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 Cosh

 http://functions.wolfram.com/01.20.21.4302.01

 Input Form

 Integrate[(A Sinh[z]^2 + B Sinh[2 z] + C Cosh[z]^2 + \[Delta])/ (a Sinh[z]^2 + b Sinh[2 z] + c Cosh[z]^2), z] == ((-4 b B + a (A + C) + c (A + C)) z + (1/Sqrt[-b^2 + a c]) ((2 b B c + A (2 b^2 - c (a + c)) - 2 b^2 C - 4 b^2 \[Delta] + c^2 \[Delta] + a^2 (C + \[Delta]) + a (-2 b B + c C + 2 c \[Delta])) ArcTan[(b + a Tanh[z])/Sqrt[-b^2 + a c]]) + ((-A) b + a B + B c - b C) Log[-a + c + (a + c) Cosh[2 z] + 2 b Sinh[2 z]])/ ((a - 2 b + c) (a + 2 b + c))

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List["\[Integral]", RowBox[List[FractionBox[RowBox[List[" ", RowBox[List[RowBox[List["A", " ", SuperscriptBox[RowBox[List["Sinh", "[", "z", "]"]], "2"]]], "+", RowBox[List["B", " ", RowBox[List["Sinh", "[", RowBox[List["2", " ", "z"]], "]"]]]], "+", RowBox[List["C", " ", SuperscriptBox[RowBox[List["Cosh", "[", "z", "]"]], "2"]]], "+", "\[Delta]"]]]], RowBox[List[RowBox[List["a", " ", SuperscriptBox[RowBox[List["Sinh", "[", "z", "]"]], "2"]]], "+", RowBox[List["b", " ", RowBox[List["Sinh", "[", RowBox[List["2", " ", "z"]], "]"]]]], "+", RowBox[List["c", " ", SuperscriptBox[RowBox[List["Cosh", "[", "z", "]"]], "2"]]]]]], RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "4"]], " ", "b", " ", "B"]], "+", RowBox[List["a", " ", RowBox[List["(", RowBox[List["A", "+", "C"]], ")"]]]], "+", RowBox[List["c", " ", RowBox[List["(", RowBox[List["A", "+", "C"]], ")"]]]]]], ")"]], " ", "z"]], "+", RowBox[List[FractionBox["1", SqrtBox[RowBox[List[RowBox[List["-", SuperscriptBox["b", "2"]]], "+", RowBox[List["a", " ", "c"]]]]]], RowBox[List["(", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["2", " ", "b", " ", "B", " ", "c"]], "+", RowBox[List["A", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", SuperscriptBox["b", "2"]]], "-", RowBox[List["c", " ", RowBox[List["(", RowBox[List["a", "+", "c"]], ")"]]]]]], ")"]]]], "-", RowBox[List["2", " ", SuperscriptBox["b", "2"], " ", "C"]], "-", RowBox[List["4", " ", SuperscriptBox["b", "2"], " ", "\[Delta]"]], "+", RowBox[List[SuperscriptBox["c", "2"], " ", "\[Delta]"]], "+", RowBox[List[SuperscriptBox["a", "2"], " ", RowBox[List["(", RowBox[List["C", "+", "\[Delta]"]], ")"]]]], "+", RowBox[List["a", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "b", " ", "B"]], "+", RowBox[List["c", " ", "C"]], "+", RowBox[List["2", " ", "c", " ", "\[Delta]"]]]], ")"]]]]]], ")"]], " ", RowBox[List["ArcTan", "[", FractionBox[RowBox[List["b", "+", RowBox[List["a", " ", RowBox[List["Tanh", "[", "z", "]"]]]]]], SqrtBox[RowBox[List[RowBox[List["-", SuperscriptBox["b", "2"]]], "+", RowBox[List["a", " ", "c"]]]]]], "]"]]]], ")"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "A"]], " ", "b"]], "+", RowBox[List["a", " ", "B"]], "+", RowBox[List["B", " ", "c"]], "-", RowBox[List["b", " ", "C"]]]], ")"]], " ", RowBox[List["Log", "[", RowBox[List[RowBox[List["-", "a"]], "+", "c", "+", RowBox[List[RowBox[List["(", RowBox[List["a", "+", "c"]], ")"]], " ", RowBox[List["Cosh", "[", RowBox[List["2", " ", "z"]], "]"]]]], "+", RowBox[List["2", " ", "b", " ", RowBox[List["Sinh", "[", RowBox[List["2", " ", "z"]], "]"]]]]]], "]"]]]]]], ")"]], "/", RowBox[List["(", RowBox[List[RowBox[List["(", RowBox[List["a", "-", RowBox[List["2", " ", "b"]], "+", "c"]], ")"]], " ", RowBox[List["(", RowBox[List["a", "+", RowBox[List["2", " ", "b"]], "+", "c"]], ")"]]]], ")"]]]]]]]]

 MathML Form

 C cosh 2 ( z ) + A sinh 2 ( z ) + δ + B sinh ( 2 z ) c cosh 2 ( z ) + a sinh 2 ( z ) + b sinh ( 2 z ) z 1 ( a - 2 b + c ) ( a + 2 b + c ) ( ( - 4 b B + a ( A + C ) + c ( A + C ) ) z + 1 a c - b 2 ( ( ( C + δ ) a 2 + ( - 2 b B + c C + 2 c δ ) a + 2 b B c + A ( 2 b 2 - c ( a + c ) ) - 2 b 2 C - 4 b 2 δ + c 2 δ ) tan - 1 ( b + a tanh ( z ) a c - b 2 ) ) + ( - A b - C b + a B + B c ) log ( - a + c + ( a + c ) cosh ( 2 z ) + 2 b sinh ( 2 z ) ) ) z C z 2 A z 2 δ B 2 z c z 2 a z 2 b 2 z -1 1 a -1 2 b c a 2 b c -1 -4 b B a A C c A C z 1 a c -1 b 2 1 2 -1 C δ a 2 -2 b B c C 2 c δ a 2 b B c A 2 b 2 -1 c a c -1 2 b 2 C -1 4 b 2 δ c 2 δ b a z a c -1 b 2 1 2 -1 -1 A b -1 C b a B B c -1 a c a c 2 z 2 b 2 z [/itex]

 Rule Form

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

 2002-12-18