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 Coth

 http://functions.wolfram.com/01.22.21.0281.01

 Input Form

 Integrate[Coth[c z]/Sqrt[(a + b Coth[c z]^2)^5], z] == ((-a + b + (a + b) Cosh[2 c z]) Csch[c z]^6 (-2 (a + b)^2 Cosh[4 c z] + (a + b) Cosh[2 c z] (8 a + 2 b + 3 Sqrt[2] Sqrt[-a + b + (a + b) Cosh[2 c z]] Log[Sqrt[-a + b + (a + b) Cosh[2 c z]] + Sqrt[2] Sqrt[(a + b) Sinh[c z]^2]] Sqrt[(a + b) Sinh[c z]^2]) - 3 (2 a (a + b) + Sqrt[2] (a - b) Sqrt[-a + b + (a + b) Cosh[2 c z]] Log[Sqrt[-a + b + (a + b) Cosh[2 c z]] + Sqrt[2] Sqrt[(a + b) Sinh[c z]^2]] Sqrt[(a + b) Sinh[c z]^2])))/ (24 (a + b)^3 c Sqrt[(a + b Coth[c z]^2)^5])

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List["\[Integral]", RowBox[List[FractionBox[RowBox[List["Coth", "[", RowBox[List["c", " ", "z"]], "]"]], SqrtBox[SuperscriptBox[RowBox[List["(", RowBox[List["a", "+", RowBox[List["b", " ", SuperscriptBox[RowBox[List["Coth", "[", RowBox[List["c", " ", "z"]], "]"]], "2"]]]]], ")"]], "5"]]], RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "a"]], "+", "b", "+", RowBox[List[RowBox[List["(", RowBox[List["a", "+", "b"]], ")"]], " ", RowBox[List["Cosh", "[", RowBox[List["2", " ", "c", " ", "z"]], "]"]]]]]], ")"]], " ", SuperscriptBox[RowBox[List["Csch", "[", RowBox[List["c", " ", "z"]], "]"]], "6"], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["a", "+", "b"]], ")"]], "2"], " ", RowBox[List["Cosh", "[", RowBox[List["4", " ", "c", " ", "z"]], "]"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List["a", "+", "b"]], ")"]], " ", RowBox[List["Cosh", "[", RowBox[List["2", " ", "c", " ", "z"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["8", " ", "a"]], "+", RowBox[List["2", " ", "b"]], "+", RowBox[List["3", " ", SqrtBox["2"], " ", SqrtBox[RowBox[List[RowBox[List["-", "a"]], "+", "b", "+", RowBox[List[RowBox[List["(", RowBox[List["a", "+", "b"]], ")"]], " ", RowBox[List["Cosh", "[", RowBox[List["2", " ", "c", " ", "z"]], "]"]]]]]]], " ", RowBox[List["Log", "[", RowBox[List[SqrtBox[RowBox[List[RowBox[List["-", "a"]], "+", "b", "+", RowBox[List[RowBox[List["(", RowBox[List["a", "+", "b"]], ")"]], " ", RowBox[List["Cosh", "[", RowBox[List["2", " ", "c", " ", "z"]], "]"]]]]]]], "+", RowBox[List[SqrtBox["2"], " ", SqrtBox[RowBox[List[RowBox[List["(", RowBox[List["a", "+", "b"]], ")"]], " ", SuperscriptBox[RowBox[List["Sinh", "[", RowBox[List["c", " ", "z"]], "]"]], "2"]]]]]]]], "]"]], " ", SqrtBox[RowBox[List[RowBox[List["(", RowBox[List["a", "+", "b"]], ")"]], " ", SuperscriptBox[RowBox[List["Sinh", "[", RowBox[List["c", " ", "z"]], "]"]], "2"]]]]]]]], ")"]]]], "-", RowBox[List["3", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "a", " ", RowBox[List["(", RowBox[List["a", "+", "b"]], ")"]]]], "+", RowBox[List[SqrtBox["2"], " ", RowBox[List["(", RowBox[List["a", "-", "b"]], ")"]], " ", SqrtBox[RowBox[List[RowBox[List["-", "a"]], "+", "b", "+", RowBox[List[RowBox[List["(", RowBox[List["a", "+", "b"]], ")"]], " ", RowBox[List["Cosh", "[", RowBox[List["2", " ", "c", " ", "z"]], "]"]]]]]]], " ", RowBox[List["Log", "[", RowBox[List[SqrtBox[RowBox[List[RowBox[List["-", "a"]], "+", "b", "+", RowBox[List[RowBox[List["(", RowBox[List["a", "+", "b"]], ")"]], " ", RowBox[List["Cosh", "[", RowBox[List["2", " ", "c", " ", "z"]], "]"]]]]]]], "+", RowBox[List[SqrtBox["2"], " ", SqrtBox[RowBox[List[RowBox[List["(", RowBox[List["a", "+", "b"]], ")"]], " ", SuperscriptBox[RowBox[List["Sinh", "[", RowBox[List["c", " ", "z"]], "]"]], "2"]]]]]]]], "]"]], " ", SqrtBox[RowBox[List[RowBox[List["(", RowBox[List["a", "+", "b"]], ")"]], " ", SuperscriptBox[RowBox[List["Sinh", "[", RowBox[List["c", " ", "z"]], "]"]], "2"]]]]]]]], ")"]]]]]], ")"]]]], ")"]], "/", RowBox[List["(", RowBox[List["24", " ", SuperscriptBox[RowBox[List["(", RowBox[List["a", "+", "b"]], ")"]], "3"], " ", "c", " ", SqrtBox[SuperscriptBox[RowBox[List["(", RowBox[List["a", "+", RowBox[List["b", " ", SuperscriptBox[RowBox[List["Coth", "[", RowBox[List["c", " ", "z"]], "]"]], "2"]]]]], ")"]], "5"]]]], ")"]]]]]]]]

