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 Coth

 http://functions.wolfram.com/01.22.21.0282.01

 Input Form

 Integrate[(a + b Sqrt[Coth[c z]])^\[Beta], z] == ((a + b Sqrt[Coth[c z]])^(1 + \[Beta]) ((a^3 + a^2 b + a b^2 + b^3) Hypergeometric2F1[1 + \[Beta], 1, 2 + \[Beta], (a + b Sqrt[Coth[c z]])/(a - b)] - (a - b) ((a^2 + (1 + I) a b + I b^2) Hypergeometric2F1[1 + \[Beta], 1, 2 + \[Beta], (a + b Sqrt[Coth[c z]])/(a - I b)] + (a - I b) ((a + b) Hypergeometric2F1[1 + \[Beta], 1, 2 + \[Beta], (a + b Sqrt[Coth[c z]])/(a + I b)] - (a + I b) Hypergeometric2F1[1 + \[Beta], 1, 2 + \[Beta], (a + b Sqrt[Coth[c z]])/(a + b)]))))/(2 (a^4 - b^4) c (1 + \[Beta]))

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List["\[Integral]", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["a", "+", RowBox[List["b", " ", SqrtBox[RowBox[List["Coth", "[", RowBox[List["c", " ", "z"]], "]"]]]]]]], ")"]], "\[Beta]"], RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["a", "+", RowBox[List["b", " ", SqrtBox[RowBox[List["Coth", "[", RowBox[List["c", " ", "z"]], "]"]]]]]]], ")"]], RowBox[List["1", "+", "\[Beta]"]]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["a", "3"], "+", RowBox[List[SuperscriptBox["a", "2"], " ", "b"]], "+", RowBox[List["a", " ", SuperscriptBox["b", "2"]]], "+", SuperscriptBox["b", "3"]]], ")"]], " ", RowBox[List["Hypergeometric2F1", "[", RowBox[List[RowBox[List["1", "+", "\[Beta]"]], ",", "1", ",", RowBox[List["2", "+", "\[Beta]"]], ",", FractionBox[RowBox[List["a", "+", RowBox[List["b", " ", SqrtBox[RowBox[List["Coth", "[", RowBox[List["c", " ", "z"]], "]"]]]]]]], RowBox[List["a", "-", "b"]]]]], "]"]]]], "-", RowBox[List[RowBox[List["(", RowBox[List["a", "-", "b"]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["a", "2"], "+", RowBox[List[RowBox[List["(", RowBox[List["1", "+", "\[ImaginaryI]"]], ")"]], " ", "a", " ", "b"]], "+", RowBox[List["\[ImaginaryI]", " ", SuperscriptBox["b", "2"]]]]], ")"]], " ", RowBox[List["Hypergeometric2F1", "[", RowBox[List[RowBox[List["1", "+", "\[Beta]"]], ",", "1", ",", RowBox[List["2", "+", "\[Beta]"]], ",", FractionBox[RowBox[List["a", "+", RowBox[List["b", " ", SqrtBox[RowBox[List["Coth", "[", RowBox[List["c", " ", "z"]], "]"]]]]]]], RowBox[List["a", "-", RowBox[List["\[ImaginaryI]", " ", "b"]]]]]]], "]"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List["a", "-", RowBox[List["\[ImaginaryI]", " ", "b"]]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["a", "+", "b"]], ")"]], " ", RowBox[List["Hypergeometric2F1", "[", RowBox[List[RowBox[List["1", "+", "\[Beta]"]], ",", "1", ",", RowBox[List["2", "+", "\[Beta]"]], ",", FractionBox[RowBox[List["a", "+", RowBox[List["b", " ", SqrtBox[RowBox[List["Coth", "[", RowBox[List["c", " ", "z"]], "]"]]]]]]], RowBox[List["a", "+", RowBox[List["\[ImaginaryI]", " ", "b"]]]]]]], "]"]]]], "-", RowBox[List[RowBox[List["(", RowBox[List["a", "+", RowBox[List["\[ImaginaryI]", " ", "b"]]]], ")"]], " ", RowBox[List["Hypergeometric2F1", "[", RowBox[List[RowBox[List["1", "+", "\[Beta]"]], ",", "1", ",", RowBox[List["2", "+", "\[Beta]"]], ",", FractionBox[RowBox[List["a", "+", RowBox[List["b", " ", SqrtBox[RowBox[List["Coth", "[", RowBox[List["c", " ", "z"]], "]"]]]]]]], RowBox[List["a", "+", "b"]]]]], "]"]]]]]], ")"]]]]]], ")"]]]]]], ")"]]]], ")"]], "/", RowBox[List["(", RowBox[List["2", " ", RowBox[List["(", RowBox[List[SuperscriptBox["a", "4"], "-", SuperscriptBox["b", "4"]]], ")"]], " ", "c", " ", RowBox[List["(", RowBox[List["1", "+", "\[Beta]"]], ")"]]]], ")"]]]]]]]]

