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 Coth

 http://functions.wolfram.com/01.22.21.0253.01

 Input Form

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

 Standard Form

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

 MathML Form

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

 Rule Form

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

 2002-12-18