 MathML Form

 coth ( c z ) ( a + b coth 2 ( c z ) ) 5 z ( ( - a + b + ( a + b ) cosh ( 2 c z ) ) csch 6 ( c z ) ( - 2 cosh ( 4 c z ) ( a + b ) 2 + cosh ( 2 c z ) ( 8 a + 2 b + 3 2 - a + b + ( a + b ) cosh ( 2 c z ) log ( - a + b + ( a + b ) cosh ( 2 c z ) + 2 ( a + b ) sinh 2 ( c z ) ) ( a + b ) sinh 2 ( c z ) ) ( a + b ) - 3 ( 2 a ( a + b ) + 2 ( a - b ) - a + b + ( a + b ) cosh ( 2 c z ) log ( - a + b + ( a + b ) cosh ( 2 c z ) + 2 ( a + b ) sinh 2 ( c z ) ) ( a + b ) sinh 2 ( c z ) ) ) ) / ( 24 ( a + b ) 3 c ( b coth 2 ( c z ) + a ) 5 ) z c z a b c z 2 5 1 2 -1 -1 a b a b 2 c z c z 6 -2 4 c z a b 2 2 c z 8 a 2 b 3 2 1 2 -1 a b a b 2 c z 1 2 -1 a b a b 2 c z 1 2 2 1 2 a b c z 2 1 2 a b c z 2 1 2 a b -1 3 2 a a b 2 1 2 a -1 b -1 a b a b 2 c z 1 2 -1 a b a b 2 c z 1 2 2 1 2 a b c z 2 1 2 a b c z 2 1 2 24 a b 3 c b c z 2 a 5 1 2 -1 [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["\[Integral]", RowBox[List[FractionBox[RowBox[List["Coth", "[", RowBox[List["c_", " ", "z_"]], "]"]], SqrtBox[SuperscriptBox[RowBox[List["(", RowBox[List["a_", "+", RowBox[List["b_", " ", SuperscriptBox[RowBox[List["Coth", "[", RowBox[List["c_", " ", "z_"]], "]"]], "2"]]]]], ")"]], "5"]]], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "a"]], "+", "b", "+", RowBox[List[RowBox[List["(", RowBox[List["a", "+", "b"]], ")"]], " ", RowBox[List["Cosh", "[", RowBox[List["2", " ", "c", " ", "z"]], "]"]]]]]], ")"]], " ", SuperscriptBox[RowBox[List["Csch", "[", RowBox[List["c", " ", "z"]], "]"]], "6"], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["a", "+", "b"]], ")"]], "2"], " ", RowBox[List["Cosh", "[", RowBox[List["4", " ", "c", " ", "z"]], "]"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List["a", "+", "b"]], ")"]], " ", RowBox[List["Cosh", "[", RowBox[List["2", " ", "c", " ", "z"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["8", " ", "a"]], "+", RowBox[List["2", " ", "b"]], "+", RowBox[List["3", " ", SqrtBox["2"], " ", SqrtBox[RowBox[List[RowBox[List["-", "a"]], "+", "b", "+", RowBox[List[RowBox[List["(", RowBox[List["a", "+", "b"]], ")"]], " ", RowBox[List["Cosh", "[", RowBox[List["2", " ", "c", " ", "z"]], "]"]]]]]]], " ", RowBox[List["Log", "[", RowBox[List[SqrtBox[RowBox[List[RowBox[List["-", "a"]], "+", "b", "+", RowBox[List[RowBox[List["(", RowBox[List["a", "+", "b"]], ")"]], " ", RowBox[List["Cosh", "[", RowBox[List["2", " ", "c", " ", "z"]], "]"]]]]]]], "+", RowBox[List[SqrtBox["2"], " ", SqrtBox[RowBox[List[RowBox[List["(", RowBox[List["a", "+", "b"]], ")"]], " ", SuperscriptBox[RowBox[List["Sinh", "[", RowBox[List["c", " ", "z"]], "]"]], "2"]]]]]]]], "]"]], " ", SqrtBox[RowBox[List[RowBox[List["(", RowBox[List["a", "+", "b"]], ")"]], " ", SuperscriptBox[RowBox[List["Sinh", "[", RowBox[List["c", " ", "z"]], "]"]], "2"]]]]]]]], ")"]]]], "-", RowBox[List["3", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "a", " ", RowBox[List["(", RowBox[List["a", "+", "b"]], ")"]]]], "+", RowBox[List[SqrtBox["2"], " ", RowBox[List["(", RowBox[List["a", "-", "b"]], ")"]], " ", SqrtBox[RowBox[List[RowBox[List["-", "a"]], "+", "b", "+", RowBox[List[RowBox[List["(", RowBox[List["a", "+", "b"]], ")"]], " ", RowBox[List["Cosh", "[", RowBox[List["2", " ", "c", " ", "z"]], "]"]]]]]]], " ", RowBox[List["Log", "[", RowBox[List[SqrtBox[RowBox[List[RowBox[List["-", "a"]], "+", "b", "+", RowBox[List[RowBox[List["(", RowBox[List["a", "+", "b"]], ")"]], " ", RowBox[List["Cosh", "[", RowBox[List["2", " ", "c", " ", "z"]], "]"]]]]]]], "+", RowBox[List[SqrtBox["2"], " ", SqrtBox[RowBox[List[RowBox[List["(", RowBox[List["a", "+", "b"]], ")"]], " ", SuperscriptBox[RowBox[List["Sinh", "[", RowBox[List["c", " ", "z"]], "]"]], "2"]]]]]]]], "]"]], " ", SqrtBox[RowBox[List[RowBox[List["(", RowBox[List["a", "+", "b"]], ")"]], " ", SuperscriptBox[RowBox[List["Sinh", "[", RowBox[List["c", " ", "z"]], "]"]], "2"]]]]]]]], ")"]]]]]], ")"]]]], RowBox[List["24", " ", SuperscriptBox[RowBox[List["(", RowBox[List["a", "+", "b"]], ")"]], "3"], " ", "c", " ", SqrtBox[SuperscriptBox[RowBox[List["(", RowBox[List["a", "+", RowBox[List["b", " ", SuperscriptBox[RowBox[List["Coth", "[", RowBox[List["c", " ", "z"]], "]"]], "2"]]]]], ")"]], "5"]]]]]]]]]

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

 2002-12-18