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 Cosh

 http://functions.wolfram.com/01.20.21.4362.01

 Input Form

 Integrate[(Sinh[c z]^4 Cosh[c z]^2)/Sqrt[a + b Cosh[2 c z]], z] == (-4 I (4 a^3 + 9 a^2 b + 2 a b^2 - 3 b^3) Sqrt[(a + b Cosh[2 c z])/(a + b)] EllipticE[I c z, (2 b)/(a + b)] + 4 I (4 a^3 + 5 a^2 b - 4 a b^2 - 5 b^3) Sqrt[(a + b Cosh[2 c z])/(a + b)] EllipticF[I c z, (2 b)/(a + b)] + b (-8 a^2 - 10 a b + 3 b^2 - 2 b (a + 5 b) Cosh[2 c z] + 3 b^2 Cosh[4 c z]) Sinh[2 c z])/(240 b^3 c Sqrt[a + b Cosh[2 c z]])

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List["\[Integral]", RowBox[List[FractionBox[RowBox[List[SuperscriptBox[RowBox[List["Sinh", "[", RowBox[List["c", " ", "z"]], "]"]], "4"], SuperscriptBox[RowBox[List["Cosh", "[", RowBox[List["c", " ", "z"]], "]"]], "2"]]], SqrtBox[RowBox[List["a", "+", RowBox[List["b", " ", RowBox[List["Cosh", "[", RowBox[List["2", " ", "c", " ", "z"]], "]"]]]]]]]], RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "4"]], " ", "\[ImaginaryI]", " ", RowBox[List["(", RowBox[List[RowBox[List["4", " ", SuperscriptBox["a", "3"]]], "+", RowBox[List["9", " ", SuperscriptBox["a", "2"], " ", "b"]], "+", RowBox[List["2", " ", "a", " ", SuperscriptBox["b", "2"]]], "-", RowBox[List["3", " ", SuperscriptBox["b", "3"]]]]], ")"]], " ", SqrtBox[FractionBox[RowBox[List["a", "+", RowBox[List["b", " ", RowBox[List["Cosh", "[", RowBox[List["2", " ", "c", " ", "z"]], "]"]]]]]], RowBox[List["a", "+", "b"]]]], " ", RowBox[List["EllipticE", "[", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "c", " ", "z"]], ",", FractionBox[RowBox[List["2", " ", "b"]], RowBox[List["a", "+", "b"]]]]], "]"]]]], "+", RowBox[List["4", " ", "\[ImaginaryI]", " ", RowBox[List["(", RowBox[List[RowBox[List["4", " ", SuperscriptBox["a", "3"]]], "+", RowBox[List["5", " ", SuperscriptBox["a", "2"], " ", "b"]], "-", RowBox[List["4", " ", "a", " ", SuperscriptBox["b", "2"]]], "-", RowBox[List["5", " ", SuperscriptBox["b", "3"]]]]], ")"]], " ", SqrtBox[FractionBox[RowBox[List["a", "+", RowBox[List["b", " ", RowBox[List["Cosh", "[", RowBox[List["2", " ", "c", " ", "z"]], "]"]]]]]], RowBox[List["a", "+", "b"]]]], " ", RowBox[List["EllipticF", "[", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "c", " ", "z"]], ",", FractionBox[RowBox[List["2", " ", "b"]], RowBox[List["a", "+", "b"]]]]], "]"]]]], "+", RowBox[List["b", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "8"]], " ", SuperscriptBox["a", "2"]]], "-", RowBox[List["10", " ", "a", " ", "b"]], "+", RowBox[List["3", " ", SuperscriptBox["b", "2"]]], "-", RowBox[List["2", " ", "b", " ", RowBox[List["(", RowBox[List["a", "+", RowBox[List["5", " ", "b"]]]], ")"]], " ", RowBox[List["Cosh", "[", RowBox[List["2", " ", "c", " ", "z"]], "]"]]]], "+", RowBox[List["3", " ", SuperscriptBox["b", "2"], " ", RowBox[List["Cosh", "[", RowBox[List["4", " ", "c", " ", "z"]], "]"]]]]]], ")"]], " ", RowBox[List["Sinh", "[", RowBox[List["2", " ", "c", " ", "z"]], "]"]]]]]], ")"]], "/", RowBox[List["(", RowBox[List["240", " ", SuperscriptBox["b", "3"], " ", "c", " ", SqrtBox[RowBox[List["a", "+", RowBox[List["b", " ", RowBox[List["Cosh", "[", RowBox[List["2", " ", "c", " ", "z"]], "]"]]]]]]]]], ")"]]]]]]]]

