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 Cot

 http://functions.wolfram.com/01.09.21.0076.01

 Input Form

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

 Standard Form

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"+", "m"]], ")"]]]]]], ")"]]]]]], ")"]]]]]]]]]]]], "/;", RowBox[List[RowBox[List["m", "\[Element]", "Integers"]], "\[And]", RowBox[List["m", ">", "0"]]]]]]]]

 MathML Form

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

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["\[Integral]", RowBox[List[RowBox[List[SuperscriptBox[RowBox[List["Cos", "[", RowBox[List["b_", " ", "z_"]], "]"]], "m_"], " ", RowBox[List["Cot", "[", RowBox[List["c_", " ", "z_"]], "]"]]]], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[FractionBox[RowBox[List[SuperscriptBox["2", RowBox[List["-", "m"]]], " ", RowBox[List["Binomial", "[", RowBox[List["m", ",", FractionBox["m", "2"]]], "]"]], " ", RowBox[List["Log", "[", RowBox[List["Sin", "[", RowBox[List["c", " ", "z"]], "]"]], "]"]], " ", RowBox[List["(", RowBox[List["1", "-", RowBox[List["Mod", "[", RowBox[List["m", ",", "2"]], "]"]]]], ")"]]]], "c"], "-", RowBox[List[SuperscriptBox["2", RowBox[List["-", "m"]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["Floor", "[", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "m"]], ")"]]]], "]"]]], RowBox[List[RowBox[List["Binomial", "[", RowBox[List["m", ",", "k"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", FractionBox[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["\[ImaginaryI]", " ", "b", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "k"]], "-", "m"]], ")"]], " ", "z"]]], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List["1", ",", RowBox[List[FractionBox[RowBox[List["b", " ", "k"]], "c"], "-", FractionBox[RowBox[List["b", " ", "m"]], RowBox[List["2", " ", "c"]]]]]]], "}"]], ",", RowBox[List["{", RowBox[List["1", "+", FractionBox[RowBox[List["b", " ", "k"]], "c"], "-", FractionBox[RowBox[List["b", " ", "m"]], RowBox[List["2", " ", "c"]]]]], "}"]], ",", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "\[ImaginaryI]", " ", "c", " ", "z"]]]]], "]"]]]], RowBox[List["b", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "m"]], ")"]]]]]]], "+", FractionBox[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["\[ImaginaryI]", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "c"]], "+", RowBox[List["2", " ", "b", " ", "k"]], "-", RowBox[List["b", " ", "m"]]]], ")"]], " ", "z"]]], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List["1", ",", RowBox[List["1", "+", FractionBox[RowBox[List["b", " ", "k"]], "c"], "-", FractionBox[RowBox[List["b", " ", "m"]], RowBox[List["2", " ", "c"]]]]]]], "}"]], ",", RowBox[List["{", RowBox[List["2", "+", FractionBox[RowBox[List["b", " ", "k"]], "c"], "-", FractionBox[RowBox[List["b", " ", "m"]], RowBox[List["2", " ", "c"]]]]], "}"]], ",", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "\[ImaginaryI]", " ", "c", " ", "z"]]]]], "]"]]]], RowBox[List[RowBox[List["2", " ", "c"]], "-", RowBox[List["b", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "m"]], ")"]]]]]]], "+", FractionBox[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "b", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "k"]], "-", "m"]], ")"]], " ", "z"]]], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List["1", ",", RowBox[List[RowBox[List["-", FractionBox[RowBox[List["b", " ", "k"]], "c"]]], "+", FractionBox[RowBox[List["b", " ", "m"]], RowBox[List["2", " ", "c"]]]]]]], "}"]], ",", RowBox[List["{", RowBox[List["1", "-", FractionBox[RowBox[List["b", " ", "k"]], "c"], "+", FractionBox[RowBox[List["b", " ", "m"]], RowBox[List["2", " ", "c"]]]]], "}"]], ",", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "\[ImaginaryI]", " ", "c", " ", "z"]]]]], "]"]]]], RowBox[List["b", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "m"]], ")"]]]]], "+", FractionBox[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["\[ImaginaryI]", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "c"]], "+", RowBox[List["b", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "m"]], ")"]]]]]], ")"]], " ", "z"]]], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List["1", ",", RowBox[List["1", "-", FractionBox[RowBox[List["b", " ", "k"]], "c"], "+", FractionBox[RowBox[List["b", " ", "m"]], RowBox[List["2", " ", "c"]]]]]]], "}"]], ",", RowBox[List["{", RowBox[List["2", "-", FractionBox[RowBox[List["b", " ", "k"]], "c"], "+", FractionBox[RowBox[List["b", " ", "m"]], RowBox[List["2", " ", "c"]]]]], "}"]], ",", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "\[ImaginaryI]", " ", "c", " ", "z"]]]]], "]"]]]], RowBox[List[RowBox[List["2", " ", "c"]], "+", RowBox[List["b", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "m"]], ")"]]]]]]]]], ")"]]]]]]]]]], "/;", RowBox[List[RowBox[List["m", "\[Element]", "Integers"]], "&&", RowBox[List["m", ">", "0"]]]]]]]]]]

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

 2002-12-18