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 Coth

 http://functions.wolfram.com/01.22.21.0252.01

 Input Form

 Integrate[(A + B Coth[z])/(a + b Coth[z])^3, z] == ((A + B Coth[z]) Csch[z]^2 (b Cosh[z] + a Sinh[z]) ((b^2 (A b - a B))/((a - b)^2 (a + b)^2) - (2 (-3 a A b + 2 a^2 B + b^2 B) Sinh[z] (b Cosh[z] + a Sinh[z]))/ ((a - b)^2 (a + b)^2) + (2 (a^3 A + 3 a A b^2 - 3 a^2 b B - b^3 B) z (b Cosh[z] + a Sinh[z])^2)/((a - b)^3 (a + b)^3) + (1/(a^2 - b^2)^3) (2 (-3 a^2 A b - A b^3 + a^3 B + 3 a b^2 B) Log[b Cosh[z] + a Sinh[z]] (b Cosh[z] + a Sinh[z])^2)))/ (2 (a + b Coth[z])^3 (B Cosh[z] + A Sinh[z]))

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List["\[Integral]", RowBox[List[FractionBox[RowBox[List["A", "+", RowBox[List["B", " ", RowBox[List["Coth", "[", "z", "]"]]]]]], SuperscriptBox[RowBox[List["(", RowBox[List["a", "+", RowBox[List["b", " ", RowBox[List["Coth", "[", "z", "]"]]]]]], ")"]], "3"]], RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["(", RowBox[List["A", "+", RowBox[List["B", " ", RowBox[List["Coth", "[", "z", "]"]]]]]], ")"]], " ", SuperscriptBox[RowBox[List["Csch", "[", "z", "]"]], "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["b", " ", RowBox[List["Cosh", "[", "z", "]"]]]], "+", RowBox[List["a", " ", RowBox[List["Sinh", "[", "z", "]"]]]]]], ")"]], " ", RowBox[List["(", RowBox[List[FractionBox[RowBox[List[SuperscriptBox["b", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["A", " ", "b"]], "-", RowBox[List["a", " ", "B"]]]], ")"]]]], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["a", "-", "b"]], ")"]], "2"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["a", "+", "b"]], ")"]], "2"]]]], "-", FractionBox[RowBox[List["2", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "3"]], " ", "a", " ", "A", " ", "b"]], "+", RowBox[List["2", " ", SuperscriptBox["a", "2"], " ", "B"]], "+", RowBox[List[SuperscriptBox["b", "2"], " ", "B"]]]], ")"]], " ", RowBox[List["Sinh", "[", "z", "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["b", " ", RowBox[List["Cosh", "[", "z", "]"]]]], "+", RowBox[List["a", " ", RowBox[List["Sinh", "[", "z", "]"]]]]]], ")"]]]], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["a", "-", "b"]], ")"]], "2"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["a", "+", "b"]], ")"]], "2"]]]], "+", FractionBox[RowBox[List["2", " ", RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox["a", "3"], " ", "A"]], "+", RowBox[List["3", " ", "a", " ", "A", " ", SuperscriptBox["b", "2"]]], "-", RowBox[List["3", " ", SuperscriptBox["a", "2"], " ", "b", " ", "B"]], "-", RowBox[List[SuperscriptBox["b", "3"], " ", "B"]]]], ")"]], " ", "z", " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["b", " ", RowBox[List["Cosh", "[", "z", "]"]]]], "+", RowBox[List["a", " ", RowBox[List["Sinh", "[", "z", "]"]]]]]], ")"]], "2"]]], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["a", "-", "b"]], ")"]], "3"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["a", "+", "b"]], ")"]], "3"]]]], "+", RowBox[List[FractionBox["1", SuperscriptBox[RowBox[List["(", RowBox[List[SuperscriptBox["a", "2"], "-", SuperscriptBox["b", "2"]]], ")"]], "3"]], RowBox[List["(", RowBox[List["2", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "3"]], " ", SuperscriptBox["a", "2"], " ", "A", " ", "b"]], "-", RowBox[List["A", " ", SuperscriptBox["b", "3"]]], "+", RowBox[List[SuperscriptBox["a", "3"], " ", "B"]], "+", RowBox[List["3", " ", "a", " ", SuperscriptBox["b", "2"], " ", "B"]]]], ")"]], " ", RowBox[List["Log", "[", RowBox[List[RowBox[List["b", " ", RowBox[List["Cosh", "[", "z", "]"]]]], "+", RowBox[List["a", " ", RowBox[List["Sinh", "[", "z", "]"]]]]]], "]"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["b", " ", RowBox[List["Cosh", "[", "z", "]"]]]], "+", RowBox[List["a", " ", RowBox[List["Sinh", "[", "z", "]"]]]]]], ")"]], "2"]]], ")"]]]]]], ")"]]]], ")"]], "/", RowBox[List["(", RowBox[List["2", " ", SuperscriptBox[RowBox[List["(", RowBox[List["a", "+", RowBox[List["b", " ", RowBox[List["Coth", "[", "z", "]"]]]]]], ")"]], "3"], " ", RowBox[List["(", RowBox[List[RowBox[List["B", " ", RowBox[List["Cosh", "[", "z", "]"]]]], "+", RowBox[List["A", " ", RowBox[List["Sinh", "[", "z", "]"]]]]]], ")"]]]], ")"]]]]]]]]

