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 Cos

 http://functions.wolfram.com/01.07.21.2475.01

 Input Form

 Integrate[z^n Sin[b z^2 + e] Cos[c z^2 + f z]^v, z] == 2^(-2 - v) ((I z^(1 + n) (b^2 z^4)^((1/2) (-1 - n)) Binomial[v, v/2] (E^(2 I e) (I b z^2)^((1 + n)/2) Gamma[(1 + n)/2, (-I) b z^2] - ((-I) b z^2)^((1 + n)/2) Gamma[(1 + n)/2, I b z^2]) (1 - Mod[v, 2]))/ E^(I e) + I E^(I e) Sum[Binomial[v, s] (E^(I (-2 e + Pi - (-2 f s + f v)^2/(-4 b - 8 c s + 4 c v))) (I (-b - 2 c s + c v))^(-1 - n) Sum[2^(-n + q) ((-I) (-2 f s + f v))^ (n - q) (I (f (-2 s + v) + 2 (-b - 2 c s + c v) z))^(1 + q) (-((I (f (-2 s + v) + 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 (f (-2 s + v) + 2 (-b - 2 c s + c v) z)^2)/(-4 b - 8 c s + 4 c v))], {q, 0, n}] + E^((I (-2 f s + f v)^2)/(-4 b - 8 c s + 4 c v)) ((-I) (-b - 2 c s + c v))^(-1 - n) Sum[2^(-n + q) (I (-2 f s + f v))^(n - q) ((-I) (f (-2 s + v) + 2 (-b - 2 c s + c v) z))^(1 + q) ((I (f (-2 s + v) + 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 (f (-2 s + v) + 2 (-b - 2 c s + c v) z)^2)/(-4 b - 8 c s + 4 c v)], {q, 0, n}] + E^(I (-2 e + Pi - (f^2 (-2 s + v)^2)/(-4 b + 8 c s - 4 c v))) (I (-b + 2 c s - c v))^(-1 - n) Sum[2^(-n + q) ((-I) (2 f s - f v))^ (n - q) (I (2 f s - f v - 2 b z + 4 c s z - 2 c v z))^(1 + q) (-((I (f (-2 s + v) + 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 (f (-2 s + v) + 2 (b - 2 c s + c v) z)^2)/(-4 b + 8 c s - 4 c v))], {q, 0, n}] + E^((I f^2 (-2 s + v)^2)/(-4 b + 8 c s - 4 c v)) ((-I) (-b + 2 c s - c v))^(-1 - n) Sum[2^(-n + q) (I (2 f s - f v))^(n - q) ((-I) (2 f s - f v - 2 b z + 4 c s z - 2 c v z))^(1 + q) ((I (f (-2 s + v) + 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 (f (-2 s + v) + 2 (b - 2 c s + c v) z)^2)/(-4 b + 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 ( b z 2 + e ) cos v ( c z 2 + f z ) z 2 - v - 2 ( - e z n + 1 ( 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 e ( b z 2 ) n + 1 2 Γ ( n + 1 2 , - b z 2 ) - ( - b z 2 ) n + 1 2 Γ ( n + 1 2 , b z 2 ) ) ( 1 - v mod 2 \$CellContext`v 2 ) ( b 2 z 4 ) 1 2 ( - n - 1 ) + e 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]]]]] ( ( f v - 2 f s ) 2 - 4 b - 8 c s + 4 c v ( q = 0 n 2 q - n ( ( f v - 2 f s ) ) n - q ( - ( f ( v - 2 s ) + 2 ( - b - 2 c s + c v ) z ) ) q + 1 ( ( f ( v - 2 s ) + 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 , ( f ( v - 2 s ) + 2 ( - b - 2 c s + c v ) z ) 2 - 4 b - 8 c s + 4 c v ) ) ( - ( - b - 2 c s + c v ) ) - n - 1 + ( - ( f v - 2 f s ) 2 - 4 b - 8 c s + 4 c v - 2 e + π ) ( ( - b - 2 c s + c v ) ) - n - 1 q = 0 n 2 q - n ( - ( f v - 2 f s ) ) n - q ( ( f ( v - 2 s ) + 2 ( - b - 2 c s + c v ) z ) ) q + 1 ( - ( f ( v - 2 s ) + 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 , - ( f ( v - 2 s ) + 2 ( - b - 2 c s + c v ) z ) 2 - 4 b - 8 c s + 4 c v ) + f 2 ( v - 2 s ) 2 - 4 b + 8 c s - 4 c v ( - ( - b + 2 c s - c v ) ) - n - 1 q = 0 n 2 q - n ( ( 2 f s - f v ) ) n - q ( - ( 2 f s + 4 c z s - f v - 2 b z - 2 c v z ) ) q + 1 ( ( f ( v - 2 s ) + 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 , ( f ( v - 2 s ) + 2 ( b - 2 c s + c v ) z ) 2 - 4 b + 8 c s - 4 c v ) + ( - f 2 ( v - 2 s ) 2 - 4 b + 8 c s - 4 c v - 2 e + π ) ( ( - b + 2 c s - c v ) ) - n - 1 q = 0 n 2 q - n ( - ( 2 f s - f v ) ) n - q ( ( 2 f s + 4 c z s - f v - 2 b z - 2 c v z ) ) q + 1 ( - ( f ( v - 2 s ) + 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 , - ( f ( v - 2 s ) + 2 ( b - 2 c s + c v ) z ) 2 - 4 b + 8 c s - 4 c v ) ) ) /; n v + Condition z z n b z 2 e c z 2 f z v 2 -1 v -2 -1 e z n 1 Binomial v v 2 -1 2 e b z 2 n 1 2 -1 Gamma n 1 2 -1 -1 b z 2 -1 -1 b z 2 n 1 2 -1 Gamma n 1 2 -1 b z 2 1 -1 \$CellContext`v 2 b 2 z 4 1 2 -1 n -1 e s 0 v -1 2 -1 Binomial v s f v -1 2 f s 2 -4 b -1 8 c s 4 c v -1 q 0 n 2 q -1 n f v -1 2 f s n -1 q -1 f v -1 2 s 2 -1 b -1 2 c s c v z q 1 f v -1 2 s 2 -1 b -1 2 c s c v z 2 -1 b -1 2 c s c v -1 1 2 -1 q -1 Binomial n q Gamma q 1 2 -1 f v -1 2 s 2 -1 b -1 2 c s c v z 2 -4 b -1 8 c s 4 c v -1 -1 -1 b -1 2 c s c v -1 n -1 -1 f v -1 2 f s 2 -4 b -1 8 c s 4 c v -1 -1 2 e -1 b -1 2 c s c v -1 n -1 q 0 n 2 q -1 n -1 f v -1 2 f s n -1 q f v -1 2 s 2 -1 b -1 2 c s c v z q 1 -1 f v -1 2 s 2 -1 b -1 2 c s c v z 2 -1 b -1 2 c s c v -1 1 2 -1 q -1 Binomial n q Gamma q 1 2 -1 -1 f v -1 2 s 2 -1 b -1 2 c s c v z 2 -4 b -1 8 c s 4 c v -1 f 2 v -1 2 s 2 -4 b 8 c s -1 4 c v -1 -1 -1 b 2 c s -1 c v -1 n -1 q 0 n 2 q -1 n 2 f s -1 f v n -1 q -1 2 f s 4 c z s -1 f v -1 2 b z -1 2 c v z q 1 f v -1 2 s 2 b -1 2 c s c v z 2 -1 b 2 c s -1 c v -1 1 2 -1 q -1 Binomial n q Gamma q 1 2 -1 f v -1 2 s 2 b -1 2 c s c v z 2 -4 b 8 c s -1 4 c v -1 -1 f 2 v -1 2 s 2 -4 b 8 c s -1 4 c v -1 -1 2 e -1 b 2 c s -1 c v -1 n -1 q 0 n 2 q -1 n -1 2 f s -1 f v n -1 q 2 f s 4 c z s -1 f v -1 2 b z -1 2 c v z q 1 -1 f v -1 2 s 2 b -1 2 c s c v z 2 -1 b 2 c s -1 c v -1 1 2 -1 q -1 Binomial n q Gamma q 1 2 -1 -1 f v -1 2 s 2 b -1 2 c s c v z 2 -4 b 8 c s -1 4 c v -1 n v SuperPlus [/itex]

 Rule Form

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

 2002-12-18