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 Coth

 http://functions.wolfram.com/01.22.21.0269.01

 Input Form

 Integrate[Sqrt[a + b Coth[c z]^2], z] == -(Sqrt[a + b Coth[c z]^2] (Sqrt[b] Log[-((1/b^(3/2)) (4 (2 a + b + Coth[(c z)/2]^2 (b + (1/Sqrt[2]) (Sqrt[b] Sqrt[(-a + b + (a + b) Cosh[2 c z]) Sech[(c z)/2]^4])))))] + Sqrt[b] Log[(1/Sqrt[b]) (4 a + 2 b + Sqrt[2] Sqrt[b] Sqrt[(-a + b + (a + b) Cosh[2 c z]) Sech[(c z)/2]^4] + 2 b Tanh[(c z)/2]^2)] - 2 Sqrt[a + b] Log[-((1/(a + b)^(3/2)) (Cosh[(c z)/2]^2 (a + b + (1/Sqrt[2]) (Sqrt[a + b] Sqrt[(-a + b + (a + b) Cosh[2 c z]) Sech[(c z)/2]^4]) + (a + b) Tanh[(c z)/2]^2)))]) Tanh[(c z)/2])/(Sqrt[2] c Sqrt[(-a + b + (a + b) Cosh[2 c z])/ (1 + Cosh[c z])^2])

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List["\[Integral]", RowBox[List[SqrtBox[RowBox[List["a", "+", RowBox[List["b", " ", SuperscriptBox[RowBox[List["Coth", "[", RowBox[List["c", " ", "z"]], "]"]], "2"]]]]]], RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", RowBox[List[RowBox[List["-", RowBox[List["(", RowBox[List[SqrtBox[RowBox[List["a", "+", RowBox[List["b", " ", SuperscriptBox[RowBox[List["Coth", "[", RowBox[List["c", " ", "z"]], "]"]], "2"]]]]]], " ", RowBox[List["(", RowBox[List[RowBox[List[SqrtBox["b"], " ", RowBox[List["Log", "[", RowBox[List["-", RowBox[List[FractionBox["1", SuperscriptBox["b", RowBox[List["3", "/", "2"]]]], RowBox[List["(", RowBox[List["4", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "a"]], "+", "b", "+", RowBox[List[SuperscriptBox[RowBox[List["Coth", "[", FractionBox[RowBox[List["c", " ", "z"]], "2"], "]"]], "2"], " ", RowBox[List["(", RowBox[List["b", "+", RowBox[List[FractionBox["1", SqrtBox["2"]], RowBox[List["(", RowBox[List[SqrtBox["b"], " ", RowBox[List["\[Sqrt]", 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["Sech", "[", FractionBox[RowBox[List["c", " ", "z"]], "2"], "]"]], "4"]]], ")"]]]]]], ")"]]]]]], ")"]]]]]], ")"]]]], ")"]]]]]], "]"]]]], "+", RowBox[List[SqrtBox["b"], " ", RowBox[List["Log", "[", RowBox[List[FractionBox["1", SqrtBox["b"]], RowBox[List["(", RowBox[List[RowBox[List["4", " ", "a"]], "+", RowBox[List["2", " ", "b"]], "+", RowBox[List[SqrtBox["2"], " ", SqrtBox["b"], " ", SqrtBox[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["Sech", "[", FractionBox[RowBox[List["c", " ", "z"]], "2"], "]"]], "4"]]]]]], "+", RowBox[List["2", " ", "b", " ", SuperscriptBox[RowBox[List["Tanh", "[", FractionBox[RowBox[List["c", " ", "z"]], "2"], "]"]], "2"]]]]], ")"]]]], "]"]]]], "-", RowBox[List["2", " ", SqrtBox[RowBox[List["a", "+", "b"]]], " ", RowBox[List["Log", "[", RowBox[List["-", RowBox[List[FractionBox["1", SuperscriptBox[RowBox[List["(", RowBox[List["a", "+", "b"]], ")"]], RowBox[List["3", "/", "2"]]]], RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["Cosh", "[", FractionBox[RowBox[List["c", " ", "z"]], "2"], "]"]], "2"], " ", RowBox[List["(", RowBox[List["a", "+", "b", "+", RowBox[List[FractionBox["1", SqrtBox["2"]], RowBox[List["(", RowBox[List[SqrtBox[RowBox[List["a", "+", "b"]]], " ", SqrtBox[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["Sech", "[", FractionBox[RowBox[List["c", " ", "z"]], "2"], "]"]], "4"]]]]]], ")"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List["a", "+", "b"]], ")"]], " ", SuperscriptBox[RowBox[List["Tanh", "[", FractionBox[RowBox[List["c", " ", "z"]], "2"], "]"]], "2"]]]]], ")"]]]], ")"]]]]]], "]"]]]]]], ")"]], " ", RowBox[List["Tanh", "[", FractionBox[RowBox[List["c", " ", "z"]], "2"], "]"]]]], ")"]]]], "/", RowBox[List["(", RowBox[List[SqrtBox["2"], " ", "c", " ", SqrtBox[FractionBox[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["(", RowBox[List["1", "+", RowBox[List["Cosh", "[", RowBox[List["c", " ", "z"]], "]"]]]], ")"]], "2"]]]]], ")"]]]]]]]]

