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 Cos

 http://functions.wolfram.com/01.07.21.2617.01

 Input Form

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

 Standard Form

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 MathML Form

 p z sin μ ( c z ) cos v ( b + a z ) z 2 - v ( ( 1 - 2 c z ) - μ sin μ ( c z ) s = 0 v - 1 2 2 b s - b v ( v s ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["v", Identity]], List[TagBox["s", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( ( 2 b ( v - 2 s ) + ( p + a ( v - 2 s ) ) z 2 F 1 ( - ( p - 2 a s + a v - c μ ) 2 c , - μ ; 1 2 ( - ( p - 2 a s + a v ) c - μ + 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[RowBox[List["-", FractionBox[RowBox[List["\[ImaginaryI]", " ", RowBox[List["(", RowBox[List["p", "-", RowBox[List["2", " ", "\[ImaginaryI]", " ", "a", " ", "s"]], "+", RowBox[List["\[ImaginaryI]", " ", "a", " ", "v"]], "-", RowBox[List["\[ImaginaryI]", " ", "c", " ", "\[Mu]"]]]], ")"]]]], RowBox[List["2", " ", "c"]]]]], Hypergeometric2F1], ",", TagBox[RowBox[List["-", "\[Mu]"]], Hypergeometric2F1]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1], ";", TagBox[TagBox[TagBox[RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", FractionBox[RowBox[List["\[ImaginaryI]", " ", RowBox[List["(", RowBox[List["p", "-", RowBox[List["2", " ", "\[ImaginaryI]", " ", "a", " ", "s"]], "+", RowBox[List["\[ImaginaryI]", " ", "a", " ", "v"]]]], ")"]]]], "c"]]], "-", "\[Mu]", "+", "2"]], ")"]]]], Hypergeometric2F1], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1], ";", TagBox[SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "\[ImaginaryI]", " ", "c", " ", "z"]]], Hypergeometric2F1]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]]], Hypergeometric2F1] ) / ( p - 2 a s + a v - c μ ) + ( ( p + 2 a s - a v ) z 2 F 1 ( - ( p + 2 a s - a v - c μ ) 2 c , - μ ; - ( p + 2 a s - a v - c ( μ - 2 ) ) 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[RowBox[List["-", FractionBox[RowBox[List["\[ImaginaryI]", " ", RowBox[List["(", RowBox[List["p", "+", RowBox[List["2", " ", "\[ImaginaryI]", " ", "a", " ", "s"]], "-", RowBox[List["\[ImaginaryI]", " ", "a", " ", "v"]], "-", RowBox[List["\[ImaginaryI]", " ", "c", " ", "\[Mu]"]]]], ")"]]]], RowBox[List["2", " ", "c"]]]]], Hypergeometric2F1], ",", TagBox[RowBox[List["-", "\[Mu]"]], Hypergeometric2F1]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1], ";", TagBox[TagBox[TagBox[RowBox[List["-", FractionBox[RowBox[List["\[ImaginaryI]", " ", RowBox[List["(", RowBox[List["p", "+", RowBox[List["2", " ", "\[ImaginaryI]", " ", "a", " ", "s"]], "-", RowBox[List["\[ImaginaryI]", " ", "a", " ", "v"]], "-", RowBox[List["\[ImaginaryI]", " ", "c", " ", RowBox[List["(", RowBox[List["\[Mu]", "-", "2"]], ")"]]]]]], ")"]]]], RowBox[List["2", " ", "c"]]]]], Hypergeometric2F1], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1], ";", TagBox[SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "\[ImaginaryI]", " ", "c", " ", "z"]]], Hypergeometric2F1]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]]], Hypergeometric2F1] ) / ( p + 2 a s - a v - c μ ) ) - 1 p - c μ ( p z ( 1 - 2 c z ) - μ ( v v 2 ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["v", Identity]], List[TagBox[FractionBox["v", "2"], Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] 2 F 1 ( - p + c μ 2 c , - μ ; 1 2 ( - p c - μ + 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[RowBox[List["-", FractionBox[RowBox[List[RowBox[List["\[ImaginaryI]", " ", "p"]], "+", RowBox[List["c", " ", "\[Mu]"]]]], RowBox[List["2", " ", "c"]]]]], Hypergeometric2F1], ",", TagBox[RowBox[List["-", "\[Mu]"]], Hypergeometric2F1]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1], ";", TagBox[TagBox[TagBox[RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", FractionBox[RowBox[List["\[ImaginaryI]", " ", "p"]], "c"]]], "-", "\[Mu]", "+", "2"]], ")"]]]], Hypergeometric2F1], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1], ";", TagBox[SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "\[ImaginaryI]", " ", "c", " ", "z"]]], Hypergeometric2F1]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]]], Hypergeometric2F1] ( v mod 2 \$CellContext`v 2 - 1 ) sin μ ( c z ) ) ) /; v + Condition z p z c z μ b a z v 2 -1 v 1 -1 2 c z -1 μ c z μ s 0 v -1 2 -1 2 b s -1 b v Binomial v s 2 b v -1 2 s p a v -1 2 s z Hypergeometric2F1 -1 p -1 2 a s a v -1 c μ 2 c -1 -1 μ 1 2 -1 p -1 2 a s a v c -1 -1 μ 2 2 c z p -1 2 a s a v -1 c μ -1 p 2 a s -1 a v z Hypergeometric2F1 -1 p 2 a s -1 a v -1 c μ 2 c -1 -1 μ -1 p 2 a s -1 a v -1 c μ -2 2 c -1 2 c z p 2 a s -1 a v -1 c μ -1 -1 1 p -1 c μ -1 p z 1 -1 2 c z -1 μ Binomial v v 2 -1 Hypergeometric2F1 -1 p c μ 2 c -1 -1 μ 1 2 -1 p c -1 -1 μ 2 2 c z \$CellContext`v 2 -1 c z μ v SuperPlus [/itex]

 Rule Form

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

 2002-12-18