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 Cosh

 http://functions.wolfram.com/01.20.21.4241.01

 Input Form

 Integrate[Sqrt[a + b Sinh[c z]] Cosh[c z]^2, z] == (8 ((-I) a^3 - a^2 b + 3 I a b^2 + 3 b^3) EllipticE[(1/4) (Pi - 2 I c z), -((2 I b)/(a - I b))] Sqrt[(a + b Sinh[c z])/(a - I b)] + 8 I a (a^2 + b^2) EllipticF[(1/4) (Pi - 2 I c z), -((2 I b)/(a - I b))] Sqrt[(a + b Sinh[c z])/(a - I b)] + 2 b Cosh[c z] (2 a^2 - 3 b^2 + 3 b^2 Cosh[2 c z] + 8 a b Sinh[c z]))/ (30 b^2 c Sqrt[a + b Sinh[c z]])

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List["\[Integral]", RowBox[List[SqrtBox[RowBox[List["a", "+", RowBox[List["b", " ", RowBox[List["Sinh", "[", RowBox[List["c", " ", "z"]], "]"]]]]]]], SuperscriptBox[RowBox[List["Cosh", "[", RowBox[List["c", " ", "z"]], "]"]], "2"], " ", RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["8", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", SuperscriptBox["a", "3"]]], "-", RowBox[List[SuperscriptBox["a", "2"], " ", "b"]], "+", RowBox[List["3", " ", "\[ImaginaryI]", " ", "a", " ", SuperscriptBox["b", "2"]]], "+", RowBox[List["3", " ", SuperscriptBox["b", "3"]]]]], ")"]], " ", RowBox[List["EllipticE", "[", RowBox[List[RowBox[List[FractionBox["1", "4"], " ", RowBox[List["(", RowBox[List["\[Pi]", "-", RowBox[List["2", " ", "\[ImaginaryI]", " ", "c", " ", "z"]]]], ")"]]]], ",", RowBox[List["-", FractionBox[RowBox[List["2", " ", "\[ImaginaryI]", " ", "b"]], RowBox[List["a", "-", RowBox[List["\[ImaginaryI]", " ", "b"]]]]]]]]], "]"]], " ", SqrtBox[FractionBox[RowBox[List["a", "+", RowBox[List["b", " ", RowBox[List["Sinh", "[", RowBox[List["c", " ", "z"]], "]"]]]]]], RowBox[List["a", "-", RowBox[List["\[ImaginaryI]", " ", "b"]]]]]]]], "+", RowBox[List["8", " ", "\[ImaginaryI]", " ", "a", " ", RowBox[List["(", RowBox[List[SuperscriptBox["a", "2"], "+", SuperscriptBox["b", "2"]]], ")"]], " ", RowBox[List["EllipticF", "[", RowBox[List[RowBox[List[FractionBox["1", "4"], " ", RowBox[List["(", RowBox[List["\[Pi]", "-", RowBox[List["2", " ", "\[ImaginaryI]", " ", "c", " ", "z"]]]], ")"]]]], ",", RowBox[List["-", FractionBox[RowBox[List["2", " ", "\[ImaginaryI]", " ", "b"]], RowBox[List["a", "-", RowBox[List["\[ImaginaryI]", " ", "b"]]]]]]]]], "]"]], " ", SqrtBox[FractionBox[RowBox[List["a", "+", RowBox[List["b", " ", RowBox[List["Sinh", "[", RowBox[List["c", " ", "z"]], "]"]]]]]], RowBox[List["a", "-", RowBox[List["\[ImaginaryI]", " ", "b"]]]]]]]], "+", RowBox[List["2", " ", "b", " ", RowBox[List["Cosh", "[", RowBox[List["c", " ", "z"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", SuperscriptBox["a", "2"]]], "-", RowBox[List["3", " ", SuperscriptBox["b", "2"]]], "+", RowBox[List["3", " ", SuperscriptBox["b", "2"], " ", RowBox[List["Cosh", "[", RowBox[List["2", " ", "c", " ", "z"]], "]"]]]], "+", RowBox[List["8", " ", "a", " ", "b", " ", RowBox[List["Sinh", "[", RowBox[List["c", " ", "z"]], "]"]]]]]], ")"]]]]]], ")"]], "/", RowBox[List["(", RowBox[List["30", " ", SuperscriptBox["b", "2"], " ", "c", " ", SqrtBox[RowBox[List["a", "+", RowBox[List["b", " ", RowBox[List["Sinh", "[", RowBox[List["c", " ", "z"]], "]"]]]]]]]]], ")"]]]]]]]]

