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/01.22.21.0288.01

 Input Form

 Integrate[Coth[c z]/(a + b Sqrt[Coth[c z]])^2, z] == (-(1/(4 c))) (-((2 I ArcTan[Sqrt[Coth[c z]]])/(a - I b)^2) + (2 I ArcTan[Sqrt[Coth[c z]]])/(a + I b)^2 + (8 a^3)/((a^4 - b^4) (a + b Sqrt[Coth[c z]])) + (2 Log[-1 + Sqrt[Coth[c z]]])/(a + b)^2 + (2 Log[1 + Sqrt[Coth[c z]]])/ (a - b)^2 - (8 (a^6 + 3 a^2 b^4) Log[a + b Sqrt[Coth[c z]]])/ (a^4 - b^4)^2 + Log[1 + Coth[c z]]/(a - I b)^2 + Log[1 + Coth[c z]]/(a + I b)^2)

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List["\[Integral]", RowBox[List[FractionBox[RowBox[List["Coth", "[", RowBox[List["c", " ", "z"]], "]"]], SuperscriptBox[RowBox[List["(", RowBox[List["a", "+", RowBox[List["b", SqrtBox[RowBox[List["Coth", "[", RowBox[List["c", " ", "z"]], "]"]]]]]]], ")"]], "2"]], RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", RowBox[List[RowBox[List["-", FractionBox["1", RowBox[List["4", " ", "c"]]]]], RowBox[List["(", RowBox[List[RowBox[List["-", FractionBox[RowBox[List["2", " ", "\[ImaginaryI]", " ", RowBox[List["ArcTan", "[", SqrtBox[RowBox[List["Coth", "[", RowBox[List["c", " ", "z"]], "]"]]], "]"]]]], SuperscriptBox[RowBox[List["(", RowBox[List["a", "-", RowBox[List["\[ImaginaryI]", " ", "b"]]]], ")"]], "2"]]]], "+", FractionBox[RowBox[List["2", " ", "\[ImaginaryI]", " ", RowBox[List["ArcTan", "[", SqrtBox[RowBox[List["Coth", "[", RowBox[List["c", " ", "z"]], "]"]]], "]"]]]], SuperscriptBox[RowBox[List["(", RowBox[List["a", "+", RowBox[List["\[ImaginaryI]", " ", "b"]]]], ")"]], "2"]], "+", FractionBox[RowBox[List["8", " ", SuperscriptBox["a", "3"]]], RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["a", "4"], "-", SuperscriptBox["b", "4"]]], ")"]], " ", RowBox[List["(", RowBox[List["a", "+", RowBox[List["b", " ", SqrtBox[RowBox[List["Coth", "[", RowBox[List["c", " ", "z"]], "]"]]]]]]], ")"]]]]], "+", FractionBox[RowBox[List["2", " ", RowBox[List["Log", "[", RowBox[List[RowBox[List["-", "1"]], "+", SqrtBox[RowBox[List["Coth", "[", RowBox[List["c", " ", "z"]], "]"]]]]], "]"]]]], SuperscriptBox[RowBox[List["(", RowBox[List["a", "+", "b"]], ")"]], "2"]], "+", FractionBox[RowBox[List["2", " ", RowBox[List["Log", "[", RowBox[List["1", "+", SqrtBox[RowBox[List["Coth", "[", RowBox[List["c", " ", "z"]], "]"]]]]], "]"]]]], SuperscriptBox[RowBox[List["(", RowBox[List["a", "-", "b"]], ")"]], "2"]], "-", FractionBox[RowBox[List["8", " ", RowBox[List["(", RowBox[List[SuperscriptBox["a", "6"], "+", RowBox[List["3", " ", SuperscriptBox["a", "2"], " ", SuperscriptBox["b", "4"]]]]], ")"]], " ", RowBox[List["Log", "[", RowBox[List["a", "+", RowBox[List["b", " ", SqrtBox[RowBox[List["Coth", "[", RowBox[List["c", " ", "z"]], "]"]]]]]]], "]"]]]], SuperscriptBox[RowBox[List["(", RowBox[List[SuperscriptBox["a", "4"], "-", SuperscriptBox["b", "4"]]], ")"]], "2"]], "+", FractionBox[RowBox[List["Log", "[", RowBox[List["1", "+", RowBox[List["Coth", "[", RowBox[List["c", " ", "z"]], "]"]]]], "]"]], SuperscriptBox[RowBox[List["(", RowBox[List["a", "-", RowBox[List["\[ImaginaryI]", " ", "b"]]]], ")"]], "2"]], "+", FractionBox[RowBox[List["Log", "[", RowBox[List["1", "+", RowBox[List["Coth", "[", RowBox[List["c", " ", "z"]], "]"]]]], "]"]], SuperscriptBox[RowBox[List["(", RowBox[List["a", "+", RowBox[List["\[ImaginaryI]", " ", "b"]]]], ")"]], "2"]]]], ")"]]]]]]]]

