html, body, form { margin: 0; padding: 0; width: 100%; } #calculate { position: relative; width: 177px; height: 110px; background: transparent url(/images/alphabox/embed_functions_inside.gif) no-repeat scroll 0 0; } #i { position: relative; left: 18px; top: 44px; width: 133px; border: 0 none; outline: 0; font-size: 11px; } #eq { width: 9px; height: 10px; background: transparent; position: absolute; top: 47px; right: 18px; cursor: pointer; }

 Coth

 http://functions.wolfram.com/01.22.21.0400.01

 Input Form

 Integrate[Sinh[c z]/Sqrt[a + b Coth[c z]^2], z] == ((1 + Cosh[c z]) Sqrt[(-a + b + (a + b) Cosh[2 c z])/(1 + Cosh[c z])^2] Csch[c z] (Sqrt[(2 a + b - 2 Sqrt[a] Sqrt[a + b])/b] (-a + b + (a + b) Cosh[2 c z]) Sech[(c z)/2]^2 Tanh[(c z)/2] - 2 I (2 a + b - 2 Sqrt[a] Sqrt[a + b]) EllipticE[I ArcSinh[Sqrt[(2 a + b - 2 Sqrt[a] Sqrt[a + b])/b] Tanh[(c z)/2]], (2 a + b + 2 Sqrt[a] Sqrt[a + b])/ (2 a + b - 2 Sqrt[a] Sqrt[a + b])] Sqrt[(b + (2 a + b - 2 Sqrt[a] Sqrt[a + b]) Tanh[(c z)/2]^2)/b] Sqrt[(b + (2 a + b + 2 Sqrt[a] Sqrt[a + b]) Tanh[(c z)/2]^2)/b] + 4 I (a + b - Sqrt[a] Sqrt[a + b]) EllipticF[ I ArcSinh[Sqrt[(2 a + b - 2 Sqrt[a] Sqrt[a + b])/b] Tanh[(c z)/2]], (2 a + b + 2 Sqrt[a] Sqrt[a + b])/(2 a + b - 2 Sqrt[a] Sqrt[a + b])] Sqrt[(b + (2 a + b - 2 Sqrt[a] Sqrt[a + b]) Tanh[(c z)/2]^2)/b] Sqrt[(b + (2 a + b + 2 Sqrt[a] Sqrt[a + b]) Tanh[(c z)/2]^2)/b]))/ (2 Sqrt[2] (a + b) Sqrt[(2 a + b - 2 Sqrt[a] Sqrt[a + b])/b] c Sqrt[a + b Coth[c z]^2] Sqrt[4 a Tanh[(c z)/2]^2 + b (1 + Tanh[(c z)/2]^2)^2])

