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 Cot

 http://functions.wolfram.com/01.09.21.0046.01

 Input Form

 Integrate[Sin[b z]^m Cot[c z], z] == (Binomial[m, m/2] (1 - Mod[m, 2]) (Log[Sin[c z]]/c))/2^m + (1/(2^m b)) Sum[((-1)^k Binomial[m, k] ((E^((I m Pi)/2) (4 c^2 - b^2 (-2 k + m)^2) Hypergeometric2F1[ -((b (-2 k + m))/(2 c)), 1, 1 - (b (-2 k + m))/(2 c), E^(2 I c z)] + E^((-(1/2)) I m Pi + 2 I b (-2 k + m) z) (-4 c^2 + b^2 (-2 k + m)^2) Hypergeometric2F1[(b (-2 k + m))/(2 c), 1, 1 + (b (-2 k + m))/(2 c), E^(2 I c z)] - b E^((I m Pi)/2 + 2 I c z) (-2 k + m) (2 c + b (-2 k + m)) Hypergeometric2F1[1 - (b (-2 k + m))/(2 c), 1, 2 - (b (-2 k + m))/(2 c), E^(2 I c z)] - b E^((-(1/2)) I m Pi + 2 I (c + b (-2 k + m)) z) (-2 k + m) (2 c - b (-2 k + m)) Hypergeometric2F1[1 + (b (-2 k + m))/(2 c), 1, 2 + (b (-2 k + m))/(2 c), E^(2 I c z)])/ ((-2 k + m) (4 c^2 - b^2 (-2 k + m)^2))))/E^(I b (-2 k + m) z), {k, 0, Floor[(1/2) (-1 + m)]}] /; Element[m, Integers] && m > 0

 Standard Form

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 MathML Form

 sin m ( b z ) cot ( c z ) z 2 - m log ( sin ( 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]] ( 1 - m mod 2 \$CellContext`m 2 ) + 2 - m b k = 0 m - 1 2 ( ( - 1 ) k ( m k ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["m", Identity, Rule[Editable, True]]], List[TagBox["k", Identity, Rule[Editable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False]] - b ( m - 2 k ) z ( 2 b ( m - 2 k ) z - m π 2 ( b 2 ( m - 2 k ) 2 - 4 c 2 ) 2 F 1 ( b ( m - 2 k ) 2 c , 1 ; b ( m - 2 k ) 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["b", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "k"]]]], ")"]]]], 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", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "k"]]]], ")"]]]], RowBox[List["2", " ", "c"]]], "+", "1"]], Hypergeometric2F1, Rule[Editable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1, Rule[Editable, False]], ";", TagBox[SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "\[ImaginaryI]", " ", "c", " ", "z"]]], Hypergeometric2F1, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], Hypergeometric2F1] + m π 2 ( 4 c 2 - b 2 ( m - 2 k ) 2 ) 2 F 1 ( - b ( m - 2 k ) 2 c , 1 ; 1 - b ( m - 2 k ) 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["b", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "k"]]]], ")"]]]], 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["b", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "k"]]]], ")"]]]], RowBox[List["2", " ", "c"]]]]], Hypergeometric2F1, Rule[Editable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1, Rule[Editable, False]], ";", TagBox[SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "\[ImaginaryI]", " ", "c", " ", "z"]]], Hypergeometric2F1, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], Hypergeometric2F1] - b 2 ( c + b ( m - 2 k ) ) z - m π 2 ( m - 2 k ) ( 2 c - b ( m - 2 k ) ) 2 F 1 ( b ( m - 2 k ) 2 c + 1 , 1 ; b ( m - 2 k ) 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", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "k"]]]], ")"]]]], 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", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "k"]]]], ")"]]]], RowBox[List["2", " ", "c"]]], "+", "2"]], Hypergeometric2F1, Rule[Editable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1, Rule[Editable, False]], ";", TagBox[SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "\[ImaginaryI]", " ", "c", " ", "z"]]], Hypergeometric2F1, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], Hypergeometric2F1] - b π m 2 + 2 c z ( m - 2 k ) ( 2 c + b ( m - 2 k ) ) 2 F 1 ( 1 - b ( m - 2 k ) 2 c , 1 ; 2 - b ( m - 2 k ) 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["b", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "k"]]]], ")"]]]], 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["b", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "k"]]]], ")"]]]], RowBox[List["2", " ", "c"]]]]], Hypergeometric2F1, Rule[Editable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1, Rule[Editable, False]], ";", TagBox[SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "\[ImaginaryI]", " ", "c", " ", "z"]]], Hypergeometric2F1, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], Hypergeometric2F1] ) ) / ( ( m - 2 k ) ( 4 c 2 - b 2 ( m - 2 k ) 2 ) ) /; m + Condition z b z m c z 2 -1 m c z c -1 Binomial m m 2 -1 1 -1 \$CellContext`m 2 2 -1 m b -1 k 0 m -1 2 -1 -1 k Binomial m k -1 b m -1 2 k z 2 b m -1 2 k z -1 m 2 -1 b 2 m -1 2 k 2 -1 4 c 2 Hypergeometric2F1 b m -1 2 k 2 c -1 1 b m -1 2 k 2 c -1 1 2 c z m 2 -1 4 c 2 -1 b 2 m -1 2 k 2 Hypergeometric2F1 -1 b m -1 2 k 2 c -1 1 1 -1 b m -1 2 k 2 c -1 2 c z -1 b 2 c b m -1 2 k z -1 m 2 -1 m -1 2 k 2 c -1 b m -1 2 k Hypergeometric2F1 b m -1 2 k 2 c -1 1 1 b m -1 2 k 2 c -1 2 2 c z -1 b m 2 -1 2 c z m -1 2 k 2 c b m -1 2 k Hypergeometric2F1 1 -1 b m -1 2 k 2 c -1 1 2 -1 b m -1 2 k 2 c -1 2 c z m -1 2 k 4 c 2 -1 b 2 m -1 2 k 2 -1 m SuperPlus [/itex]

 Rule Form

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

 2002-12-18