 MathML Form

 A + B coth ( z ) ( a + b coth ( z ) ) 3 z ( ( A + B coth ( z ) ) csch 2 ( z ) ( b cosh ( z ) + a sinh ( z ) ) ( ( A b - a B ) b 2 ( a - b ) 2 ( a + b ) 2 + 2 ( A a 3 - 3 b B a 2 + 3 A b 2 a - b 3 B ) z ( b cosh ( z ) + a sinh ( z ) ) 2 ( a - b ) 3 ( a + b ) 3 + 1 ( a 2 - b 2 ) 3 ( 2 ( B a 3 - 3 A b a 2 + 3 b 2 B a - A b 3 ) log ( b cosh ( z ) + a sinh ( z ) ) ( b cosh ( z ) + a sinh ( z ) ) 2 ) - 2 ( 2 B a 2 - 3 A b a + b 2 B ) sinh ( z ) ( b cosh ( z ) + a sinh ( z ) ) ( a - b ) 2 ( a + b ) 2 ) ) / ( 2 ( a + b coth ( z ) ) 3 ( B cosh ( z ) + A sinh ( z ) ) ) z A B z a b z 3 -1 A B z z 2 b z a z A b -1 a B b 2 a -1 b 2 a b 2 -1 2 A a 3 -1 3 b B a 2 3 A b 2 a -1 b 3 B z b z a z 2 a -1 b 3 a b 3 -1 1 a 2 -1 b 2 3 -1 2 B a 3 -1 3 A b a 2 3 b 2 B a -1 A b 3 b z a z b z a z 2 -1 2 2 B a 2 -1 3 A b a b 2 B z b z a z a -1 b 2 a b 2 -1 2 a b z 3 B z A z -1 [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["\[Integral]", RowBox[List[FractionBox[RowBox[List["A_", "+", RowBox[List["B_", " ", RowBox[List["Coth", "[", "z_", "]"]]]]]], SuperscriptBox[RowBox[List["(", RowBox[List["a_", "+", RowBox[List["b_", " ", RowBox[List["Coth", "[", "z_", "]"]]]]]], ")"]], "3"]], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["A", "+", RowBox[List["B", " ", RowBox[List["Coth", "[", "z", "]"]]]]]], ")"]], " ", SuperscriptBox[RowBox[List["Csch", "[", "z", "]"]], "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["b", " ", RowBox[List["Cosh", "[", "z", "]"]]]], "+", RowBox[List["a", " ", RowBox[List["Sinh", "[", "z", "]"]]]]]], ")"]], " ", RowBox[List["(", RowBox[List[FractionBox[RowBox[List[SuperscriptBox["b", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["A", " ", "b"]], "-", RowBox[List["a", " ", "B"]]]], ")"]]]], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["a", "-", "b"]], ")"]], "2"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["a", "+", "b"]], ")"]], "2"]]]], "-", FractionBox[RowBox[List["2", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "3"]], " ", "a", " ", "A", " ", "b"]], "+", RowBox[List["2", " ", SuperscriptBox["a", "2"], " ", "B"]], "+", RowBox[List[SuperscriptBox["b", "2"], " ", "B"]]]], ")"]], " ", RowBox[List["Sinh", "[", "z", "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["b", " ", RowBox[List["Cosh", "[", "z", "]"]]]], "+", RowBox[List["a", " ", RowBox[List["Sinh", "[", "z", "]"]]]]]], ")"]]]], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["a", "-", "b"]], ")"]], "2"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["a", "+", "b"]], ")"]], "2"]]]], "+", FractionBox[RowBox[List["2", " ", RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox["a", "3"], " ", "A"]], "+", RowBox[List["3", " ", "a", " ", "A", " ", SuperscriptBox["b", "2"]]], "-", RowBox[List["3", " ", SuperscriptBox["a", "2"], " ", "b", " ", "B"]], "-", RowBox[List[SuperscriptBox["b", "3"], " ", "B"]]]], ")"]], " ", "z", " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["b", " ", RowBox[List["Cosh", "[", "z", "]"]]]], "+", RowBox[List["a", " ", RowBox[List["Sinh", "[", "z", "]"]]]]]], ")"]], "2"]]], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["a", "-", "b"]], ")"]], "3"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["a", "+", "b"]], ")"]], "3"]]]], "+", FractionBox[RowBox[List["2", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "3"]], " ", SuperscriptBox["a", "2"], " ", "A", " ", "b"]], "-", RowBox[List["A", " ", SuperscriptBox["b", "3"]]], "+", RowBox[List[SuperscriptBox["a", "3"], " ", "B"]], "+", RowBox[List["3", " ", "a", " ", SuperscriptBox["b", "2"], " ", "B"]]]], ")"]], " ", RowBox[List["Log", "[", RowBox[List[RowBox[List["b", " ", RowBox[List["Cosh", "[", "z", "]"]]]], "+", RowBox[List["a", " ", RowBox[List["Sinh", "[", "z", "]"]]]]]], "]"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["b", " ", RowBox[List["Cosh", "[", "z", "]"]]]], "+", RowBox[List["a", " ", RowBox[List["Sinh", "[", "z", "]"]]]]]], ")"]], "2"]]], SuperscriptBox[RowBox[List["(", RowBox[List[SuperscriptBox["a", "2"], "-", SuperscriptBox["b", "2"]]], ")"]], "3"]]]], ")"]]]], RowBox[List["2", " ", SuperscriptBox[RowBox[List["(", RowBox[List["a", "+", RowBox[List["b", " ", RowBox[List["Coth", "[", "z", "]"]]]]]], ")"]], "3"], " ", RowBox[List["(", RowBox[List[RowBox[List["B", " ", RowBox[List["Cosh", "[", "z", "]"]]]], "+", RowBox[List["A", " ", RowBox[List["Sinh", "[", "z", "]"]]]]]], ")"]]]]]]]]]

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

 2002-12-18