 Sqrt[Tanh[z/2]] + 690 (-1)^(1/4) Sqrt[1 + Coth[z/2]^2] EllipticPi[(-1)^(5/6), I ArcSinh[(-1)^(1/4)/Sqrt[Tanh[z/2]]], -1] Sinh[z]^3 Sqrt[Tanh[z/2]]))

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List["\[Integral]", RowBox[List[FractionBox[RowBox[List[SuperscriptBox[RowBox[List["Cosh", "[", "z", "]"]], "3"], " ", RowBox[List["(", RowBox[List[RowBox[List["Cosh", "[", RowBox[List["2", " ", "z"]], "]"]], "-", RowBox[List["3", " ", RowBox[List["Tanh", "[", "z", "]"]]]]]], ")"]]]], RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["Sinh", "[", "z", "]"]], "2"], "-", RowBox[List["Sinh", "[", RowBox[List["2", " ", "z"]], "]"]]]], ")"]], " ", SqrtBox[SuperscriptBox[RowBox[List["Sinh", "[", RowBox[List["2", " ", "z"]], "]"]], "5"]]]]], RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", RowBox[List[FractionBox["1", RowBox[List["240", " ", SqrtBox[SuperscriptBox[RowBox[List["Sinh", "[", RowBox[List["2", " ", "z"]], "]"]], "5"]]]]], RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["Cosh", "[", "z", "]"]], "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["51", " ", RowBox[List["Cosh", "[", "z", "]"]]]], "-", RowBox[List["3", " ", RowBox[List["Cosh", "[", RowBox[List["3", " ", "z"]], "]"]]]], "-", RowBox[List["50", " ", RowBox[List["Sinh", "[", "z", "]"]]]], "-", RowBox[List["50", " ", RowBox[List["Sinh", "[", RowBox[List["3", " ", "z"]], "]"]]]], "-", RowBox[List["690", " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["1", "/", "4"]]], " ", SqrtBox[RowBox[List["1", "+", SuperscriptBox[RowBox[List["Coth", "[", FractionBox["z", "2"], "]"]], "2"]]]], " ", RowBox[List["EllipticF", "[", RowBox[List[RowBox[List["\[ImaginaryI]", " ", RowBox[List["ArcSinh", "[", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["1", "/", "4"]]], SqrtBox[RowBox[List["Tanh", "[", FractionBox["z", "2"], "]"]]]], "]"]]]], ",", RowBox[List["-", "1"]]]], "]"]], " ", SuperscriptBox[RowBox[List["Sinh", "[", "z", "]"]], "3"], " ", SqrtBox[RowBox[List["Tanh", "[", FractionBox["z", "2"], "]"]]]]], "+", RowBox[List["690", " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["1", "/", "4"]]], " ", SqrtBox[RowBox[List["1", "+", SuperscriptBox[RowBox[List["Coth", "[", FractionBox["z", "2"], "]"]], "2"]]]], " ", RowBox[List["EllipticPi", "[", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["1", "/", "6"]]], ",", RowBox[List["\[ImaginaryI]", " ", RowBox[List["ArcSinh", "[", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["1", "/", "4"]]], SqrtBox[RowBox[List["Tanh", "[", FractionBox["z", "2"], "]"]]]], "]"]]]], ",", RowBox[List["-", "1"]]]], "]"]], " ", SuperscriptBox[RowBox[List["Sinh", "[", "z", "]"]], "3"], " ", SqrtBox[RowBox[List["Tanh", "[", FractionBox["z", "2"], "]"]]]]], "+", RowBox[List["690", " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["1", "/", "4"]]], " ", SqrtBox[RowBox[List["1", "+", SuperscriptBox[RowBox[List["Coth", "[", FractionBox["z", "2"], "]"]], "2"]]]], " ", RowBox[List["EllipticPi", "[", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["5", "/", "6"]]], ",", RowBox[List["\[ImaginaryI]", " ", RowBox[List["ArcSinh", "[", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["1", "/", "4"]]], SqrtBox[RowBox[List["Tanh", "[", FractionBox["z", "2"], "]"]]]], "]"]]]], ",", RowBox[List["-", "1"]]]], "]"]], " ", SuperscriptBox[RowBox[List["Sinh", "[", "z", "]"]], "3"], " ", SqrtBox[RowBox[List["Tanh", "[", FractionBox["z", "2"], "]"]]]]]]], ")"]]]], ")"]]]]]]]]