 MathML Form

 ( a + b coth 1 2 ( c z ) ) β z 1 2 ( a 4 - b 4 ) c ( β + 1 ) ( ( a + b coth 1 2 ( c z ) ) β + 1 ( ( a 3 + b a 2 + b 2 a + b 3 ) 2 F 1 ( β + 1 , 1 ; β + 2 ; a + b coth 1 2 ( c z ) a - b ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["2", TraditionalForm]], SubscriptBox["F", FormBox["1", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox[RowBox[List["\[Beta]", "+", "1"]], Hypergeometric2F1], ",", TagBox["1", Hypergeometric2F1]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1], ";", TagBox[TagBox[TagBox[RowBox[List["\[Beta]", "+", "2"]], Hypergeometric2F1], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1], ";", TagBox[FractionBox[RowBox[List["a", "+", RowBox[List["b", " ", RowBox[List[SuperscriptBox["coth", FractionBox["1", "2"]], "(", RowBox[List["c", " ", "z"]], ")"]]]]]], RowBox[List["a", "-", "b"]]], Hypergeometric2F1]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]]], Hypergeometric2F1] - ( a - b ) ( ( a - b ) ( ( a + b ) 2 F 1 ( β + 1 , 1 ; β + 2 ; a + b coth 1 2 ( c z ) a + b ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["2", TraditionalForm]], SubscriptBox["F", FormBox["1", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox[RowBox[List["\[Beta]", "+", "1"]], Hypergeometric2F1], ",", TagBox["1", Hypergeometric2F1]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1], ";", TagBox[TagBox[TagBox[RowBox[List["\[Beta]", "+", "2"]], Hypergeometric2F1], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1], ";", TagBox[FractionBox[RowBox[List["a", "+", RowBox[List["b", " ", RowBox[List[SuperscriptBox["coth", FractionBox["1", "2"]], "(", RowBox[List["c", " ", "z"]], ")"]]]]]], RowBox[List["a", "+", RowBox[List["\[ImaginaryI]", " ", "b", " "]]]]], Hypergeometric2F1]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]]], Hypergeometric2F1] - ( a + b ) 2 F 1 ( β + 1 , 1 ; β + 2 ; a + b coth 1 2 ( c z ) a + b ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["2", TraditionalForm]], SubscriptBox["F", FormBox["1", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox[RowBox[List["\[Beta]", "+", "1"]], Hypergeometric2F1], ",", TagBox["1", Hypergeometric2F1]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1], ";", TagBox[TagBox[TagBox[RowBox[List["\[Beta]", "+", "2"]], Hypergeometric2F1], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1], ";", TagBox[FractionBox[RowBox[List["a", "+", RowBox[List["b", " ", RowBox[List[SuperscriptBox["coth", FractionBox["1", "2"]], "(", RowBox[List["c", " ", "z"]], ")"]]]]]], RowBox[List["a", "+", "b"]]], Hypergeometric2F1]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]]], Hypergeometric2F1] ) + ( a 2 + b ( 1 + ) a + b 2 ) 2 F 1 ( β + 1 , 1 ; β + 2 ; a + b coth 1 2 ( c z ) a - b ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["2", TraditionalForm]], SubscriptBox["F", FormBox["1", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox[RowBox[List["\[Beta]", "+", "1"]], Hypergeometric2F1], ",", TagBox["1", Hypergeometric2F1]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1], ";", TagBox[TagBox[TagBox[RowBox[List["\[Beta]", "+", "2"]], Hypergeometric2F1], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1], ";", TagBox[FractionBox[RowBox[List["a", "+", RowBox[List["b", " ", RowBox[List[SuperscriptBox["coth", FractionBox["1", "2"]], "(", RowBox[List["c", " ", "z"]], ")"]]]]]], RowBox[List["a", "-", RowBox[List["\[ImaginaryI]", " ", "b"]]]]], Hypergeometric2F1]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]]], Hypergeometric2F1] ) ) ) z