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 Cos

 http://functions.wolfram.com/01.07.21.2481.01

 Input Form

 Integrate[z^n Sin[d z] Cos[c z^2 + f z + g]^v, z] == 2^(-2 - v) (-2 d^(-1 - 2 n) Binomial[v, v/2] ((I d)^n Gamma[1 + n, (-I) d z] + ((-I) d)^n Gamma[1 + n, I d z]) (1 - Mod[v, 2]) + I Sum[E^(I (-2 s + v) g) Binomial[v, s] ((-E^(-((I (-d - 2 f s + f v)^2)/(-8 c s + 4 c v)))) (I (-2 c s + c v))^(-1 - n) Sum[2^(-n + q) ((-I) (-d - 2 f s + f v))^ (n - q) (I (-d + f (-2 s + v) + 2 (-2 c s + c v) z))^(1 + q) (-((I (-d + f (-2 s + v) + 2 (-2 c s + c v) z)^2)/(-2 c s + c v)))^ ((1/2) (-1 - q)) Binomial[n, q] Gamma[(1 + q)/2, -((I (-d + f (-2 s + v) + 2 (-2 c s + c v) z)^2)/(-8 c s + 4 c v))], {q, 0, n}] + E^(-2 I g (-2 s + v) + (I (-d - 2 f s + f v)^2)/(-8 c s + 4 c v)) ((-I) (-2 c s + c v))^ (-1 - n) Sum[2^(-n + q) (I (-d - 2 f s + f v))^(n - q) ((-I) (-d + f (-2 s + v) + 2 (-2 c s + c v) z))^(1 + q) ((I (-d + f (-2 s + v) + 2 (-2 c s + c v) z)^2)/(-2 c s + c v))^ ((1/2) (-1 - q)) Binomial[n, q] Gamma[(1 + q)/2, (I (-d + f (-2 s + v) + 2 (-2 c s + c v) z)^2)/(-8 c s + 4 c v)], {q, 0, n}] - E^(I (-2 g (-2 s + v) - (d + f (-2 s + v))^2/ (8 c s - 4 c v))) (I (2 c s - c v))^(-1 - n) Sum[2^(-n + q) ((-I) (-d + 2 f s - f v))^(n - q) (I (-d + 2 f s - f v + 4 c s z - 2 c v z))^(1 + q) (-((I (d + f (-2 s + v) + 2 (-2 c s + c v) z)^2)/(2 c s - c v)))^ ((1/2) (-1 - q)) Binomial[n, q] Gamma[(1 + q)/2, -((I (d + f (-2 s + v) + 2 (-2 c s + c v) z)^2)/(8 c s - 4 c v))], {q, 0, n}] + E^((I (d + f (-2 s + v))^2)/ (8 c s - 4 c v)) ((-I) (2 c s - c v))^(-1 - n) Sum[2^(-n + q) (I (-d + 2 f s - f v))^(n - q) ((-I) (-d + 2 f s - f v + 4 c s z - 2 c v z))^(1 + q) ((I (d + f (-2 s + v) + 2 (-2 c s + c v) z)^2)/(2 c s - c v))^ ((1/2) (-1 - q)) Binomial[n, q] Gamma[(1 + q)/2, (I (d + f (-2 s + v) + 2 (-2 c s + c v) z)^2)/(8 c s - 4 c v)], {q, 0, n}]), {s, 0, Floor[(1/2) (-1 + v)]}]) /; Element[n, Integers] && n >= 0 && Element[v, Integers] && v > 0