 MathML Form

 sinh 4 ( c z ) cosh 2 ( c z ) a + b cosh ( 2 c z ) z ( - 4 ( 4 a 3 + 9 b a 2 + 2 b 2 a - 3 b 3 ) a + b cosh ( 2 c z ) a + b E ( c z 2 b a + b ) + 4 ( 4 a 3 + 5 b a 2 - 4 b 2 a - 5 b 3 ) a + b cosh ( 2 c z ) a + b F ( c z 2 b a + b ) + b ( - 8 a 2 - 10 b a + 3 b 2 - 2 b ( a + 5 b ) cosh ( 2 c z ) + 3 b 2 cosh ( 4 c z ) ) sinh ( 2 c z ) ) / ( 240 b 3 c a + b cosh ( 2 c z ) ) z c z 4 c z 2 a b 2 c z 1 2 -1 -4 4 a 3 9 b a 2 2 b 2 a -1 3 b 3 a b 2 c z a b -1 1 2 EllipticE c z 2 b a b -1 4 4 a 3 5 b a 2 -1 4 b 2 a -1 5 b 3 a b 2 c z a b -1 1 2 EllipticF c z 2 b a b -1 b -8 a 2 -1 10 b a 3 b 2 -1 2 b a 5 b 2 c z 3 b 2 4 c z 2 c z 240 b 3 c a b 2 c z 1 2 -1 [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["\[Integral]", RowBox[List[FractionBox[RowBox[List[SuperscriptBox[RowBox[List["Sinh", "[", RowBox[List["c_", " ", "z_"]], "]"]], "4"], " ", SuperscriptBox[RowBox[List["Cosh", "[", RowBox[List["c_", " ", "z_"]], "]"]], "2"]]], SqrtBox[RowBox[List["a_", "+", RowBox[List["b_", " ", RowBox[List["Cosh", "[", RowBox[List["2", " ", "c_", " ", "z_"]], "]"]]]]]]]], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", FractionBox[RowBox[List[RowBox[List[RowBox[List["-", "4"]], " ", "\[ImaginaryI]", " ", RowBox[List["(", RowBox[List[RowBox[List["4", " ", SuperscriptBox["a", "3"]]], "+", RowBox[List["9", " ", SuperscriptBox["a", "2"], " ", "b"]], "+", RowBox[List["2", " ", "a", " ", SuperscriptBox["b", "2"]]], "-", RowBox[List["3", " ", SuperscriptBox["b", "3"]]]]], ")"]], " ", SqrtBox[FractionBox[RowBox[List["a", "+", RowBox[List["b", " ", RowBox[List["Cosh", "[", RowBox[List["2", " ", "c", " ", "z"]], "]"]]]]]], RowBox[List["a", "+", "b"]]]], " ", RowBox[List["EllipticE", "[", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "c", " ", "z"]], ",", FractionBox[RowBox[List["2", " ", "b"]], RowBox[List["a", "+", "b"]]]]], "]"]]]], "+", RowBox[List["4", " ", "\[ImaginaryI]", " ", RowBox[List["(", RowBox[List[RowBox[List["4", " ", SuperscriptBox["a", "3"]]], "+", RowBox[List["5", " ", SuperscriptBox["a", "2"], " ", "b"]], "-", RowBox[List["4", " ", "a", " ", SuperscriptBox["b", "2"]]], "-", RowBox[List["5", " ", SuperscriptBox["b", "3"]]]]], ")"]], " ", SqrtBox[FractionBox[RowBox[List["a", "+", RowBox[List["b", " ", RowBox[List["Cosh", "[", RowBox[List["2", " ", "c", " ", "z"]], "]"]]]]]], RowBox[List["a", "+", "b"]]]], " ", RowBox[List["EllipticF", "[", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "c", " ", "z"]], ",", FractionBox[RowBox[List["2", " ", "b"]], RowBox[List["a", "+", "b"]]]]], "]"]]]], "+", RowBox[List["b", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "8"]], " ", SuperscriptBox["a", "2"]]], "-", RowBox[List["10", " ", "a", " ", "b"]], "+", RowBox[List["3", " ", SuperscriptBox["b", "2"]]], "-", RowBox[List["2", " ", "b", " ", RowBox[List["(", RowBox[List["a", "+", RowBox[List["5", " ", "b"]]]], ")"]], " ", RowBox[List["Cosh", "[", RowBox[List["2", " ", "c", " ", "z"]], "]"]]]], "+", RowBox[List["3", " ", SuperscriptBox["b", "2"], " ", RowBox[List["Cosh", "[", RowBox[List["4", " ", "c", " ", "z"]], "]"]]]]]], ")"]], " ", RowBox[List["Sinh", "[", RowBox[List["2", " ", "c", " ", "z"]], "]"]]]]]], RowBox[List["240", " ", SuperscriptBox["b", "3"], " ", "c", " ", SqrtBox[RowBox[List["a", "+", RowBox[List["b", " ", RowBox[List["Cosh", "[", RowBox[List["2", " ", "c", " ", "z"]], "]"]]]]]]]]]]]]]]

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

 2002-12-18