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 Sec

 http://functions.wolfram.com/01.11.21.0062.01

 Input Form

 Integrate[E^(p z) Cos[b z]^m Sec[c z], z] == ((2^(1 - m) E^((I c + p) z) (1 - Mod[m, 2]))/(I c + p)) Binomial[m, m/2] Hypergeometric2F1[1, (c - I p)/(2 c), (1/2) (3 - (I p)/c), -E^(2 I c z)] - I 2^(1 - m) Sum[Binomial[m, s] ((E^((I c + p - I b (m - 2 s)) z) Hypergeometric2F1[1, (c - b m - I p + 2 b s)/(2 c), (3 c - b m - I p + 2 b s)/(2 c), -E^(2 I c z)])/ (c - b m - I p + 2 b s) + (E^((I c + p + I b (m - 2 s)) z) Hypergeometric2F1[1, (c + b m - I p - 2 b s)/(2 c), (3 c + b m - I p - 2 b s)/(2 c), -E^(2 I c z)])/ (c - I p + b (m - 2 s))), {s, 0, Floor[(1/2) (-1 + m)]}] /; Element[m, Integers] && m > 0

 Standard Form

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 MathML Form

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

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["\[Integral]", RowBox[List[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["p_", " ", "z_"]]], " ", SuperscriptBox[RowBox[List["Cos", "[", RowBox[List["b_", " ", "z_"]], "]"]], "m_"], " ", RowBox[List["Sec", "[", RowBox[List["c_", " ", "z_"]], "]"]]]], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["2", RowBox[List["1", "-", "m"]]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "c"]], "+", "p"]], ")"]], " ", "z"]]], " ", RowBox[List["(", RowBox[List["1", "-", RowBox[List["Mod", "[", RowBox[List["m", ",", "2"]], "]"]]]], ")"]]]], ")"]], " ", RowBox[List["Binomial", "[", RowBox[List["m", ",", FractionBox["m", "2"]]], "]"]], " ", RowBox[List["Hypergeometric2F1", "[", RowBox[List["1", ",", FractionBox[RowBox[List["c", "-", RowBox[List["\[ImaginaryI]", " ", "p"]]]], RowBox[List["2", " ", "c"]]], ",", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["3", "-", FractionBox[RowBox[List["\[ImaginaryI]", " ", "p"]], "c"]]], ")"]]]], ",", RowBox[List["-", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "\[ImaginaryI]", " ", "c", " ", "z"]]]]]]], "]"]]]], RowBox[List[RowBox[List["\[ImaginaryI]", " ", "c"]], "+", "p"]]], "-", RowBox[List["\[ImaginaryI]", " ", SuperscriptBox["2", RowBox[List["1", "-", "m"]]], " ", 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[RowBox[List["Binomial", "[", RowBox[List["m", ",", "s"]], "]"]], " ", RowBox[List["(", RowBox[List[FractionBox[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "c"]], "+", "p", "-", RowBox[List["\[ImaginaryI]", " ", "b", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "s"]]]], ")"]]]]]], ")"]], " ", "z"]]], " ", RowBox[List["Hypergeometric2F1", "[", RowBox[List["1", ",", FractionBox[RowBox[List["c", "-", RowBox[List["b", " ", "m"]], "-", RowBox[List["\[ImaginaryI]", " ", "p"]], "+", RowBox[List["2", " ", "b", " ", "s"]]]], RowBox[List["2", " ", "c"]]], ",", FractionBox[RowBox[List[RowBox[List["3", " ", "c"]], "-", RowBox[List["b", " ", "m"]], "-", RowBox[List["\[ImaginaryI]", " ", "p"]], "+", RowBox[List["2", " ", "b", " ", "s"]]]], RowBox[List["2", " ", "c"]]], ",", RowBox[List["-", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "\[ImaginaryI]", " ", "c", " ", "z"]]]]]]], "]"]]]], RowBox[List["c", "-", RowBox[List["b", " ", "m"]], "-", RowBox[List["\[ImaginaryI]", " ", "p"]], "+", RowBox[List["2", " ", "b", " ", "s"]]]]], "+", FractionBox[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "c"]], "+", "p", "+", RowBox[List["\[ImaginaryI]", " ", "b", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "s"]]]], ")"]]]]]], ")"]], " ", "z"]]], " ", RowBox[List["Hypergeometric2F1", "[", RowBox[List["1", ",", FractionBox[RowBox[List["c", "+", RowBox[List["b", " ", "m"]], "-", RowBox[List["\[ImaginaryI]", " ", "p"]], "-", RowBox[List["2", " ", "b", " ", "s"]]]], RowBox[List["2", " ", "c"]]], ",", FractionBox[RowBox[List[RowBox[List["3", " ", "c"]], "+", RowBox[List["b", " ", "m"]], "-", RowBox[List["\[ImaginaryI]", " ", "p"]], "-", RowBox[List["2", " ", "b", " ", "s"]]]], RowBox[List["2", " ", "c"]]], ",", RowBox[List["-", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "\[ImaginaryI]", " ", "c", " ", "z"]]]]]]], "]"]]]], RowBox[List["c", "-", RowBox[List["\[ImaginaryI]", " ", "p"]], "+", RowBox[List["b", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "s"]]]], ")"]]]]]]]]], ")"]]]]]]]]]], "/;", RowBox[List[RowBox[List["m", "\[Element]", "Integers"]], "&&", RowBox[List["m", ">", "0"]]]]]]]]]]

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

 2002-10-15