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 Coth

 http://functions.wolfram.com/01.22.21.0397.01

 Input Form

 Integrate[Sinh[c z]/Sqrt[a + b Coth[c z]], z] == (Cosh[c z] ((-a) (a + b) Sqrt[1 - Coth[c z]] Sqrt[(b - b Coth[c z])/(a + b)] Sqrt[(a + b Coth[c z])/(a - b)] EllipticE[ ArcSin[Sqrt[(a + b Coth[c z])/(a - b)]], (a - b)/(a + b)] Sech[c z] (Cosh[c z] + Sinh[c z]) + b^2 Sqrt[1 + Coth[c z]] Sqrt[(b (1 + Coth[c z]))/(-a + b)] Sqrt[(a + b Coth[c z])/(a + b)] EllipticF[ArcSin[Sqrt[1 - Coth[c z]]/Sqrt[2]], (2 b)/(a + b)] (-1 + Tanh[c z]) + Sqrt[1 - Coth[c z]] (Sqrt[(b (1 + Coth[c z]))/(-a + b)] (a^2 - b^2 + a b Csch[c z] Sech[c z]) + a b Sqrt[(b - b Coth[c z])/(a + b)] Sqrt[(a + b Coth[c z])/(a - b)] EllipticF[ ArcSin[Sqrt[(a + b Coth[c z])/(a - b)]], (a - b)/(a + b)] (1 + Tanh[c z]))))/((a - b) (a + b) c Sqrt[1 - Coth[c z]] Sqrt[(b (1 + Coth[c z]))/(-a + b)] Sqrt[a + b Coth[c z]])

