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 Csc

 http://functions.wolfram.com/01.10.21.0048.01

 Input Form

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

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["\[Integral]", RowBox[List[SuperscriptBox[RowBox[List["Sin", "[", RowBox[List["b", " ", "z"]], "]"]], "m"], RowBox[List["Csc", "[", RowBox[List["c", " ", "z"]], "]"]], " ", RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", RowBox[List[FractionBox[RowBox[List[SuperscriptBox["2", RowBox[List["1", "-", "m"]]], " ", RowBox[List["ArcTanh", "[", SuperscriptBox["\[ExponentialE]", RowBox[List["\[ImaginaryI]", " ", "c", " ", "z"]]], "]"]], " ", RowBox[List["Binomial", "[", RowBox[List["m", ",", FractionBox["m", "2"]]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", RowBox[List["Mod", "[", RowBox[List["m", ",", "2"]], "]"]]]], ")"]]]], "c"], "-", RowBox[List[SuperscriptBox["2", RowBox[List["1", "-", "m"]]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["\[ImaginaryI]", " ", "c", " ", "z"]]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["s", "=", "0"]], RowBox[List["Floor", "[", RowBox[List[FractionBox["1", "2"], RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "m"]], ")"]]]], "]"]]], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "s"], RowBox[List["Binomial", "[", RowBox[List["m", ",", "s"]], "]"]], RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], " ", "\[ImaginaryI]", " ", "m", " ", "\[Pi]"]], "+", RowBox[List["\[ImaginaryI]", " ", "b", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "s"]]]], ")"]], " ", "z"]]]]], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[FractionBox[RowBox[List[RowBox[List["b", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", "s"]]]], ")"]]]], "+", "c"]], RowBox[List["2", " ", "c"]]], ",", "1"]], "}"]], ",", RowBox[List["{", RowBox[List["1", "+", FractionBox[RowBox[List[RowBox[List["b", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", "s"]]]], ")"]]]], "+", "c"]], RowBox[List["2", " ", "c"]]]]], "}"]], ",", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "\[ImaginaryI]", " ", "c", " ", "z"]]]]], "]"]]]], ")"]], "/", RowBox[List["(", RowBox[List[RowBox[List["b", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "s"]]]], ")"]]]], "+", "c"]], ")"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[FractionBox[RowBox[List["\[ImaginaryI]", " ", "m", " ", "\[Pi]"]], "2"], "-", RowBox[List["\[ImaginaryI]", " ", "b", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "s"]]]], ")"]], "z"]]]]], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[FractionBox[RowBox[List[RowBox[List[RowBox[List["-", "b"]], " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "s"]]]], ")"]]]], "+", "c"]], RowBox[List["2", " ", "c"]]], ",", "1"]], "}"]], ",", RowBox[List["{", RowBox[List["1", "+", FractionBox[RowBox[List[RowBox[List[RowBox[List["-", "b"]], " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "s"]]]], ")"]]]], "+", "c"]], RowBox[List["2", " ", "c"]]]]], "}"]], ",", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "\[ImaginaryI]", " ", "c", " ", "z"]]]]], "]"]]]], ")"]], "/", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "b"]], " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "s"]]]], ")"]]]], "+", "c"]], ")"]]]]]], ")"]]]]]]]]]]]], "/;", RowBox[List[RowBox[List["m", "\[Element]", "Integers"]], "\[And]", RowBox[List["m", ">", "0"]]]]]]]]

 MathML Form

 sin m ( b z ) csc ( c z ) z 2 1 - m tanh - 1 ( c z ) ( m mod 2 \$CellContext`m 2 - 1 ) 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 1 - m c z s = 0 m - 1 2 ( - 1 ) s ( m s ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["m", Identity]], List[TagBox["s", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( m π 2 - b ( m - 2 s ) z 2 F 1 ( c - b ( m - 2 s ) 2 c , 1 ; c - 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["c", "-", RowBox[List["b", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "s"]]]], ")"]]]]]], RowBox[List["2", " ", "c"]]], HypergeometricPFQ], ",", TagBox["1", HypergeometricPFQ]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ], ";", TagBox[TagBox[TagBox[RowBox[List[FractionBox[RowBox[List["c", "-", RowBox[List["b", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "s"]]]], ")"]]]]]], 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] c - b ( m - 2 s ) + b ( m - 2 s ) z - m π 2 2 F 1 ( c + b ( m - 2 s ) 2 c , 1 ; c + 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["c", "+", RowBox[List["b", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "s"]]]], ")"]]]]]], RowBox[List["2", " ", "c"]]], HypergeometricPFQ], ",", TagBox["1", HypergeometricPFQ]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ], ";", TagBox[TagBox[TagBox[RowBox[List[FractionBox[RowBox[List["c", "+", RowBox[List["b", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "s"]]]], ")"]]]]]], 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] c + b ( m - 2 s ) ) /; m + Condition z b z m c z 2 1 -1 m c z \$CellContext`m 2 -1 c -1 Binomial m m 2 -1 -1 2 1 -1 m c z s 0 m -1 2 -1 -1 s Binomial m s m 2 -1 -1 b m -1 2 s z HypergeometricPFQ c -1 b m -1 2 s 2 c -1 1 c -1 b m -1 2 s 2 c -1 1 2 c z c -1 b m -1 2 s -1 b m -1 2 s z -1 m 2 -1 HypergeometricPFQ c b m -1 2 s 2 c -1 1 c b m -1 2 s 2 c -1 1 2 c z c 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