 Standard Form

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 MathML Form

 z n sin ( d z ) cos v ( c z 2 + f z + g ) z 2 - v - 2 ( s = 0 v - 1 2 ( v - 2 s ) g ( v s ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["v", Identity, Rule[Editable, True]]], List[TagBox["s", Identity, Rule[Editable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False]] ( ( - d - 2 f s + f v ) 2 4 c v - 8 c s - 2 g ( v - 2 s ) ( q = 0 n 2 q - n ( ( - d - 2 f s + f v ) ) n - q ( - ( - d + f ( v - 2 s ) + 2 ( c v - 2 c s ) z ) ) q + 1 ( ( - d + f ( v - 2 s ) + 2 ( c v - 2 c s ) z ) 2 c v - 2 c s ) 1 2 ( - q - 1 ) ( n q ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["n", Identity, Rule[Editable, True]]], List[TagBox["q", Identity, Rule[Editable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False]] Γ ( q + 1 2 , ( - d + f ( v - 2 s ) + 2 ( c v - 2 c s ) z ) 2 4 c v - 8 c s ) ) ( - ( c v - 2 c s ) ) - n - 1 - - ( - d - 2 f s + f v ) 2 4 c v - 8 c s ( ( c v - 2 c s ) ) - n - 1 q = 0 n 2 q - n ( - ( - d - 2 f s + f v ) ) n - q ( ( - d + f ( v - 2 s ) + 2 ( c v - 2 c s ) z ) ) q + 1 ( - ( - d + f ( v - 2 s ) + 2 ( c v - 2 c s ) z ) 2 c v - 2 c s ) 1 2 ( - q - 1 ) ( n q ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["n", Identity, Rule[Editable, True]]], List[TagBox["q", Identity, Rule[Editable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False]] Γ ( q + 1 2 , - ( - d + f ( v - 2 s ) + 2 ( c v - 2 c s ) z ) 2 4 c v - 8 c s ) + ( d + f ( v - 2 s ) ) 2 8 c s - 4 c v ( - ( 2 c s - c v ) ) - n - 1 q = 0 n 2 q - n ( ( - d + 2 f s - f v ) ) n - q ( - ( - d + 2 f s - f v + 4 c s z - 2 c v z ) ) q + 1 ( ( d + f ( v - 2 s ) + 2 ( c v - 2 c s ) z ) 2 2 c s - c v ) 1 2 ( - q - 1 ) ( n q ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["n", Identity, Rule[Editable, True]]], List[TagBox["q", Identity, Rule[Editable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False]] Γ ( q + 1 2 , ( d + f ( v - 2 s ) + 2 ( c v - 2 c s ) z ) 2 8 c s - 4 c v ) - ( - ( d + f ( v - 2 s ) ) 2 8 c s - 4 c v - 2 g ( v - 2 s ) ) ( ( 2 c s - c v ) ) - n - 1 q = 0 n 2 q - n ( - ( - d + 2 f s - f v ) ) n - q ( ( - d + 2 f s - f v + 4 c s z - 2 c v z ) ) q + 1 ( - ( d + f ( v - 2 s ) + 2 ( c v - 2 c s ) z ) 2 2 c s - c v ) 1 2 ( - q - 1 ) ( n q ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["n", Identity, Rule[Editable, True]]], List[TagBox["q", Identity, Rule[Editable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False]] Γ ( q + 1 2 , - ( d + f ( v - 2 s ) + 2 ( c v - 2 c s ) z ) 2 8 c s - 4 c v ) ) - 2 d - 2 n - 1 ( v v 2 ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["v", Identity, Rule[Editable, True]]], List[TagBox[FractionBox["v", "2"], Identity, Rule[Editable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False]] ( Γ ( n + 1 , d z ) ( - d ) n + ( d ) n Γ ( n + 1 , - d z ) ) ( 1 - v mod 2 \$CellContext`v 2 ) ) /; n v + Condition z z n d z c z 2 f z g v 2 -1 v -2 s 0 v -1 2 -1 v -1 2 s g Binomial v s -1 d -1 2 f s f v 2 4 c v -1 8 c s -1 -1 2 g v -1 2 s q 0 n 2 q -1 n -1 d -1 2 f s f v n -1 q -1 -1 d f v -1 2 s 2 c v -1 2 c s z q 1 -1 d f v -1 2 s 2 c v -1 2 c s z 2 c v -1 2 c s -1 1 2 -1 q -1 Binomial n q Gamma q 1 2 -1 -1 d f v -1 2 s 2 c v -1 2 c s z 2 4 c v -1 8 c s -1 -1 c v -1 2 c s -1 n -1 -1 -1 -1 d -1 2 f s f v 2 4 c v -1 8 c s -1 c v -1 2 c s -1 n -1 q 0 n 2 q -1 n -1 -1 d -1 2 f s f v n -1 q -1 d f v -1 2 s 2 c v -1 2 c s z q 1 -1 -1 d f v -1 2 s 2 c v -1 2 c s z 2 c v -1 2 c s -1 1 2 -1 q -1 Binomial n q Gamma q 1 2 -1 -1 -1 d f v -1 2 s 2 c v -1 2 c s z 2 4 c v -1 8 c s -1 d f v -1 2 s 2 8 c s -1 4 c v -1 -1 2 c s -1 c v -1 n -1 q 0 n 2 q -1 n -1 d 2 f s -1 f v n -1 q -1 -1 d 2 f s -1 f v 4 c s z -1 2 c v z q 1 d f v -1 2 s 2 c v -1 2 c s z 2 2 c s -1 c v -1 1 2 -1 q -1 Binomial n q Gamma q 1 2 -1 d f v -1 2 s 2 c v -1 2 c s z 2 8 c s -1 4 c v -1 -1 -1 d f v -1 2 s 2 8 c s -1 4 c v -1 -1 2 g v -1 2 s 2 c s -1 c v -1 n -1 q 0 n 2 q -1 n -1 -1 d 2 f s -1 f v n -1 q -1 d 2 f s -1 f v 4 c s z -1 2 c v z q 1 -1 d f v -1 2 s 2 c v -1 2 c s z 2 2 c s -1 c v -1 1 2 -1 q -1 Binomial n q Gamma q 1 2 -1 -1 d f v -1 2 s 2 c v -1 2 c s z 2 8 c s -1 4 c v -1 -1 2 d -2 n -1 Binomial v v 2 -1 Gamma n 1 d z -1 d n d n Gamma n 1 -1 d z 1 -1 \$CellContext`v 2 n v SuperPlus [/itex]

 Rule Form

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

 2002-12-18