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 Coth

 http://functions.wolfram.com/01.22.21.0401.01

 Input Form

 Integrate[Cosh[c z] (a + b Coth[c z])^\[Beta], z] == -(((b + (a + b)/(-1 + Coth[c z]))^\[Beta] ((-1 + \[Beta]) AppellF1[2 - \[Beta], 3/2, -\[Beta], 3 - \[Beta], -(2/(-1 + Coth[c z])), (a + b)/(b - b Coth[c z])] + (-2 + \[Beta]) AppellF1[1 - \[Beta], 3/2, -\[Beta], 2 - \[Beta], -(2/(-1 + Coth[c z])), (a + b)/(b - b Coth[c z])] (-1 + Coth[c z])) (a + b Coth[c z])^(-1 + \[Beta]) ((a + b Coth[c z])/(-1 + Coth[c z]))^ (1 - \[Beta]) Sinh[c z] Sqrt[Cosh[2 c z] + Sinh[2 c z]])/ (1 + (a + b)/(-b + b Coth[c z]))^\[Beta])/ (c (-2 + \[Beta]) (-1 + \[Beta]))

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List["\[Integral]", RowBox[List[RowBox[List["Cosh", "[", RowBox[List["c", " ", "z"]], "]"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["a", "+", RowBox[List["b", " ", RowBox[List["Coth", "[", RowBox[List["c", " ", "z"]], "]"]]]]]], ")"]], "\[Beta]"], RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", RowBox[List[RowBox[List["-", RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["b", "+", FractionBox[RowBox[List["a", "+", "b"]], RowBox[List[RowBox[List["-", "1"]], "+", RowBox[List["Coth", "[", RowBox[List["c", " ", "z"]], "]"]]]]]]], ")"]], "\[Beta]"], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "\[Beta]"]], ")"]], " ", RowBox[List["AppellF1", "[", RowBox[List[RowBox[List["2", "-", "\[Beta]"]], ",", FractionBox["3", "2"], ",", RowBox[List["-", "\[Beta]"]], ",", RowBox[List["3", "-", "\[Beta]"]], ",", RowBox[List["-", FractionBox["2", RowBox[List[RowBox[List["-", "1"]], "+", RowBox[List["Coth", "[", RowBox[List["c", " ", "z"]], "]"]]]]]]], ",", FractionBox[RowBox[List["a", "+", "b"]], RowBox[List["b", "-", RowBox[List["b", " ", RowBox[List["Coth", "[", RowBox[List["c", " ", "z"]], "]"]]]]]]]]], "]"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "2"]], "+", "\[Beta]"]], ")"]], " ", RowBox[List["AppellF1", "[", RowBox[List[RowBox[List["1", "-", "\[Beta]"]], ",", FractionBox["3", "2"], ",", RowBox[List["-", "\[Beta]"]], ",", RowBox[List["2", "-", "\[Beta]"]], ",", RowBox[List["-", FractionBox["2", RowBox[List[RowBox[List["-", "1"]], "+", RowBox[List["Coth", "[", RowBox[List["c", " ", "z"]], "]"]]]]]]], ",", FractionBox[RowBox[List["a", "+", "b"]], RowBox[List["b", "-", RowBox[List["b", " ", RowBox[List["Coth", "[", RowBox[List["c", " ", "z"]], "]"]]]]]]]]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", RowBox[List["Coth", "[", RowBox[List["c", " ", "z"]], "]"]]]], ")"]]]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["a", "+", RowBox[List["b", " ", RowBox[List["Coth", "[", RowBox[List["c", " ", "z"]], "]"]]]]]], ")"]], RowBox[List[RowBox[List["-", "1"]], "+", "\[Beta]"]]], " ", SuperscriptBox[RowBox[List["(", FractionBox[RowBox[List["a", "+", RowBox[List["b", " ", RowBox[List["Coth", "[", RowBox[List["c", " ", "z"]], "]"]]]]]], RowBox[List[RowBox[List["-", "1"]], "+", RowBox[List["Coth", "[", RowBox[List["c", " ", "z"]], "]"]]]]], ")"]], RowBox[List["1", "-", "\[Beta]"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "+", FractionBox[RowBox[List["a", "+", "b"]], RowBox[List[RowBox[List["-", "b"]], "+", RowBox[List["b", " ", RowBox[List["Coth", "[", RowBox[List["c", " ", "z"]], "]"]]]]]]]]], ")"]], RowBox[List["-", "\[Beta]"]]], " ", RowBox[List["Sinh", "[", RowBox[List["c", " ", "z"]], "]"]], " ", SqrtBox[RowBox[List[RowBox[List["Cosh", "[", RowBox[List["2", " ", "c", " ", "z"]], "]"]], "+", RowBox[List["Sinh", "[", RowBox[List["2", " ", "c", " ", "z"]], "]"]]]]]]], ")"]]]], "/", RowBox[List["(", RowBox[List["c", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "2"]], "+", "\[Beta]"]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "\[Beta]"]], ")"]]]], ")"]]]]]]]]