a b c z 1 2 β 1 2 a 4 -1 b 4 c β 1 -1 a b c z 1 2 β 1 a 3 b a 2 b 2 a b 3 Hypergeometric2F1 β 1 1 β 2 a b c z 1 2 a -1 b -1 -1 a -1 b a -1 b a b Hypergeometric2F1 β 1 1 β 2 a b c z 1 2 a b -1 -1 a b Hypergeometric2F1 β 1 1 β 2 a b c z 1 2 a b -1 a 2 b 1 a b 2 Hypergeometric2F1 β 1 1 β 2 a b c z 1 2 a -1 b -1 [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["\[Integral]", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["a_", "+", RowBox[List["b_", " ", SqrtBox[RowBox[List["Coth", "[", RowBox[List["c_", " ", "z_"]], "]"]]]]]]], ")"]], "\[Beta]_"], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["a", "+", RowBox[List["b", " ", SqrtBox[RowBox[List["Coth", "[", RowBox[List["c", " ", "z"]], "]"]]]]]]], ")"]], RowBox[List["1", "+", "\[Beta]"]]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["a", "3"], "+", RowBox[List[SuperscriptBox["a", "2"], " ", "b"]], "+", RowBox[List["a", " ", SuperscriptBox["b", "2"]]], "+", SuperscriptBox["b", "3"]]], ")"]], " ", RowBox[List["Hypergeometric2F1", "[", RowBox[List[RowBox[List["1", "+", "\[Beta]"]], ",", "1", ",", RowBox[List["2", "+", "\[Beta]"]], ",", FractionBox[RowBox[List["a", "+", RowBox[List["b", " ", SqrtBox[RowBox[List["Coth", "[", RowBox[List["c", " ", "z"]], "]"]]]]]]], RowBox[List["a", "-", "b"]]]]], "]"]]]], "-", RowBox[List[RowBox[List["(", RowBox[List["a", "-", "b"]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["a", "2"], "+", RowBox[List[RowBox[List["(", RowBox[List["1", "+", "\[ImaginaryI]"]], ")"]], " ", "a", " ", "b"]], "+", RowBox[List["\[ImaginaryI]", " ", SuperscriptBox["b", "2"]]]]], ")"]], " ", RowBox[List["Hypergeometric2F1", "[", RowBox[List[RowBox[List["1", "+", "\[Beta]"]], ",", "1", ",", RowBox[List["2", "+", "\[Beta]"]], ",", FractionBox[RowBox[List["a", "+", RowBox[List["b", " ", SqrtBox[RowBox[List["Coth", "[", RowBox[List["c", " ", "z"]], "]"]]]]]]], RowBox[List["a", "-", RowBox[List["\[ImaginaryI]", " ", "b"]]]]]]], "]"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List["a", "-", RowBox[List["\[ImaginaryI]", " ", "b"]]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["a", "+", "b"]], ")"]], " ", RowBox[List["Hypergeometric2F1", "[", RowBox[List[RowBox[List["1", "+", "\[Beta]"]], ",", "1", ",", RowBox[List["2", "+", "\[Beta]"]], ",", FractionBox[RowBox[List["a", "+", RowBox[List["b", " ", SqrtBox[RowBox[List["Coth", "[", RowBox[List["c", " ", "z"]], "]"]]]]]]], RowBox[List["a", "+", RowBox[List["\[ImaginaryI]", " ", "b"]]]]]]], "]"]]]], "-", RowBox[List[RowBox[List["(", RowBox[List["a", "+", RowBox[List["\[ImaginaryI]", " ", "b"]]]], ")"]], " ", RowBox[List["Hypergeometric2F1", "[", RowBox[List[RowBox[List["1", "+", "\[Beta]"]], ",", "1", ",", RowBox[List["2", "+", "\[Beta]"]], ",", FractionBox[RowBox[List["a", "+", RowBox[List["b", " ", SqrtBox[RowBox[List["Coth", "[", RowBox[List["c", " ", "z"]], "]"]]]]]]], RowBox[List["a", "+", "b"]]]]], "]"]]]]]], ")"]]]]]], ")"]]]]]], ")"]]]], RowBox[List["2", " ", RowBox[List["(", RowBox[List[SuperscriptBox["a", "4"], "-", SuperscriptBox["b", "4"]]], ")"]], " ", "c", " ", RowBox[List["(", RowBox[List["1", "+", "\[Beta]"]], ")"]]]]]]]]]

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

 2002-12-18