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List["\[Integral]", RowBox[List[FractionBox[RowBox[List["Sinh", "[", RowBox[List["c", " ", "z"]], "]"]], 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["(", RowBox[List["1", "+", RowBox[List["Cosh", "[", RowBox[List["c", " ", "z"]], "]"]]]], ")"]], " ", 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"]]], " ", RowBox[List["Csch", "[", RowBox[List["c", " ", "z"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List[SqrtBox[FractionBox[RowBox[List[RowBox[List["2", " ", "a"]], "+", "b", "-", RowBox[List["2", " ", SqrtBox["a"], " ", SqrtBox[RowBox[List["a", "+", "b"]]]]]]], "b"]], " ", 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"], "]"]], "2"], " ", RowBox[List["Tanh", "[", FractionBox[RowBox[List["c", " ", "z"]], "2"], "]"]]]], "-", RowBox[List["2", " ", "\[ImaginaryI]", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "a"]], "+", "b", "-", RowBox[List["2", " ", SqrtBox["a"], " ", SqrtBox[RowBox[List["a", "+", "b"]]]]]]], ")"]], " ", RowBox[List["EllipticE", "[", RowBox[List[RowBox[List["\[ImaginaryI]", " ", RowBox[List["ArcSinh", "[", RowBox[List[SqrtBox[FractionBox[RowBox[List[RowBox[List["2", " ", "a"]], "+", "b", "-", RowBox[List["2", " ", SqrtBox["a"], " ", SqrtBox[RowBox[List["a", "+", "b"]]]]]]], "b"]], " ", RowBox[List["Tanh", "[", FractionBox[RowBox[List["c", " ", "z"]], "2"], "]"]]]], "]"]]]], ",", FractionBox[RowBox[List[RowBox[List["2", " ", "a"]], "+", "b", "+", RowBox[List["2", " ", SqrtBox["a"], " ", SqrtBox[RowBox[List["a", "+", "b"]]]]]]], RowBox[List[RowBox[List["2", " ", "a"]], "+", "b", "-", RowBox[List["2", " ", SqrtBox["a"], " ", SqrtBox[RowBox[List["a", "+", "b"]]]]]]]]]], "]"]], " ", SqrtBox[FractionBox[RowBox[List["b", "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["2", " ", "a"]], "+", "b", "-", RowBox[List["2", " ", SqrtBox["a"], " ", SqrtBox[RowBox[List["a", "+", "b"]]]]]]], ")"]], " ", SuperscriptBox[RowBox[List["Tanh", "[", FractionBox[RowBox[List["c", " ", "z"]], "2"], "]"]], "2"]]]]], "b"]], " ", SqrtBox[FractionBox[RowBox[List["b", "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["2", " ", "a"]], "+", "b", "+", RowBox[List["2", " ", SqrtBox["a"], " ", SqrtBox[RowBox[List["a", "+", "b"]]]]]]], ")"]], " ", SuperscriptBox[RowBox[List["Tanh", "[", FractionBox[RowBox[List["c", " ", "z"]], "2"], "]"]], "2"]]]]], "b"]]]], "+", RowBox[List["4", " ", "\[ImaginaryI]", " ", RowBox[List["(", RowBox[List["a", "+", "b", "-", RowBox[List[SqrtBox["a"], " ", SqrtBox[RowBox[List["a", "+", "b"]]]]]]], ")"]], " ", RowBox[List["EllipticF", "[", RowBox[List[RowBox[List["\[ImaginaryI]", " ", RowBox[List["ArcSinh", "[", RowBox[List[SqrtBox[FractionBox[RowBox[List[RowBox[List["2", " ", "a"]], "+", "b", "-", RowBox[List["2", " ", SqrtBox["a"], " ", SqrtBox[RowBox[List["a", "+", "b"]]]]]]], "b"]], " ", RowBox[List["Tanh", "[", FractionBox[RowBox[List["c", " ", "z"]], "2"], "]"]]]], "]"]]]], ",", FractionBox[RowBox[List[RowBox[List["2", " ", "a"]], "+", "b", "+", RowBox[List["2", " ", SqrtBox["a"], " ", SqrtBox[RowBox[List["a", "+", "b"]]]]]]], RowBox[List[RowBox[List["2", " ", "a"]], "+", "b", "-", RowBox[List["2", " ", SqrtBox["a"], " ", SqrtBox[RowBox[List["a", "+", "b"]]]]]]]]]], "]"]], " ", SqrtBox[FractionBox[RowBox[List["b", "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["2", " ", "a"]], "+", "b", "-", RowBox[List["2", " ", SqrtBox["a"], " ", SqrtBox[RowBox[List["a", "+", "b"]]]]]]], ")"]], " ", SuperscriptBox[RowBox[List["Tanh", "[", FractionBox[RowBox[List["c", " ", "z"]], "2"], "]"]], "2"]]]]], "b"]], " ", SqrtBox[FractionBox[RowBox[List["b", "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["2", " ", "a"]], "+", "b", "+", RowBox[List["2", " ", SqrtBox["a"], " ", SqrtBox[RowBox[List["a", "+", "b"]]]]]]], ")"]], " ", SuperscriptBox[RowBox[List["Tanh", "[", FractionBox[RowBox[List["c", " ", "z"]], "2"], "]"]], "2"]]]]], "b"]]]]]], ")"]]]], ")"]], "/", RowBox[List["(", RowBox[List["2", " ", SqrtBox["2"], " ", RowBox[List["(", RowBox[List["a", "+", "b"]], ")"]], " ", SqrtBox[FractionBox[RowBox[List[RowBox[List["2", " ", "a"]], "+", "b", "-", RowBox[List["2", " ", SqrtBox["a"], " ", SqrtBox[RowBox[List["a", "+", "b"]]]]]]], "b"]], " ", "c", " ", SqrtBox[RowBox[List["a", "+", RowBox[List["b", " ", SuperscriptBox[RowBox[List["Coth", "[", RowBox[List["c", " ", "z"]], "]"]], "2"]]]]]], " ", SqrtBox[RowBox[List[RowBox[List["4", " ", "a", " ", SuperscriptBox[RowBox[List["Tanh", "[", FractionBox[RowBox[List["c", " ", "z"]], "2"], "]"]], "2"]]], "+", RowBox[List["b", " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "+", SuperscriptBox[RowBox[List["Tanh", "[", FractionBox[RowBox[List["c", " ", "z"]], "2"], "]"]], "2"]]], ")"]], "2"]]]]]]]], ")"]]]]]]]]