 MathML Form

 cosh ( c z ) ( a + b coth ( c z ) ) β z - 1 c ( β - 2 ) ( β - 1 ) ( ( b + a + b coth ( c z ) - 1 ) β ( ( β - 1 ) F 1 AppellF1 ( 2 - β ; 3 2 , - β ; 3 - β ; - 2 coth ( c z ) - 1 , a + b b - b coth ( c z ) ) + ( β - 2 ) F 1 AppellF1 ( 1 - β ; 3 2 , - β ; 2 - β ; - 2 coth ( c z ) - 1 , a + b b - b coth ( c z ) ) ( coth ( c z ) - 1 ) ) ( a + b coth ( c z ) ) β - 1 ( a + b coth ( c z ) coth ( c z ) - 1 ) 1 - β ( a + b b coth ( c z ) - b + 1 ) - β sinh ( c z ) cosh ( 2 c z ) + sinh ( 2 c z ) ) z c z a b c z β -1 1 c β -2 β -1 -1 b a b c z -1 -1 β β -1 AppellF1 2 -1 β 3 2 -1 β 3 -1 β -1 2 c z -1 -1 a b b -1 b c z -1 β -2 AppellF1 1 -1 β 3 2 -1 β 2 -1 β -1 2 c z -1 -1 a b b -1 b c z -1 c z -1 a b c z β -1 a b c z c z -1 -1 1 -1 β a b b c z -1 b -1 1 -1 β c z 2 c z 2 c z 1 2 [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["\[Integral]", RowBox[List[RowBox[List[RowBox[List["Cosh", "[", RowBox[List["c_", " ", "z_"]], "]"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["a_", "+", RowBox[List["b_", " ", RowBox[List["Coth", "[", RowBox[List["c_", " ", "z_"]], "]"]]]]]], ")"]], "\[Beta]_"]]], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List["-", FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["b", "+", FractionBox[RowBox[List["a", "+", "b"]], RowBox[List[RowBox[List["-", "1"]], "+", RowBox[List["Coth", "[", RowBox[List["c", " ", "z"]], "]"]]]]]]], ")"]], "\[Beta]"], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "\[Beta]"]], ")"]], " ", RowBox[List["AppellF1", "[", RowBox[List[RowBox[List["2", "-", "\[Beta]"]], ",", FractionBox["3", "2"], ",", RowBox[List["-", "\[Beta]"]], ",", RowBox[List["3", "-", "\[Beta]"]], ",", RowBox[List["-", FractionBox["2", RowBox[List[RowBox[List["-", "1"]], "+", RowBox[List["Coth", "[", RowBox[List["c", " ", "z"]], "]"]]]]]]], ",", FractionBox[RowBox[List["a", "+", "b"]], RowBox[List["b", "-", RowBox[List["b", " ", RowBox[List["Coth", "[", RowBox[List["c", " ", "z"]], "]"]]]]]]]]], "]"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "2"]], "+", "\[Beta]"]], ")"]], " ", RowBox[List["AppellF1", "[", RowBox[List[RowBox[List["1", "-", "\[Beta]"]], ",", FractionBox["3", "2"], ",", RowBox[List["-", "\[Beta]"]], ",", RowBox[List["2", "-", "\[Beta]"]], ",", RowBox[List["-", FractionBox["2", RowBox[List[RowBox[List["-", "1"]], "+", RowBox[List["Coth", "[", RowBox[List["c", " ", "z"]], "]"]]]]]]], ",", FractionBox[RowBox[List["a", "+", "b"]], RowBox[List["b", "-", RowBox[List["b", " ", RowBox[List["Coth", "[", RowBox[List["c", " ", "z"]], "]"]]]]]]]]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", RowBox[List["Coth", "[", RowBox[List["c", " ", "z"]], "]"]]]], ")"]]]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["a", "+", RowBox[List["b", " ", RowBox[List["Coth", "[", RowBox[List["c", " ", "z"]], "]"]]]]]], ")"]], RowBox[List[RowBox[List["-", "1"]], "+", "\[Beta]"]]], " ", SuperscriptBox[RowBox[List["(", FractionBox[RowBox[List["a", "+", RowBox[List["b", " ", RowBox[List["Coth", "[", RowBox[List["c", " ", "z"]], "]"]]]]]], RowBox[List[RowBox[List["-", "1"]], "+", RowBox[List["Coth", "[", RowBox[List["c", " ", "z"]], "]"]]]]], ")"]], RowBox[List["1", "-", "\[Beta]"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "+", FractionBox[RowBox[List["a", "+", "b"]], RowBox[List[RowBox[List["-", "b"]], "+", RowBox[List["b", " ", RowBox[List["Coth", "[", RowBox[List["c", " ", "z"]], "]"]]]]]]]]], ")"]], RowBox[List["-", "\[Beta]"]]], " ", RowBox[List["Sinh", "[", RowBox[List["c", " ", "z"]], "]"]], " ", SqrtBox[RowBox[List[RowBox[List["Cosh", "[", RowBox[List["2", " ", "c", " ", "z"]], "]"]], "+", RowBox[List["Sinh", "[", RowBox[List["2", " ", "c", " ", "z"]], "]"]]]]]]], RowBox[List["c", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "2"]], "+", "\[Beta]"]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "\[Beta]"]], ")"]]]]]]]]]]]

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

 2002-12-18