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 Sec

 http://functions.wolfram.com/01.11.21.0058.01

 Input Form

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

 Standard Form

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"Integers"]], "\[And]", RowBox[List["m", ">", "0"]]]]]]]]

 MathML Form

 p z sin m ( b z ) sec ( c z ) z 2 1 - m ( 1 - m mod 2 \$CellContext`m 2 ) c + p ( c + p ) z ( 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]] 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, Rule[Editable, True]], ",", TagBox[RowBox[List["-", FractionBox[RowBox[List["\[ImaginaryI]", " ", RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "c"]], "+", "p"]], ")"]]]], RowBox[List["2", " ", "c"]]]]], Hypergeometric2F1, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1, Rule[Editable, False]], ";", TagBox[TagBox[TagBox[RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["3", "-", FractionBox[RowBox[List["\[ImaginaryI]", " ", "p"]], "c"]]], ")"]]]], Hypergeometric2F1, Rule[Editable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1, Rule[Editable, False]], ";", TagBox[RowBox[List["-", 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] + 2 1 - m 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]] ( ( c - 2 b k + p ) z - 1 2 m ( π - 2 b z ) c - 2 b k + b m + p 2 F 1 ( 1 , - ( c - 2 b k + b m + p ) 2 c ; - ( 3 c - 2 b k + b m + p ) 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, Rule[Editable, True]], ",", TagBox[RowBox[List["-", FractionBox[RowBox[List["\[ImaginaryI]", " ", RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "c"]], "-", RowBox[List["2", " ", "\[ImaginaryI]", " ", "b", " ", "k"]], "+", RowBox[List["\[ImaginaryI]", " ", "b", " ", "m"]], "+", "p"]], ")"]]]], RowBox[List["2", " ", "c"]]]]], Hypergeometric2F1, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1, Rule[Editable, False]], ";", TagBox[TagBox[TagBox[RowBox[List["-", FractionBox[RowBox[List["\[ImaginaryI]", " ", RowBox[List["(", RowBox[List[RowBox[List["3", " ", "\[ImaginaryI]", " ", "c"]], "-", RowBox[List["2", " ", "\[ImaginaryI]", " ", "b", " ", "k"]], "+", RowBox[List["\[ImaginaryI]", " ", "b", " ", "m"]], "+", "p"]], ")"]]]], RowBox[List["2", " ", "c"]]]]], Hypergeometric2F1, Rule[Editable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1, Rule[Editable, False]], ";", TagBox[RowBox[List["-", 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] + ( c + 2 b k + p ) z + 1 2 m ( π - 2 b z ) c + 2 b k - b m + p 2 F 1 ( 1 , - ( c + 2 b k - b m + p ) 2 c ; - ( 3 c + 2 b k - b m + p ) 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, Rule[Editable, True]], ",", TagBox[RowBox[List["-", FractionBox[RowBox[List["\[ImaginaryI]", " ", RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "c"]], "+", RowBox[List["2", " ", "\[ImaginaryI]", " ", "b", " ", "k"]], "-", RowBox[List["\[ImaginaryI]", " ", "b", " ", "m"]], "+", "p"]], ")"]]]], RowBox[List["2", " ", "c"]]]]], Hypergeometric2F1, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1, Rule[Editable, False]], ";", TagBox[TagBox[TagBox[RowBox[List["-", FractionBox[RowBox[List["\[ImaginaryI]", " ", RowBox[List["(", RowBox[List[RowBox[List["3", " ", "\[ImaginaryI]", " ", "c"]], "+", RowBox[List["2", " ", "\[ImaginaryI]", " ", "b", " ", "k"]], "-", RowBox[List["\[ImaginaryI]", " ", "b", " ", "m"]], "+", "p"]], ")"]]]], RowBox[List["2", " ", "c"]]]]], Hypergeometric2F1, Rule[Editable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1, Rule[Editable, False]], ";", TagBox[RowBox[List["-", 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 + TagBox[SuperscriptBox["\[DoubleStruckCapitalN]", "+"], Function[Integers]] Condition z p z b z m c z 2 1 -1 m 1 -1 \$CellContext`m 2 c p -1 c p z Binomial m m 2 -1 Hypergeometric2F1 1 -1 c p 2 c -1 1 2 3 -1 p c -1 -1 2 c z 2 1 -1 m k 0 m -1 2 -1 -1 k Binomial m k c -1 2 b k p z -1 1 2 m -1 2 b z c -1 2 b k b m p -1 Hypergeometric2F1 1 -1 c -1 2 b k b m p 2 c -1 -1 3 c -1 2 b k b m p 2 c -1 -1 2 c z c 2 b k p z 1 2 m -1 2 b z c 2 b k -1 b m p -1 Hypergeometric2F1 1 -1 c 2 b k -1 b m p 2 c -1 -1 3 c 2 b k -1 b m p 2 c -1 -1 2 c z m [/itex]

 Rule Form

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

 2002-10-15