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 Cosh

 http://functions.wolfram.com/01.20.21.4252.01

 Input Form

 Integrate[(Sinh[3 z] Cosh[z] Cosh[2 z])/Sqrt[(a + b Sinh[z]^2)^5], z] == ((2 a - b + b Cosh[2 z]) (64 a^2 - 68 a b + 21 b^2 + 48 a b Cosh[2 z] - 27 b^2 Cosh[2 z] - 24 a^2 Sqrt[1 + (b Cosh[2 z])/(2 a - b)] + 24 a b Sqrt[1 + (b Cosh[2 z])/(2 a - b)] - 6 b^2 Sqrt[1 + (b Cosh[2 z])/(2 a - b)] - 12 a b Cosh[2 z] Sqrt[1 + (b Cosh[2 z])/(2 a - b)] + 6 b^2 Cosh[2 z] Sqrt[1 + (b Cosh[2 z])/(2 a - b)] + 3 b^2 Cosh[4 z]))/ (6 b^3 Sqrt[(a + b Sinh[z]^2)^5])

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List["\[Integral]", RowBox[List[FractionBox[RowBox[List[" ", RowBox[List[RowBox[List["Sinh", "[", RowBox[List["3", " ", "z"]], "]"]], RowBox[List["Cosh", "[", "z", "]"]], " ", RowBox[List["Cosh", "[", RowBox[List["2", " ", "z"]], "]"]]]]]], SqrtBox[SuperscriptBox[RowBox[List["(", RowBox[List["a", "+", RowBox[List["b", " ", SuperscriptBox[RowBox[List["Sinh", "[", "z", "]"]], "2"]]]]], ")"]], "5"]]], RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["2", " ", "a"]], "-", "b", "+", RowBox[List["b", " ", RowBox[List["Cosh", "[", RowBox[List["2", " ", "z"]], "]"]]]]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["64", " ", SuperscriptBox["a", "2"]]], "-", RowBox[List["68", " ", "a", " ", "b"]], "+", RowBox[List["21", " ", SuperscriptBox["b", "2"]]], "+", RowBox[List["48", " ", "a", " ", "b", " ", RowBox[List["Cosh", "[", RowBox[List["2", " ", "z"]], "]"]]]], "-", RowBox[List["27", " ", SuperscriptBox["b", "2"], " ", RowBox[List["Cosh", "[", RowBox[List["2", " ", "z"]], "]"]]]], "-", RowBox[List["24", " ", SuperscriptBox["a", "2"], " ", SqrtBox[RowBox[List["1", "+", FractionBox[RowBox[List["b", " ", RowBox[List["Cosh", "[", RowBox[List["2", " ", "z"]], "]"]]]], RowBox[List[RowBox[List["2", " ", "a"]], "-", "b"]]]]]]]], "+", RowBox[List["24", " ", "a", " ", "b", " ", SqrtBox[RowBox[List["1", "+", FractionBox[RowBox[List["b", " ", RowBox[List["Cosh", "[", RowBox[List["2", " ", "z"]], "]"]]]], RowBox[List[RowBox[List["2", " ", "a"]], "-", "b"]]]]]]]], "-", RowBox[List["6", " ", SuperscriptBox["b", "2"], " ", SqrtBox[RowBox[List["1", "+", FractionBox[RowBox[List["b", " ", RowBox[List["Cosh", "[", RowBox[List["2", " ", "z"]], "]"]]]], RowBox[List[RowBox[List["2", " ", "a"]], "-", "b"]]]]]]]], "-", RowBox[List["12", " ", "a", " ", "b", " ", RowBox[List["Cosh", "[", RowBox[List["2", " ", "z"]], "]"]], " ", SqrtBox[RowBox[List["1", "+", FractionBox[RowBox[List["b", " ", RowBox[List["Cosh", "[", RowBox[List["2", " ", "z"]], "]"]]]], RowBox[List[RowBox[List["2", " ", "a"]], "-", "b"]]]]]]]], "+", RowBox[List["6", " ", SuperscriptBox["b", "2"], " ", RowBox[List["Cosh", "[", RowBox[List["2", " ", "z"]], "]"]], " ", SqrtBox[RowBox[List["1", "+", FractionBox[RowBox[List["b", " ", RowBox[List["Cosh", "[", RowBox[List["2", " ", "z"]], "]"]]]], RowBox[List[RowBox[List["2", " ", "a"]], "-", "b"]]]]]]]], "+", RowBox[List["3", " ", SuperscriptBox["b", "2"], " ", RowBox[List["Cosh", "[", RowBox[List["4", " ", "z"]], "]"]]]]]], ")"]]]], ")"]], "/", RowBox[List["(", RowBox[List["6", " ", SuperscriptBox["b", "3"], " ", SqrtBox[SuperscriptBox[RowBox[List["(", RowBox[List["a", "+", RowBox[List["b", " ", SuperscriptBox[RowBox[List["Sinh", "[", "z", "]"]], "2"]]]]], ")"]], "5"]]]], ")"]]]]]]]]

 MathML Form

 sinh ( 3 z ) cosh ( z ) cosh ( 2 z ) ( b sinh 2 ( z ) + a ) 5 z 1 6 b 3 ( b sinh 2 ( z ) + a ) 5 ( ( 2 a - b + b cosh ( 2 z ) ) ( - 24 b cosh ( 2 z ) 2 a - b + 1 a 2 + 64 a 2 + 24 b cosh ( 2 z ) 2 a - b + 1 b a - 68 b a + 48 b cosh ( 2 z ) a - 12 b cosh ( 2 z ) b cosh ( 2 z ) 2 a - b + 1 a + 21 b 2 + 6 b cosh ( 2 z ) 2 a - b + 1 b 2 cosh ( 2 z ) - 27 b 2 cosh ( 2 z ) + 3 b 2 cosh ( 4 z ) - 6 b 2 b cosh ( 2 z ) 2 a - b + 1 ) ) z 3 z z 2 z b z 2 a 5 1 2 -1 1 6 b 3 b z 2 a 5 1 2 -1 2 a -1 b b 2 z -24 b 2 z 2 a -1 b -1 1 1 2 a 2 64 a 2 24 b 2 z 2 a -1 b -1 1 1 2 b a -1 68 b a 48 b 2 z a -1 12 b 2 z b 2 z 2 a -1 b -1 1 1 2 a 21 b 2 6 b 2 z 2 a -1 b -1 1 1 2 b 2 2 z -1 27 b 2 2 z 3 b 2 4 z -1 6 b 2 b 2 z 2 a -1 b -1 1 1 2 [/itex]

 Rule Form

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

 2002-12-18