 MathML Form

 sinh ( c z ) a + b coth 2 ( c z ) z ( ( cosh ( c z ) + 1 ) - a + b + ( a + b ) cosh ( 2 c z ) ( cosh ( c z ) + 1 ) 2 csch ( c z ) ( 2 a - 2 a + b a + b b ( - a + b + ( a + b ) cosh ( 2 c z ) ) tanh ( c z 2 ) sech 2 ( c z 2 ) + 4 ( 2 a + 2 a + b a + b ) tanh 2 ( c z 2 ) + b b ( 2 a - 2 a + b a + b ) tanh 2 ( c z 2 ) + b b ( a - a + b a + b ) F ( sinh - 1 ( 2 a - 2 a + b a + b b tanh ( c z 2 ) ) 2 a + 2 a + b a + b 2 a - 2 a + b a + b ) - 2 ( 2 a - 2 a + b a + b ) E ( sinh - 1 ( 2 a - 2 a + b a + b b tanh ( c z 2 ) ) 2 a + 2 a + b a + b 2 a - 2 a + b a + b ) ( 2 a - 2 a + b a + b ) tanh 2 ( c z 2 ) + b b ( 2 a + 2 a + b a + b ) tanh 2 ( c z 2 ) + b b ) ) / ( 2 2 ( a + b ) 2 a - 2 a + b a + b b c b coth 2 ( c z ) + a 4 a tanh 2 ( c z 2 ) + b ( tanh 2 ( c z 2 ) + 1 ) 2 ) z c z a b c z 2 1 2 -1 c z 1 -1 a b a b 2 c z c z 1 2 -1 1 2 c z 2 a -1 2 a b 1 2 a 1 2 b b -1 1 2 -1 a b a b 2 c z c z 2 -1 c z 2 -1 2 4 2 a 2 a b 1 2 a 1 2 b c z 2 -1 2 b b -1 1 2 2 a -1 2 a b 1 2 a 1 2 b c z 2 -1 2 b b -1 1 2 a -1 a b 1 2 a 1 2 b EllipticF 2 a -1 2 a b 1 2 a 1 2 b b -1 1 2 c z 2 -1 2 a 2 a b 1 2 a 1 2 b 2 a -1 2 a b 1 2 a 1 2 b -1 -1 2 2 a -1 2 a b 1 2 a 1 2 b EllipticE 2 a -1 2 a b 1 2 a 1 2 b b -1 1 2 c z 2 -1 2 a 2 a b 1 2 a 1 2 b 2 a -1 2 a b 1 2 a 1 2 b -1 2 a -1 2 a b 1 2 a 1 2 b c z 2 -1 2 b b -1 1 2 2 a 2 a b 1 2 a 1 2 b c z 2 -1 2 b b -1 1 2 2 2 1 2 a b 2 a -1 2 a b 1 2 a 1 2 b b -1 1 2 c b c z 2 a 1 2 4 a c z 2 -1 2 b c z 2 -1 2 1 2 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_", " ", SuperscriptBox[RowBox[List["Coth", "[", RowBox[List["c_", " ", "z_"]], "]"]], "2"]]]]]]], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["1", "+", RowBox[List["Cosh", "[", RowBox[List["c", " ", "z"]], "]"]]]], ")"]], " ", 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"]]], " ", RowBox[List["Csch", "[", RowBox[List["c", " ", "z"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List[SqrtBox[FractionBox[RowBox[List[RowBox[List["2", " ", "a"]], "+", "b", "-", RowBox[List["2", " ", SqrtBox["a"], " ", SqrtBox[RowBox[List["a", "+", "b"]]]]]]], "b"]], " ", 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"], "]"]], "2"], " ", RowBox[List["Tanh", "[", FractionBox[RowBox[List["c", " ", "z"]], "2"], "]"]]]], "-", RowBox[List["2", " ", "\[ImaginaryI]", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "a"]], "+", "b", "-", RowBox[List["2", " ", SqrtBox["a"], " ", SqrtBox[RowBox[List["a", "+", "b"]]]]]]], ")"]], " ", RowBox[List["EllipticE", "[", RowBox[List[RowBox[List["\[ImaginaryI]", " ", RowBox[List["ArcSinh", "[", RowBox[List[SqrtBox[FractionBox[RowBox[List[RowBox[List["2", " ", "a"]], "+", "b", "-", RowBox[List["2", " ", SqrtBox["a"], " ", SqrtBox[RowBox[List["a", "+", "b"]]]]]]], "b"]], " ", RowBox[List["Tanh", "[", FractionBox[RowBox[List["c", " ", "z"]], "2"], "]"]]]], "]"]]]], ",", FractionBox[RowBox[List[RowBox[List["2", " ", "a"]], "+", "b", "+", RowBox[List["2", " ", SqrtBox["a"], " ", SqrtBox[RowBox[List["a", "+", "b"]]]]]]], RowBox[List[RowBox[List["2", " ", "a"]], "+", "b", "-", RowBox[List["2", " ", SqrtBox["a"], " ", SqrtBox[RowBox