 MathML Form

 a + b sinh ( c z ) cosh 2 ( c z ) z 1 30 b 2 c a + b sinh ( c z ) ( 8 a + b sinh ( c z ) a - b ( - a 3 - b a 2 + 3 b 2 a + 3 b 3 ) E ( 1 4 ( π - 2 c z ) - 2 b a - b ) + 8 a a + b sinh ( c z ) a - b ( a 2 + b 2 ) F ( 1 4 ( π - 2 c z ) - 2 b a - b ) + 2 b cosh ( c z ) ( 2 a 2 + 8 b sinh ( c z ) a - 3 b 2 + 3 b 2 cosh ( 2 c z ) ) ) z a b c z 1 2 c z 2 1 30 b 2 c a b c z 1 2 -1 8 a b c z a -1 b -1 1 2 -1 a 3 -1 b a 2 3 b 2 a 3 b 3 EllipticE 1 4 -1 2 c z -1 2 b a -1 b -1 8 a a b c z a -1 b -1 1 2 a 2 b 2 EllipticF 1 4 -1 2 c z -1 2 b a -1 b -1 2 b c z 2 a 2 8 b c z a -1 3 b 2 3 b 2 2 c z [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["\[Integral]", RowBox[List[RowBox[List[SqrtBox[RowBox[List["a_", "+", RowBox[List["b_", " ", RowBox[List["Sinh", "[", RowBox[List["c_", " ", "z_"]], "]"]]]]]]], " ", SuperscriptBox[RowBox[List["Cosh", "[", RowBox[List["c_", " ", "z_"]], "]"]], "2"]]], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", FractionBox[RowBox[List[RowBox[List["8", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", SuperscriptBox["a", "3"]]], "-", RowBox[List[SuperscriptBox["a", "2"], " ", "b"]], "+", RowBox[List["3", " ", "\[ImaginaryI]", " ", "a", " ", SuperscriptBox["b", "2"]]], "+", RowBox[List["3", " ", SuperscriptBox["b", "3"]]]]], ")"]], " ", RowBox[List["EllipticE", "[", RowBox[List[RowBox[List[FractionBox["1", "4"], " ", RowBox[List["(", RowBox[List["\[Pi]", "-", RowBox[List["2", " ", "\[ImaginaryI]", " ", "c", " ", "z"]]]], ")"]]]], ",", RowBox[List["-", FractionBox[RowBox[List["2", " ", "\[ImaginaryI]", " ", "b"]], RowBox[List["a", "-", RowBox[List["\[ImaginaryI]", " ", "b"]]]]]]]]], "]"]], " ", SqrtBox[FractionBox[RowBox[List["a", "+", RowBox[List["b", " ", RowBox[List["Sinh", "[", RowBox[List["c", " ", "z"]], "]"]]]]]], RowBox[List["a", "-", RowBox[List["\[ImaginaryI]", " ", "b"]]]]]]]], "+", RowBox[List["8", " ", "\[ImaginaryI]", " ", "a", " ", RowBox[List["(", RowBox[List[SuperscriptBox["a", "2"], "+", SuperscriptBox["b", "2"]]], ")"]], " ", RowBox[List["EllipticF", "[", RowBox[List[RowBox[List[FractionBox["1", "4"], " ", RowBox[List["(", RowBox[List["\[Pi]", "-", RowBox[List["2", " ", "\[ImaginaryI]", " ", "c", " ", "z"]]]], ")"]]]], ",", RowBox[List["-", FractionBox[RowBox[List["2", " ", "\[ImaginaryI]", " ", "b"]], RowBox[List["a", "-", RowBox[List["\[ImaginaryI]", " ", "b"]]]]]]]]], "]"]], " ", SqrtBox[FractionBox[RowBox[List["a", "+", RowBox[List["b", " ", RowBox[List["Sinh", "[", RowBox[List["c", " ", "z"]], "]"]]]]]], RowBox[List["a", "-", RowBox[List["\[ImaginaryI]", " ", "b"]]]]]]]], "+", RowBox[List["2", " ", "b", " ", RowBox[List["Cosh", "[", RowBox[List["c", " ", "z"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", SuperscriptBox["a", "2"]]], "-", RowBox[List["3", " ", SuperscriptBox["b", "2"]]], "+", RowBox[List["3", " ", SuperscriptBox["b", "2"], " ", RowBox[List["Cosh", "[", RowBox[List["2", " ", "c", " ", "z"]], "]"]]]], "+", RowBox[List["8", " ", "a", " ", "b", " ", RowBox[List["Sinh", "[", RowBox[List["c", " ", "z"]], "]"]]]]]], ")"]]]]]], RowBox[List["30", " ", SuperscriptBox["b", "2"], " ", "c", " ", SqrtBox[RowBox[List["a", "+", RowBox[List["b", " ", RowBox[List["Sinh", "[", RowBox[List["c", " ", "z"]], "]"]]]]]]]]]]]]]]

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

 2002-12-18