 MathML Form

 a + b coth 2 ( c z ) z - ( b coth 2 ( c z ) + a ( b log ( - 4 ( ( b + b ( - a + b + ( a + b ) cosh ( 2 c z ) ) sech 4 ( c z 2 ) 2 ) coth 2 ( c z 2 ) + 2 a + b ) b 3 / 2 ) + b log ( 2 b tanh 2 ( c z 2 ) + 4 a + 2 b + 2 b ( - a + b + ( a + b ) cosh ( 2 c z ) ) sech 4 ( c z 2 ) b ) - 2 a + b log ( - cosh 2 ( c z 2 ) ( ( a + b ) tanh 2 ( c z 2 ) + a + b + a + b ( - a + b + ( a + b ) cosh ( 2 c z ) ) sech 4 ( c z 2 ) 2 ) ( a + b ) 3 / 2 ) ) tanh ( c z 2 ) ) / ( 2 c - a + b + ( a + b ) cosh ( 2 c z ) ( cosh ( c z ) + 1 ) 2 ) z a b c z 2 1 2 -1 b c z 2 a 1 2 b 1 2 -1 4 b b 1 2 -1 a b a b 2 c z c z 2 -1 4 1 2 2 1 2 -1 c z 2 -1 2 2 a b b 3 2 -1 b 1 2 2 b c z 2 -1 2 4 a 2 b 2 1 2 b 1 2 -1 a b a b 2 c z c z 2 -1 4 1 2 b 1 2 -1 -1 2 a b 1 2 -1 c z 2 -1 2 a b c z 2 -1 2 a b a b 1 2 -1 a b a b 2 c z c z 2 -1 4 1 2 2 1 2 -1 a b 3 2 -1 c z 2 -1 2 1 2 c -1 a b a b 2 c z c z 1 2 -1 1 2 -1 [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["\[Integral]", RowBox[List[SqrtBox[RowBox[List["a_", "+", RowBox[List["b_", " ", SuperscriptBox[RowBox[List["Coth", "[", RowBox[List["c_", " ", "z_"]], "]"]], "2"]]]]]], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List["-", FractionBox[RowBox[List[SqrtBox[RowBox[List["a", "+", RowBox[List["b", " ", SuperscriptBox[RowBox[List["Coth", "[", RowBox[List["c", " ", "z"]], "]"]], "2"]]]]]], " ", RowBox[List["(", RowBox[List[RowBox[List[SqrtBox["b"], " ", RowBox[List["Log", "[", RowBox[List["-", FractionBox[RowBox[List["4", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "a"]], "+", "b", "+", RowBox[List[SuperscriptBox[RowBox[List["Coth", "[", FractionBox[RowBox[List["c", " ", "z"]], "2"], "]"]], "2"], " ", RowBox[List["(", RowBox[List["b", "+", FractionBox[RowBox[List[SqrtBox["b"], " ", SqrtBox[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["Sech", "[", FractionBox[RowBox[List["c", " ", "z"]], "2"], "]"]], "4"]]]]]], SqrtBox["2"]]]], ")"]]]]]], ")"]]]], SuperscriptBox["b", RowBox[List["3", "/", "2"]]]]]], "]"]]]], "+", RowBox[List[SqrtBox["b"], " ", RowBox[List["Log", "[", FractionBox[RowBox[List[RowBox[List["4", " ", "a"]], "+", RowBox[List["2", " ", "b"]], "+", RowBox[List[SqrtBox["2"], " ", SqrtBox["b"], " ", SqrtBox[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["Sech", "[", FractionBox[RowBox[List["c", " ", "z"]], "2"], "]"]], "4"]]]]]], "+", RowBox[List["2", " ", "b", " ", SuperscriptBox[RowBox[List["Tanh", "[", FractionBox[RowBox[List["c", " ", "z"]], "2"], "]"]], "2"]]]]], SqrtBox["b"]], "]"]]]], "-", RowBox[List["2", " ", SqrtBox[RowBox[List["a", "+", "b"]]], " ", RowBox[List["Log", "[", RowBox[List["-", FractionBox[RowBox[List[SuperscriptBox[RowBox[List["Cosh", "[", FractionBox[RowBox[List["c", " ", "z"]], "2"], "]"]], "2"], " ", RowBox[List["(", RowBox[List["a", "+", "b", "+", FractionBox[RowBox[List[SqrtBox[RowBox[List["a", "+", "b"]]], " ", SqrtBox[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["Sech", "[", FractionBox[RowBox[List["c", " ", "z"]], "2"], "]"]], "4"]]]]]], SqrtBox["2"]], "+", RowBox[List[RowBox[List["(", RowBox[List["a", "+", "b"]], ")"]], " ", SuperscriptBox[RowBox[List["Tanh", "[", FractionBox[RowBox[List["c", " ", "z"]], "2"], "]"]], "2"]]]]], ")"]]]], SuperscriptBox[RowBox[List["(", RowBox[List["a", "+", "b"]], ")"]], RowBox[List["3", "/", "2"]]]]]], "]"]]]]]], ")"]], " ", RowBox[List["Tanh", "[", FractionBox[RowBox[List["c", " ", "z"]], "2"], "]"]]]], RowBox[List[SqrtBox["2"], " ", "c", " ", SqrtBox[FractionBox[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["(", RowBox[List["1", "+", RowBox[List["Cosh", "[", RowBox[List["c", " ", "z"]], "]"]]]], ")"]], "2"]]]]]]]]]]]]

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

 2002-12-18