[List["a", "+", "b"]]]]]]]]]], "]"]], " ", SqrtBox[FractionBox[RowBox[List["b", "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["2", " ", "a"]], "+", "b", "-", RowBox[List["2", " ", SqrtBox["a"], " ", SqrtBox[RowBox[List["a", "+", "b"]]]]]]], ")"]], " ", SuperscriptBox[RowBox[List["Tanh", "[", FractionBox[RowBox[List["c", " ", "z"]], "2"], "]"]], "2"]]]]], "b"]], " ", SqrtBox[FractionBox[RowBox[List["b", "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["2", " ", "a"]], "+", "b", "+", RowBox[List["2", " ", SqrtBox["a"], " ", SqrtBox[RowBox[List["a", "+", "b"]]]]]]], ")"]], " ", SuperscriptBox[RowBox[List["Tanh", "[", FractionBox[RowBox[List["c", " ", "z"]], "2"], "]"]], "2"]]]]], "b"]]]], "+", RowBox[List["4", " ", "\[ImaginaryI]", " ", RowBox[List["(", RowBox[List["a", "+", "b", "-", RowBox[List[SqrtBox["a"], " ", SqrtBox[RowBox[List["a", "+", "b"]]]]]]], ")"]], " ", RowBox[List["EllipticF", "[", RowBox[List[RowBox[List["\[ImaginaryI]", " ", RowBox[List["ArcSinh", "[", RowBox[List[SqrtBox[FractionBox[RowBox[List[RowBox[List["2", " ", "a"]], "+", "b", "-", RowBox[List["2", " ", SqrtBox["a"], " ", SqrtBox[RowBox[List["a", "+", "b"]]]]]]], "b"]], " ", RowBox[List["Tanh", "[", FractionBox[RowBox[List["c", " ", "z"]], "2"], "]"]]]], "]"]]]], ",", FractionBox[RowBox[List[RowBox[List["2", " ", "a"]], "+", "b", "+", RowBox[List["2", " ", SqrtBox["a"], " ", SqrtBox[RowBox[List["a", "+", "b"]]]]]]], RowBox[List[RowBox[List["2", " ", "a"]], "+", "b", "-", RowBox[List["2", " ", SqrtBox["a"], " ", SqrtBox[RowBox[List["a", "+", "b"]]]]]]]]]], "]"]], " ", SqrtBox[FractionBox[RowBox[List["b", "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["2", " ", "a"]], "+", "b", "-", RowBox[List["2", " ", SqrtBox["a"], " ", SqrtBox[RowBox[List["a", "+", "b"]]]]]]], ")"]], " ", SuperscriptBox[RowBox[List["Tanh", "[", FractionBox[RowBox[List["c", " ", "z"]], "2"], "]"]], "2"]]]]], "b"]], " ", SqrtBox[FractionBox[RowBox[List["b", "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["2", " ", "a"]], "+", "b", "+", RowBox[List["2", " ", SqrtBox["a"], " ", SqrtBox[RowBox[List["a", "+", "b"]]]]]]], ")"]], " ", SuperscriptBox[RowBox[List["Tanh", "[", FractionBox[RowBox[List["c", " ", "z"]], "2"], "]"]], "2"]]]]], "b"]]]]]], ")"]]]], RowBox[List["2", " ", SqrtBox["2"], " ", RowBox[List["(", RowBox[List["a", "+", "b"]], ")"]], " ", SqrtBox[FractionBox[RowBox[List[RowBox[List["2", " ", "a"]], "+", "b", "-", RowBox[List["2", " ", SqrtBox["a"], " ", SqrtBox[RowBox[List["a", "+", "b"]]]]]]], "b"]], " ", "c", " ", SqrtBox[RowBox[List["a", "+", RowBox[List["b", " ", SuperscriptBox[RowBox[List["Coth", "[", RowBox[List["c", " ", "z"]], "]"]], "2"]]]]]], " ", SqrtBox[RowBox[List[RowBox[List["4", " ", "a", " ", SuperscriptBox[RowBox[List["Tanh", "[", FractionBox[RowBox[List["c", " ", "z"]], "2"], "]"]], "2"]]], "+", RowBox[List["b", " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "+", SuperscriptBox[RowBox[List["Tanh", "[", FractionBox[RowBox[List["c", " ", "z"]], "2"], "]"]], "2"]]], ")"]], "2"]]]]]]]]]]]]]

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

 2002-12-18