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List["\[Integral]", RowBox[List[FractionBox[RowBox[List["Sinh", "[", RowBox[List["c", " ", "z"]], "]"]], RowBox[List[" ", SqrtBox[RowBox[List["a", "+", RowBox[List["b", " ", RowBox[List["Coth", "[", RowBox[List["c", " ", "z"]], "]"]]]]]]]]]], RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Cosh", "[", RowBox[List["c", " ", "z"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "a"]], " ", RowBox[List["(", RowBox[List["a", "+", "b"]], ")"]], " ", SqrtBox[RowBox[List["1", "-", RowBox[List["Coth", "[", RowBox[List["c", " ", "z"]], "]"]]]]], " ", SqrtBox[FractionBox[RowBox[List["b", "-", RowBox[List["b", " ", RowBox[List["Coth", "[", RowBox[List["c", " ", "z"]], "]"]]]]]], RowBox[List["a", "+", "b"]]]], " ", SqrtBox[FractionBox[RowBox[List["a", "+", RowBox[List["b", " ", RowBox[List["Coth", "[", RowBox[List["c", " ", "z"]], "]"]]]]]], RowBox[List["a", "-", "b"]]]], " ", RowBox[List["EllipticE", "[", RowBox[List[RowBox[List["ArcSin", "[", SqrtBox[FractionBox[RowBox[List["a", "+", RowBox[List["b", " ", RowBox[List["Coth", "[", RowBox[List["c", " ", "z"]], "]"]]]]]], RowBox[List["a", "-", "b"]]]], "]"]], ",", FractionBox[RowBox[List["a", "-", "b"]], RowBox[List["a", "+", "b"]]]]], "]"]], " ", RowBox[List["Sech", "[", RowBox[List["c", " ", "z"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["Cosh", "[", RowBox[List["c", " ", "z"]], "]"]], "+", RowBox[List["Sinh", "[", RowBox[List["c", " ", "z"]], "]"]]]], ")"]]]], "+", RowBox[List[SuperscriptBox["b", "2"], " ", SqrtBox[RowBox[List["1", "+", RowBox[List["Coth", "[", RowBox[List["c", " ", "z"]], "]"]]]]], " ", SqrtBox[FractionBox[RowBox[List["b", " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List["Coth", "[", RowBox[List["c", " ", "z"]], "]"]]]], ")"]]]], RowBox[List[RowBox[List["-", "a"]], "+", "b"]]]], " ", SqrtBox[FractionBox[RowBox[List["a", "+", RowBox[List["b", " ", RowBox[List["Coth", "[", RowBox[List["c", " ", "z"]], "]"]]]]]], RowBox[List["a", "+", "b"]]]], " ", RowBox[List["EllipticF", "[", RowBox[List[RowBox[List["ArcSin", "[", FractionBox[SqrtBox[RowBox[List["1", "-", RowBox[List["Coth", "[", RowBox[List["c", " ", "z"]], "]"]]]]], SqrtBox["2"]], "]"]], ",", FractionBox[RowBox[List["2", " ", "b"]], RowBox[List["a", "+", "b"]]]]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", RowBox[List["Tanh", "[", RowBox[List["c", " ", "z"]], "]"]]]], ")"]]]], "+", RowBox[List[SqrtBox[RowBox[List["1", "-", RowBox[List["Coth", "[", RowBox[List["c", " ", "z"]], "]"]]]]], " ", RowBox[List["(", RowBox[List[RowBox[List[SqrtBox[FractionBox[RowBox[List["b", " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List["Coth", "[", RowBox[List["c", " ", "z"]], "]"]]]], ")"]]]], RowBox[List[RowBox[List["-", "a"]], "+", "b"]]]], " ", RowBox[List["(", RowBox[List[SuperscriptBox["a", "2"], "-", SuperscriptBox["b", "2"], "+", RowBox[List["a", " ", "b", " ", RowBox[List["Csch", "[", RowBox[List["c", " ", "z"]], "]"]], " ", RowBox[List["Sech", "[", RowBox[List["c", " ", "z"]], "]"]]]]]], ")"]]]], "+", RowBox[List["a", " ", "b", " ", SqrtBox[FractionBox[RowBox[List["b", "-", RowBox[List["b", " ", RowBox[List["Coth", "[", RowBox[List["c", " ", "z"]], "]"]]]]]], RowBox[List["a", "+", "b"]]]], " ", SqrtBox[FractionBox[RowBox[List["a", "+", RowBox[List["b", " ", RowBox[List["Coth", "[", RowBox[List["c", " ", "z"]], "]"]]]]]], RowBox[List["a", "-", "b"]]]], " ", RowBox[List["EllipticF", "[", RowBox[List[RowBox[List["ArcSin", "[", SqrtBox[FractionBox[RowBox[List["a", "+", RowBox[List["b", " ", RowBox[List["Coth", "[", RowBox[List["c", " ", "z"]], "]"]]]]]], RowBox[List["a", "-", "b"]]]], "]"]], ",", FractionBox[RowBox[List["a", "-", "b"]], RowBox[List["a", "+", "b"]]]]], "]"]], " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List["Tanh", "[", RowBox[List["c", " ", "z"]], "]"]]]], ")"]]]]]], ")"]]]]]], ")"]]]], ")"]], "/", RowBox[List["(", RowBox[List[RowBox[List["(", RowBox[List["a", "-", "b"]], ")"]], " ", RowBox[List["(", RowBox[List["a", "+", "b"]], ")"]], " ", "c", " ", SqrtBox[RowBox[List["1", "-", RowBox[List["Coth", "[", RowBox[List["c", " ", "z"]], "]"]]]]], " ", SqrtBox[FractionBox[RowBox[List["b", " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List["Coth", "[", RowBox[List["c", " ", "z"]], "]"]]]], ")"]]]], RowBox[List[RowBox[List["-", "a"]], "+", "b"]]]], " ", SqrtBox[RowBox[List["a", "+", RowBox[List["b", " ", RowBox[List["Coth", "[", RowBox[List["c", " ", "z"]], "]"]]]]]]]]], ")"]]]]]]]]

