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 Cos

 http://functions.wolfram.com/01.07.21.2453.01

 Input Form

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

 Standard Form

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 MathML Form

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

 Rule Form

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

 2002-12-18