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 Cosh

 http://functions.wolfram.com/01.20.21.4331.01

 Input Form

 Integrate[Sinh[c z]^4 Sqrt[a + b Cosh[2 c z]], z] == (8 I (a^3 + 6 a^2 b - 7 a b^2 - 12 b^3) Sqrt[(a + b Cosh[2 c z])/(a + b)] EllipticE[I c z, (2 b)/(a + b)] - 8 I (a^3 + 5 a^2 b - a b^2 - 5 b^3) Sqrt[(a + b Cosh[2 c z])/(a + b)] EllipticF[I c z, (2 b)/(a + b)] + 2 b (2 a^2 - 20 a b + 3 b^2 + 4 (2 a - 5 b) b Cosh[2 c z] + 3 b^2 Cosh[4 c z]) Sinh[2 c z])/(240 b^2 c Sqrt[a + b Cosh[2 c z]])

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List["\[Integral]", RowBox[List[SuperscriptBox[RowBox[List["Sinh", "[", RowBox[List["c", " ", "z"]], "]"]], "4"], 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["8", " ", "\[ImaginaryI]", " ", RowBox[List["(", RowBox[List[SuperscriptBox["a", "3"], "+", RowBox[List["6", " ", SuperscriptBox["a", "2"], " ", "b"]], "-", RowBox[List["7", " ", "a", " ", SuperscriptBox["b", "2"]]], "-", RowBox[List["12", " ", 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["8", " ", "\[ImaginaryI]", " ", RowBox[List["(", RowBox[List[SuperscriptBox["a", "3"], "+", RowBox[List["5", " ", SuperscriptBox["a", "2"], " ", "b"]], "-", RowBox[List["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["2", " ", "b", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", SuperscriptBox["a", "2"]]], "-", RowBox[List["20", " ", "a", " ", "b"]], "+", RowBox[List["3", " ", SuperscriptBox["b", "2"]]], "+", RowBox[List["4", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "a"]], "-", RowBox[List["5", " ", "b"]]]], ")"]], " ", "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", "2"], " ", "c", " ", SqrtBox[RowBox[List["a", "+", RowBox[List["b", " ", RowBox[List["Cosh", "[", RowBox[List["2", " ", "c", " ", "z"]], "]"]]]]]]]]], ")"]]]]]]]]

 MathML Form

 sinh 4 ( c z ) a + b cosh ( 2 c z ) z ( 8 ( a 3 + 6 b a 2 - 7 b 2 a - 12 b 3 ) a + b cosh ( 2 c z ) a + b E ( c z 2 b a + b ) - 8 ( a 3 + 5 b a 2 - b 2 a - 5 b 3 ) a + b cosh ( 2 c z ) a + b F ( c z 2 b a + b ) + 2 b ( 2 a 2 - 20 b a + 3 b 2 + 4 ( 2 a - 5 b ) b cosh ( 2 c z ) + 3 b 2 cosh ( 4 c z ) ) sinh ( 2 c z ) ) / ( 240 b 2 c a + b cosh ( 2 c z ) ) z c z 4 a b 2 c z 1 2 8 a 3 6 b a 2 -1 7 b 2 a -1 12 b 3 a b 2 c z a b -1 1 2 EllipticE c z 2 b a b -1 -1 8 a 3 5 b a 2 -1 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 2 b 2 a 2 -1 20 b a 3 b 2 4 2 a -1 5 b b 2 c z 3 b 2 4 c z 2 c z 240 b 2 c a b 2 c z 1 2 -1 [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["\[Integral]", RowBox[List[RowBox[List[SuperscriptBox[RowBox[List["Sinh", "[", RowBox[List["c_", " ", "z_"]], "]"]], "4"], " ", 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["8", " ", "\[ImaginaryI]", " ", RowBox[List["(", RowBox[List[SuperscriptBox["a", "3"], "+", RowBox[List["6", " ", SuperscriptBox["a", "2"], " ", "b"]], "-", RowBox[List["7", " ", "a", " ", SuperscriptBox["b", "2"]]], "-", RowBox[List["12", " ", 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["8", " ", "\[ImaginaryI]", " ", RowBox[List["(", RowBox[List[SuperscriptBox["a", "3"], "+", RowBox[List["5", " ", SuperscriptBox["a", "2"], " ", "b"]], "-", RowBox[List["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["2", " ", "b", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", SuperscriptBox["a", "2"]]], "-", RowBox[List["20", " ", "a", " ", "b"]], "+", RowBox[List["3", " ", SuperscriptBox["b", "2"]]], "+", RowBox[List["4", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "a"]], "-", RowBox[List["5", " ", "b"]]]], ")"]], " ", "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", "2"], " ", "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