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 Coth

 http://functions.wolfram.com/01.22.21.0038.01

 Input Form

 Integrate[Sin[b z]^m Coth[c z], z] == (Binomial[m, m/2] (1 - Mod[m, 2]) (Log[Sinh[c z]]/c))/2^m + (I^(1 - m) Sum[(-1)^s Binomial[m, s] (-(((-1)^m ((2 c - I b (-2 s + m)) Hypergeometric2F1[ -((I b (-2 s + m))/(2 c)), 1, 1 - (I b (-2 s + m))/(2 c), E^ (2 c z)] - I b E^(2 c z) (-2 s + m) Hypergeometric2F1[1 - (I b (-2 s + m))/(2 c), 1, 2 - (I b (-2 s + m))/(2 c), E^ (2 c z)]))/E^(I b (-2 s + m) z))/(b (-2 s + m) (2 c - I b (-2 s + m))) + (E^(I b (-2 s + m) z) ((2 c + I b (-2 s + m)) Hypergeometric2F1[(I b (-2 s + m))/(2 c), 1, 1 + (I b (-2 s + m))/(2 c), E^(2 c z)] + I b E^(2 c z) (-2 s + m) Hypergeometric2F1[1 + (I b (-2 s + m))/(2 c), 1, 2 + (I b (-2 s + m))/(2 c), E^(2 c z)]))/(b (-2 s + m) (2 c + I b (-2 s + m)))), {s, 0, Floor[(1/2) (-1 + m)]}])/2^m /; Element[m, Integers] && m > 0

 Standard Form

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 MathML Form

 sin m ( b z ) coth ( c z ) z 2 - m ( 1 - m mod 2 \$CellContext`m 2 ) log ( sinh ( c z ) ) c ( m m 2 ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["m", Identity, Rule[Editable, True]]], List[TagBox[FractionBox["m", "2"], Identity, Rule[Editable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False]] + 2 - m 1 - m s = 0 m - 1 2 ( - 1 ) s ( m s ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["m", Identity, Rule[Editable, True]]], List[TagBox["s", Identity, Rule[Editable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False]] ( ( b ( m - 2 s ) z ( ( 2 c + b ( m - 2 s ) ) 2 F 1 ( b ( m - 2 s ) 2 c , 1 ; b ( m - 2 s ) 2 c + 1 ; 2 c z ) 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[FractionBox[RowBox[List["\[ImaginaryI]", " ", "b", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "s"]]]], ")"]]]], RowBox[List["2", " ", "c"]]], Hypergeometric2F1, Rule[Editable, True]], ",", TagBox["1", Hypergeometric2F1, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1, Rule[Editable, False]], ";", TagBox[TagBox[TagBox[RowBox[List[FractionBox[RowBox[List["b", " ", "\[ImaginaryI]", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "s"]]]], ")"]]]], RowBox[List["2", " ", "c"]]], "+", "1"]], Hypergeometric2F1, Rule[Editable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1, Rule[Editable, False]], ";", TagBox[SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "c", " ", "z"]]], Hypergeometric2F1, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], Hypergeometric2F1] + b 2 c z ( m - 2 s ) 2 F 1 ( b ( m - 2 s ) 2 c + 1 , 1 ; b ( m - 2 s ) 2 c + 2 ; 2 c z ) 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[FractionBox[RowBox[List["b", " ", "\[ImaginaryI]", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "s"]]]], ")"]]]], RowBox[List["2", " ", "c"]]], "+", "1"]], Hypergeometric2F1, Rule[Editable, True]], ",", TagBox["1", Hypergeometric2F1, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1, Rule[Editable, False]], ";", TagBox[TagBox[TagBox[RowBox[List[FractionBox[RowBox[List["b", " ", "\[ImaginaryI]", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "s"]]]], ")"]]]], RowBox[List["2", " ", "c"]]], "+", "2"]], Hypergeometric2F1, Rule[Editable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1, Rule[Editable, False]], ";", TagBox[SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "c", " ", "z"]]], Hypergeometric2F1, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], Hypergeometric2F1] ) ) / ( b ( m - 2 s ) ( 2 c + b ( m - 2 s ) ) ) - ( ( - 1 ) m - b ( m - 2 s ) z ( ( 2 c - b ( m - 2 s ) ) 2 F 1 ( - b ( m - 2 s ) 2 c , 1 ; 1 - b ( m - 2 s ) 2 c ; 2 c z ) 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["-", FractionBox[RowBox[List["\[ImaginaryI]", " ", "b", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "s"]]]], ")"]]]], RowBox[List["2", " ", "c"]]]]], Hypergeometric2F1, Rule[Editable, True]], ",", TagBox["1", Hypergeometric2F1, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1, Rule[Editable, False]], ";", TagBox[TagBox[TagBox[RowBox[List["1", "-", FractionBox[RowBox[List["\[ImaginaryI]", " ", "b", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "s"]]]], ")"]]]], RowBox[List["2", " ", "c"]]]]], Hypergeometric2F1, Rule[Editable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1, Rule[Editable, False]], ";", TagBox[SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "c", " ", "z"]]], Hypergeometric2F1, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], Hypergeometric2F1] - b 2 c z ( m - 2 s ) 2 F 1 ( 1 - b ( m - 2 s ) 2 c , 1 ; 2 - b ( m - 2 s ) 2 c ; 2 c z ) 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["1", "-", FractionBox[RowBox[List["\[ImaginaryI]", " ", "b", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "s"]]]], ")"]]]], RowBox[List["2", " ", "c"]]]]], Hypergeometric2F1, Rule[Editable, True]], ",", TagBox["1", Hypergeometric2F1, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1, Rule[Editable, False]], ";", TagBox[TagBox[TagBox[RowBox[List["2", "-", FractionBox[RowBox[List["\[ImaginaryI]", " ", "b", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "s"]]]], ")"]]]], RowBox[List["2", " ", "c"]]]]], Hypergeometric2F1, Rule[Editable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1, Rule[Editable, False]], ";", TagBox[SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "c", " ", "z"]]], Hypergeometric2F1, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], Hypergeometric2F1] ) ) / ( b ( m - 2 s ) ( 2 c - b ( m - 2 s ) ) ) ) /; m + Condition z b z m c z 2 -1 m 1 -1 \$CellContext`m 2 c z c -1 Binomial m m 2 -1 2 -1 m 1 -1 m s 0 m -1 2 -1 -1 s Binomial m s b m -1 2 s z 2 c b m -1 2 s Hypergeometric2F1 b m -1 2 s 2 c -1 1 b m -1 2 s 2 c -1 1 2 c z b 2 c z m -1 2 s Hypergeometric2F1 b m -1 2 s 2 c -1 1 1 b m -1 2 s 2 c -1 2 2 c z b m -1 2 s 2 c b m -1 2 s -1 -1 -1 m -1 b m -1 2 s z 2 c -1 b m -1 2 s Hypergeometric2F1 -1 b m -1 2 s 2 c -1 1 1 -1 b m -1 2 s 2 c -1 2 c z -1 b 2 c z m -1 2 s Hypergeometric2F1 1 -1 b m -1 2 s 2 c -1 1 2 -1 b m -1 2 s 2 c -1 2 c z b m -1 2 s 2 c -1 b m -1 2 s -1 m SuperPlus [/itex]

 Rule Form

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

 2002-12-18