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 Sec

 http://functions.wolfram.com/01.11.21.0046.01

 Input Form

 Integrate[Sin[b z]^m Sec[c z], z] == ((2^(1 - m) (1 - Mod[m, 2]))/c) ArcTanh[Tan[(c z)/2]] Binomial[m, m/2] + 2^(1 - m) I^(-m - 1) Sum[(-1)^k Binomial[m, k] (((-1)^m E^(I (c + 2 b k - b m) z) Hypergeometric2F1[1, (c + 2 b k - b m)/(2 c), (3 c + 2 b k - b m)/(2 c), -E^(2 I c z)])/ (c + 2 b k - b m) + (E^(I (c + b (-2 k + m)) z) Hypergeometric2F1[1, (c - 2 b k + b m)/(2 c), (3 c - 2 b k + b m)/(2 c), -E^(2 I c z)])/ (c - 2 b k + b m)), {k, 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["Sec", "[", RowBox[List["c", " ", "z"]], "]"]], RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", RowBox[List[RowBox[List[FractionBox[RowBox[List[SuperscriptBox["2", RowBox[List["1", "-", "m"]]], " ", RowBox[List["(", RowBox[List["1", "-", RowBox[List["Mod", "[", RowBox[List["m", ",", "2"]], "]"]]]], ")"]]]], "c"], RowBox[List["ArcTanh", "[", RowBox[List["Tan", "[", FractionBox[RowBox[List["c", " ", "z"]], "2"], "]"]], "]"]], " ", RowBox[List["Binomial", "[", RowBox[List["m", ",", FractionBox["m", "2"]]], "]"]]]], "+", RowBox[List[SuperscriptBox["2", RowBox[List["1", "-", "m"]]], " ", SuperscriptBox["\[ImaginaryI]", RowBox[List[RowBox[List["-", "m"]], "-", "1"]]], 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[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "k"], " ", RowBox[List["Binomial", "[", RowBox[List["m", ",", "k"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "m"], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["\[ImaginaryI]", " ", RowBox[List["(", RowBox[List["c", "+", RowBox[List["2", " ", "b", " ", "k"]], "-", RowBox[List["b", " ", "m"]]]], ")"]], " ", "z"]]], " ", RowBox[List["Hypergeometric2F1", "[", RowBox[List["1", ",", FractionBox[RowBox[List["c", "+", RowBox[List["2", " ", "b", " ", "k"]], "-", RowBox[List["b", " ", "m"]]]], RowBox[List["2", " ", "c"]]], ",", FractionBox[RowBox[List[RowBox[List["3", "c"]], "+", RowBox[List["2", " ", "b", " ", "k"]], "-", RowBox[List["b", " ", "m"]]]], RowBox[List["2", " ", "c"]]], ",", RowBox[List["-", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "\[ImaginaryI]", " ", "c", " ", "z"]]]]]]], "]"]]]], ")"]], "/", RowBox[List["(", RowBox[List["c", "+", RowBox[List["2", " ", "b", " ", "k"]], "-", RowBox[List["b", " ", "m"]]]], ")"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["\[ImaginaryI]", " ", RowBox[List["(", RowBox[List["c", "+", RowBox[List["b", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "m"]], ")"]]]]]], ")"]], " ", "z"]]], " ", RowBox[List["Hypergeometric2F1", "[", RowBox[List["1", ",", FractionBox[RowBox[List["c", "-", RowBox[List["2", " ", "b", " ", "k"]], "+", RowBox[List["b", " ", "m"]]]], RowBox[List["2", " ", "c"]]], ",", FractionBox[RowBox[List[RowBox[List["3", " ", "c"]], "-", RowBox[List["2", " ", "b", " ", "k"]], "+", RowBox[List["b", " ", "m"]]]], RowBox[List["2", " ", "c"]]], ",", RowBox[List["-", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "\[ImaginaryI]", " ", "c", " ", "z"]]]]]]], "]"]]]], ")"]], "/", RowBox[List["(", RowBox[List["c", "-", RowBox[List["2", " ", "b", " ", "k"]], "+", RowBox[List["b", " ", "m"]]]], ")"]]]]]], ")"]]]]]]]]]]]], "/;", RowBox[List[RowBox[List["m", "\[Element]", "Integers"]], "\[And]", RowBox[List["m", ">", "0"]]]]]]]]

 MathML Form

 sin m ( b z ) sec ( c z ) z ( 2 1 - m ( 1 - m mod 2 \$CellContext`m 2 ) ) c tanh - 1 ( tan ( c z 2 ) ) ( 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 - m - 1 k = 0 m - 1 2 ( - 1 ) k ( m k ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["m", Identity]], List[TagBox["k", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( ( c + b ( - 2 k + m ) ) z 2 F 1 ( 1 , c - 2 b k + b m 2 c ; 3 c - 2 b k + b m 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["1", Hypergeometric2F1], ",", TagBox[FractionBox[RowBox[List["c", "-", RowBox[List["2", " ", "b", " ", "k"]], "+", RowBox[List["b", " ", "m"]]]], RowBox[List["2", " ", "c"]]], Hypergeometric2F1]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1], ";", TagBox[TagBox[TagBox[FractionBox[RowBox[List[RowBox[List["3", " ", "c"]], "-", RowBox[List["2", " ", "b", " ", "k"]], "+", RowBox[List["b", " ", "m"]]]], RowBox[List["2", " ", "c"]]], Hypergeometric2F1], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1], ";", TagBox[RowBox[List["-", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "\[ImaginaryI]", " ", "c", " ", "z"]]]]], Hypergeometric2F1]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]]], Hypergeometric2F1] c - 2 b k + b m + ( - 1 ) m ( c + 2 b k - b m ) z 2 F 1 ( 1 , c + 2 b k - b m 2 c ; 3 c + 2 b k - b m 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["1", Hypergeometric2F1], ",", TagBox[FractionBox[RowBox[List["c", "+", RowBox[List["2", " ", "b", " ", "k"]], "-", RowBox[List["b", " ", "m"]]]], RowBox[List["2", " ", "c"]]], Hypergeometric2F1]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1], ";", TagBox[TagBox[TagBox[FractionBox[RowBox[List[RowBox[List["3", " ", "c"]], "+", RowBox[List["2", " ", "b", " ", "k"]], "-", RowBox[List["b", " ", "m"]]]], RowBox[List["2", " ", "c"]]], Hypergeometric2F1], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1], ";", TagBox[RowBox[List["-", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "\[ImaginaryI]", " ", "c", " ", "z"]]]]], Hypergeometric2F1]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]]], Hypergeometric2F1] c + 2 b k - b m ) /; m + TagBox[SuperscriptBox["\[DoubleStruckCapitalN]", "+"], Function[Integers]] Condition z b z m c z 2 1 -1 m 1 -1 \$CellContext`m 2 c -1 c z 2 -1 Binomial m m 2 -1 2 1 -1 m -1 m -1 k 0 m -1 2 -1 -1 k Binomial m k c b -2 k m z Hypergeometric2F1 1 c -1 2 b k b m 2 c -1 3 c -1 2 b k b m 2 c -1 -1 2 c z c -1 2 b k b m -1 -1 m c 2 b k -1 b m z Hypergeometric2F1 1 c 2 b k -1 b m 2 c -1 3 c 2 b k -1 b m 2 c -1 -1 2 c z c 2 b k -1 b m -1 m [/itex]

 Rule Form

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

 2002-10-15