 MathML Form

 cosh 3 ( z ) ( cosh ( 2 z ) - 3 tanh ( z ) ) ( sinh 2 ( z ) - sinh ( 2 z ) ) sinh 5 ( 2 z ) z 1 240 sinh 5 ( 2 z ) ( cosh 2 ( z ) ( - 690 - 1 4 coth 2 ( z 2 ) + 1 F ( sinh - 1 ( - 1 4 tanh 1 2 ( z 2 ) ) - 1 ) tanh 1 2 ( z 2 ) sinh 3 ( z ) + 690 - 1 4 coth 2 ( z 2 ) + 1 Π ( - 1 6 ; sinh - 1 ( - 1 4 tanh 1 2 ( z 2 ) ) - 1 ) tanh 1 2 ( z 2 ) sinh 3 ( z ) + 690 - 1 4 coth 2 ( z 2 ) + 1 Π ( ( - 1 ) 5 / 6 ; sinh - 1 ( - 1 4 tanh 1 2 ( z 2 ) ) - 1 ) tanh 1 2 ( z 2 ) sinh 3 ( z ) - 50 sinh ( z ) + 51 cosh ( z ) - 3 cosh ( 3 z ) - 50 sinh ( 3 z ) ) ) z z 3 2 z -1 3 z z 2 -1 2 z 2 z 5 1 2 -1 1 240 2 z 5 1 2 -1 z 2 -690 -1 1 4 z 2 -1 2 1 1 2 EllipticF -1 1 4 z 2 -1 1 2 -1 -1 z 2 -1 1 2 z 3 690 -1 1 4 z 2 -1 2 1 1 2 EllipticPi -1 1 6 -1 1 4 z 2 -1 1 2 -1 -1 z 2 -1 1 2 z 3 690 -1 1 4 z 2 -1 2 1 1 2 EllipticPi -1 5 6 -1 1 4 z 2 -1 1 2 -1 -1 z 2 -1 1 2 z 3 -1 50 z 51 z -1 3 3 z -1 50 3 z [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["\[Integral]", RowBox[List[FractionBox[RowBox[List[SuperscriptBox[RowBox[List["Cosh", "[", "z_", "]"]], "3"], " ", RowBox[List["(", RowBox[List[RowBox[List["Cosh", "[", RowBox[List["2", " ", "z_"]], "]"]], "-", RowBox[List["3", " ", RowBox[List["Tanh", "[", "z_", "]"]]]]]], ")"]]]], RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["Sinh", "[", "z_", "]"]], "2"], "-", RowBox[List["Sinh", "[", RowBox[List["2", " ", "z_"]], "]"]]]], ")"]], " ", SqrtBox[SuperscriptBox[RowBox[List["Sinh", "[", RowBox[List["2", " ", "z_"]], "]"]], "5"]]]]], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", FractionBox[RowBox[List[SuperscriptBox[RowBox[List["Cosh", "[", "z", "]"]], "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["51", " ", RowBox[List["Cosh", "[", "z", "]"]]]], "-", RowBox[List["3", " ", RowBox[List["Cosh", "[", RowBox[List["3", " ", "z"]], "]"]]]], "-", RowBox[List["50", " ", RowBox[List["Sinh", "[", "z", "]"]]]], "-", RowBox[List["50", " ", RowBox[List["Sinh", "[", RowBox[List["3", " ", "z"]], "]"]]]], "-", RowBox[List["690", " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["1", "/", "4"]]], " ", SqrtBox[RowBox[List["1", "+", SuperscriptBox[RowBox[List["Coth", "[", FractionBox["z", "2"], "]"]], "2"]]]], " ", RowBox[List["EllipticF", "[", RowBox[List[RowBox[List["\[ImaginaryI]", " ", RowBox[List["ArcSinh", "[", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["1", "/", "4"]]], SqrtBox[RowBox[List["Tanh", "[", FractionBox["z", "2"], "]"]]]], "]"]]]], ",", RowBox[List["-", "1"]]]], "]"]], " ", SuperscriptBox[RowBox[List["Sinh", "[", "z", "]"]], "3"], " ", SqrtBox[RowBox[List["Tanh", "[", FractionBox["z", "2"], "]"]]]]], "+", RowBox[List["690", " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["1", "/", "4"]]], " ", SqrtBox[RowBox[List["1", "+", SuperscriptBox[RowBox[List["Coth", "[", FractionBox["z", "2"], "]"]], "2"]]]], " ", RowBox[List["EllipticPi", "[", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["1", "/", "6"]]], ",", RowBox[List["\[ImaginaryI]", " ", RowBox[List["ArcSinh", "[", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["1", "/", "4"]]], SqrtBox[RowBox[List["Tanh", "[", FractionBox["z", "2"], "]"]]]], "]"]]]], ",", RowBox[List["-", "1"]]]], "]"]], " ", SuperscriptBox[RowBox[List["Sinh", "[", "z", "]"]], "3"], " ", SqrtBox[RowBox[List["Tanh", "[", FractionBox["z", "2"], "]"]]]]], "+", RowBox[List["690", " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["1", "/", "4"]]], " ", SqrtBox[RowBox[List["1", "+", SuperscriptBox[RowBox[List["Coth", "[", FractionBox["z", "2"], "]"]], "2"]]]], " ", RowBox[List["EllipticPi", "[", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["5", "/", "6"]]], ",", RowBox[List["\[ImaginaryI]", " ", RowBox[List["ArcSinh", "[", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["1", "/", "4"]]], SqrtBox[RowBox[List["Tanh", "[", FractionBox["z", "2"], "]"]]]], "]"]]]], ",", RowBox[List["-", "1"]]]], "]"]], " ", SuperscriptBox[RowBox[List["Sinh", "[", "z", "]"]], "3"], " ", SqrtBox[RowBox[List["Tanh", "[", FractionBox["z", "2"], "]"]]]]]]], ")"]]]], RowBox[List["240", " ", SqrtBox[SuperscriptBox[RowBox[List["Sinh", "[", RowBox[List["2", " ", "z"]], "]"]], "5"]]]]]]]]]

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

 2002-12-18