 MathML Form

 sinh ( c z ) a + b coth ( c z ) z ( cosh ( c z ) ( b ( coth ( c z ) + 1 ) b - a a + b coth ( c z ) a + b coth ( c z ) + 1 F ( sin - 1 ( 1 - coth ( c z ) 2 ) 2 b a + b ) ( tanh ( c z ) - 1 ) b 2 - a a + b coth ( c z ) a - b b - b coth ( c z ) a + b ( a + b ) 1 - coth ( c z ) E ( sin - 1 ( a + b coth ( c z ) a - b ) a - b a + b ) sech ( c z ) ( cosh ( c z ) + sinh ( c z ) ) + 1 - coth ( c z ) ( b ( coth ( c z ) + 1 ) b - a ( a 2 + b csch ( c z ) sech ( c z ) a - b 2 ) + a a + b coth ( c z ) a - b b b - b coth ( c z ) a + b F ( sin - 1 ( a + b coth ( c z ) a - b ) a - b a + b ) ( tanh ( c z ) + 1 ) ) ) ) / ( ( a - b ) ( a + b ) c 1 - coth ( c z ) b ( coth ( c z ) + 1 ) b - a a + b coth ( c z ) ) z c z a b c z 1 2 -1 c z b c z 1 b -1 a -1 1 2 a b c z a b -1 1 2 c z 1 1 2 EllipticF 1 -1 c z 1 2 2 1 2 -1 2 b a b -1 c z -1 b 2 -1 a a b c z a -1 b -1 1 2 b -1 b c z a b -1 1 2 a b 1 -1 c z 1 2 EllipticE a b c z a -1 b -1 1 2 a -1 b a b -1 c z c z c z 1 -1 c z 1 2 b c z 1 b -1 a -1 1 2 a 2 b c z c z a -1 b 2 a a b c z a -1 b -1 1 2 b b -1 b c z a b -1 1 2 EllipticF a b c z a -1 b -1 1 2 a -1 b a b -1 c z 1 a -1 b a b c 1 -1 c z 1 2 b c z 1 b -1 a -1 1 2 a b c z 1 2 -1 [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["\[Integral]", RowBox[List[FractionBox[RowBox[List["Sinh", "[", RowBox[List["c_", " ", "z_"]], "]"]], SqrtBox[RowBox[List["a_", "+", RowBox[List["b_", " ", RowBox[List["Coth", "[", RowBox[List["c_", " ", "z_"]], "]"]]]]]]]], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", FractionBox[RowBox[List[RowBox[List["Cosh", "[", RowBox[List["c", " ", "z"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "a"]], " ", RowBox[List["(", RowBox[List["a", "+", "b"]], ")"]], " ", SqrtBox[RowBox[List["1", "-", RowBox[List["Coth", "[", RowBox[List["c", " ", "z"]], "]"]]]]], " ", SqrtBox[FractionBox[RowBox[List["b", "-", RowBox[List["b", " ", RowBox[List["Coth", "[", RowBox[List["c", " ", "z"]], "]"]]]]]], RowBox[List["a", "+", "b"]]]], " ", SqrtBox[FractionBox[RowBox[List["a", "+", RowBox[List["b", " ", RowBox[List["Coth", "[", RowBox[List["c", " ", "z"]], "]"]]]]]], RowBox[List["a", "-", "b"]]]], " ", RowBox[List["EllipticE", "[", RowBox[List[RowBox[List["ArcSin", "[", SqrtBox[FractionBox[RowBox[List["a", "+", RowBox[List["b", " ", RowBox[List["Coth", "[", RowBox[List["c", " ", "z"]], "]"]]]]]], RowBox[List["a", "-", "b"]]]], "]"]], ",", FractionBox[RowBox[List["a", "-", "b"]], RowBox[List["a", "+", "b"]]]]], "]"]], " ", RowBox[List["Sech", "[", RowBox[List["c", " ", "z"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["Cosh", "[", RowBox[List["c", " ", "z"]], "]"]], "+", RowBox[List["Sinh", "[", RowBox[List["c", " ", "z"]], "]"]]]], ")"]]]], "+", RowBox[List[SuperscriptBox["b", "2"], " ", SqrtBox[RowBox[List["1", "+", RowBox[List["Coth", "[", RowBox[List["c", " ", "z"]], "]"]]]]], " ", SqrtBox[FractionBox[RowBox[List["b", " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List["Coth", "[", RowBox[List["c", " ", "z"]], "]"]]]], ")"]]]], RowBox[List[RowBox[List["-", "a"]], "+", "b"]]]], " ", SqrtBox[FractionBox[RowBox[List["a", "+", RowBox[List["b", " ", RowBox[List["Coth", "[", RowBox[List["c", " ", "z"]], "]"]]]]]], RowBox[List["a", "+", "b"]]]], " ", RowBox[List["EllipticF", "[", RowBox[List[RowBox[List["ArcSin", "[", FractionBox[SqrtBox[RowBox[List["1", "-", RowBox[List["Coth", "[", RowBox[List["c", " ", "z"]], "]"]]]]], SqrtBox["2"]], "]"]], ",", FractionBox[RowBox[List["2", " ", "b"]], RowBox[List["a", "+", "b"]]]]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", RowBox[List["Tanh", "[", RowBox[List["c", " ", "z"]], "]"]]]], ")"]]]], "+", RowBox[List[SqrtBox[RowBox[List["1", "-", RowBox[List["Coth", "[", RowBox[List["c", " ", "z"]], "]"]]]]], " ", RowBox[List["(", RowBox[List[RowBox[List[SqrtBox[FractionBox[RowBox[List["b", " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List["Coth", "[", RowBox[List["c", " ", "z"]], "]"]]]], ")"]]]], RowBox[List[RowBox[List["-", "a"]], "+", "b"]]]], " ", RowBox[List["(", RowBox[List[SuperscriptBox["a", "2"], "-", SuperscriptBox["b", "2"], "+", RowBox[List["a", " ", "b", " ", RowBox[List["Csch", "[", RowBox[List["c", " ", "z"]], "]"]], " ", RowBox[List["Sech", "[", RowBox[List["c", " ", "z"]], "]"]]]]]], ")"]]]], "+", RowBox[List["a", " ", "b", " ", SqrtBox[FractionBox[RowBox[List["b", "-", RowBox[List["b", " ", RowBox[List["Coth", "[", RowBox[List["c", " ", "z"]], "]"]]]]]], RowBox[List["a", "+", "b"]]]], " ", SqrtBox[FractionBox[RowBox[List["a", "+", RowBox[List["b", " ", RowBox[List["Coth", "[", RowBox[List["c", " ", "z"]], "]"]]]]]], RowBox[List["a", "-", "b"]]]], " ", RowBox[List["EllipticF", "[", RowBox[List[RowBox[List["ArcSin", "[", SqrtBox[FractionBox[RowBox[List["a", "+", RowBox[List["b", " ", RowBox[List["Coth", "[", RowBox[List["c", " ", "z"]], "]"]]]]]], RowBox[List["a", "-", "b"]]]], "]"]], ",", FractionBox[RowBox[List["a", "-", "b"]], RowBox[List["a", "+", "b"]]]]], "]"]], " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List["Tanh", "[", RowBox[List["c", " ", "z"]], "]"]]]], ")"]]]]]], ")"]]]]]], ")"]]]], RowBox[List[RowBox[List["(", RowBox[List["a", "-", "b"]], ")"]], " ", RowBox[List["(", RowBox[List["a", "+", "b"]], ")"]], " ", "c", " ", SqrtBox[RowBox[List["1", "-", RowBox[List["Coth", "[", RowBox[List["c", " ", "z"]], "]"]]]]], " ", SqrtBox[FractionBox[RowBox[List["b", " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List["Coth", "[", RowBox[List["c", " ", "z"]], "]"]]]], ")"]]]], RowBox[List[RowBox[List["-", "a"]], "+", "b"]]]], " ", SqrtBox[RowBox[List["a", "+", RowBox[List["b", " ", RowBox[List["Coth", "[", RowBox[List["c", " ", "z"]], "]"]]]]]]]]]]]]]]

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

 2002-12-18