 MathML Form

 coth ( c z ) ( a + b coth 1 2 ( c z ) ) 2 z - 1 4 c ( 8 a 3 ( a 4 - b 4 ) ( a + b coth 1 2 ( c z ) ) + 2 tan - 1 ( coth 1 2 ( c z ) ) ( a + b ) 2 + 2 log ( coth 1 2 ( c z ) - 1 ) ( a + b ) 2 + 2 log ( coth 1 2 ( c z ) + 1 ) ( a - b ) 2 + log ( coth ( c z ) + 1 ) ( a + b ) 2 + log ( coth ( c z ) + 1 ) ( a - b ) 2 - 2 tan - 1 ( coth 1 2 ( c z ) ) ( a - b ) 2 - 8 ( a 6 + 3 b 4 a 2 ) log ( a + b coth 1 2 ( c z ) ) ( a 4 - b 4 ) 2 ) z c z a b c z 1 2 2 -1 -1 1 4 c -1 8 a 3 a 4 -1 b 4 a b c z 1 2 -1 2 c z 1 2 a b 2 -1 2 c z 1 2 -1 a b 2 -1 2 c z 1 2 1 a -1 b 2 -1 c z 1 a b 2 -1 c z 1 a -1 b 2 -1 -1 2 c z 1 2 a -1 b 2 -1 -1 8 a 6 3 b 4 a 2 a b c z 1 2 a 4 -1 b 4 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_"]], "]"]], SuperscriptBox[RowBox[List["(", RowBox[List["a_", "+", RowBox[List["b_", " ", SqrtBox[RowBox[List["Coth", "[", RowBox[List["c_", " ", "z_"]], "]"]]]]]]], ")"]], "2"]], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List["-", FractionBox[RowBox[List[RowBox[List["-", FractionBox[RowBox[List["2", " ", "\[ImaginaryI]", " ", RowBox[List["ArcTan", "[", SqrtBox[RowBox[List["Coth", "[", RowBox[List["c", " ", "z"]], "]"]]], "]"]]]], SuperscriptBox[RowBox[List["(", RowBox[List["a", "-", RowBox[List["\[ImaginaryI]", " ", "b"]]]], ")"]], "2"]]]], "+", FractionBox[RowBox[List["2", " ", "\[ImaginaryI]", " ", RowBox[List["ArcTan", "[", SqrtBox[RowBox[List["Coth", "[", RowBox[List["c", " ", "z"]], "]"]]], "]"]]]], SuperscriptBox[RowBox[List["(", RowBox[List["a", "+", RowBox[List["\[ImaginaryI]", " ", "b"]]]], ")"]], "2"]], "+", FractionBox[RowBox[List["8", " ", SuperscriptBox["a", "3"]]], RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["a", "4"], "-", SuperscriptBox["b", "4"]]], ")"]], " ", RowBox[List["(", RowBox[List["a", "+", RowBox[List["b", " ", SqrtBox[RowBox[List["Coth", "[", RowBox[List["c", " ", "z"]], "]"]]]]]]], ")"]]]]], "+", FractionBox[RowBox[List["2", " ", RowBox[List["Log", "[", RowBox[List[RowBox[List["-", "1"]], "+", SqrtBox[RowBox[List["Coth", "[", RowBox[List["c", " ", "z"]], "]"]]]]], "]"]]]], SuperscriptBox[RowBox[List["(", RowBox[List["a", "+", "b"]], ")"]], "2"]], "+", FractionBox[RowBox[List["2", " ", RowBox[List["Log", "[", RowBox[List["1", "+", SqrtBox[RowBox[List["Coth", "[", RowBox[List["c", " ", "z"]], "]"]]]]], "]"]]]], SuperscriptBox[RowBox[List["(", RowBox[List["a", "-", "b"]], ")"]], "2"]], "-", FractionBox[RowBox[List["8", " ", RowBox[List["(", RowBox[List[SuperscriptBox["a", "6"], "+", RowBox[List["3", " ", SuperscriptBox["a", "2"], " ", SuperscriptBox["b", "4"]]]]], ")"]], " ", RowBox[List["Log", "[", RowBox[List["a", "+", RowBox[List["b", " ", SqrtBox[RowBox[List["Coth", "[", RowBox[List["c", " ", "z"]], "]"]]]]]]], "]"]]]], SuperscriptBox[RowBox[List["(", RowBox[List[SuperscriptBox["a", "4"], "-", SuperscriptBox["b", "4"]]], ")"]], "2"]], "+", FractionBox[RowBox[List["Log", "[", RowBox[List["1", "+", RowBox[List["Coth", "[", RowBox[List["c", " ", "z"]], "]"]]]], "]"]], SuperscriptBox[RowBox[List["(", RowBox[List["a", "-", RowBox[List["\[ImaginaryI]", " ", "b"]]]], ")"]], "2"]], "+", FractionBox[RowBox[List["Log", "[", RowBox[List["1", "+", RowBox[List["Coth", "[", RowBox[List["c", " ", "z"]], "]"]]]], "]"]], SuperscriptBox[RowBox[List["(", RowBox[List["a", "+", RowBox[List["\[ImaginaryI]", " ", "b"]]]], ")"]], "2"]]]], RowBox[List["4", " ", "c"]]]]]]]]]

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

 2002-12-18