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 Cos

 http://functions.wolfram.com/01.07.21.2433.01

 Input Form

 Integrate[z^n Sin[b z^2 + d z] Cos[c z]^v, z] == 2^(-2 - v) (b^(-1 - 2 n) E^((I d^2)/(4 b)) Binomial[v, v/2] (-1 + Mod[v, 2]) ((((b ((-I) b)^n)/(d + 2 b z)) Sum[2^(-n + q) ((-I) d)^(n - q) (I (d + 2 b z))^q (-((I (d + 2 b z)^2)/b))^((1 - q)/2) Binomial[n, q] Gamma[(1 + q)/2, -((I (d + 2 b z)^2)/(4 b))], {q, 0, n}])/ E^((I d^2)/(2 b)) + I (I b)^n Sum[2^(-n + q) (I d)^(n - q) ((-I) (d + 2 b z))^q (d + 2 b z) ((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 b^(-1 - 2 n) Sum[E^((I (-d^2 - b Pi - c^2 (-2 s + v)^2))/(2 b)) Binomial[v, s] ((-((-I) b)^n) E^((I (d + c (-2 s + v))^2)/(4 b)) Sum[2^(-n + q) (I (-d - 2 c s + c v))^(n - q) ((-I) (-d - 2 c s + c v - 2 b z))^(1 + q) (-((I (-d - 2 c s + c v - 2 b z)^2)/b))^((1/2) (-1 - q)) Binomial[n, q] Gamma[(1 + q)/2, -((I (-d - 2 c s + c v - 2 b z)^2)/ (4 b))], {q, 0, n}] + ((I b)^n Sum[2^(-n + q) (I (d + 2 c s - c v))^(n - q) (I (-d - 2 c s + c v - 2 b z))^(1 + q) ((I (-d - 2 c s + c v - 2 b z)^2)/b)^((1/2) (-1 - q)) Binomial[n, q] Gamma[(1 + q)/2, (I (-d - 2 c s + c v - 2 b z)^2)/ (4 b)], {q, 0, n}])/E^((I (-3 d^2 - 4 b Pi - 2 c d (2 s - v) - 3 c^2 (-2 s + v)^2))/(4 b)) - ((-I) b)^n E^((I (-d - 2 c s + c v)^2)/(4 b)) Sum[2^(-n + q) (I (-d + 2 c s - c v))^(n - q) (I (d + c (-2 s + v) + 2 b z))^(1 + q) (-((I (d + c (-2 s + v) + 2 b z)^2)/b))^((1/2) (-1 - q)) Binomial[n, q] Gamma[(1 + q)/2, -((I (d + c (-2 s + v) + 2 b z)^2)/ (4 b))], {q, 0, n}] + ((I b)^n Sum[2^(-n + q) ((-I) (-d + 2 c s - c v))^(n - q) (I (-d + 2 c s - c v - 2 b z))^(1 + q) ((I (d + c (-2 s + v) + 2 b z)^2)/b)^((1/2) (-1 - q)) Binomial[n, q] Gamma[(1 + q)/2, (I (d + c (-2 s + v) + 2 b z)^2)/ (4 b)], {q, 0, n}])/E^((I (-3 d^2 - 4 b Pi + 2 c d (2 s - v) - 3 c^2 (-2 s + v)^2))/(4 b))), {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 + d z ) cos v ( c z ) z 2 - v - 2 ( b - 2 n - 1 d 2 4 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 ) ( ( b ) n q = 0 n 2 q - n ( d ) n - q ( - ( d + 2 b z ) ) q ( d + 2 b z ) ( ( 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 ) + 1 d + 2 b z ( - d 2 2 b ( b ( - b ) n ) q = 0 n 2 q - n ( - d ) n - q ( ( d + 2 b z ) ) q ( - ( d + 2 b z ) 2 b ) 1 - q 2 ( 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 ) ) ) - b - 2 n - 1 s = 0 v - 1 2 ( - d 2 - c 2 ( v - 2 s ) 2 - b π ) 2 b ( v s ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["v", Identity]], List[TagBox["s", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( - ( - d - 2 c s + c v ) 2 4 b ( q = 0 n 2 q - n ( ( - d + 2 c s - c v ) ) n - q ( ( d + c ( v - 2 s ) + 2 b z ) ) q + 1 ( - ( d + c ( v - 2 s ) + 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 + c ( v - 2 s ) + 2 b z ) 2 4 b ) ) ( - b ) n + ( b ) n - ( - 3 d 2 + 2 c ( 2 s - v ) d - 3 c 2 ( v - 2 s ) 2 - 4 b π ) 4 b q = 0 n 2 q - n ( - ( - d + 2 c s - c v ) ) n - q ( ( - d + 2 c s - c v - 2 b z ) ) q + 1 ( ( d + c ( v - 2 s ) + 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 + c ( v - 2 s ) + 2 b z ) 2 4 b ) + ( b ) n - ( - 3 d 2 - 2 c ( 2 s - v ) d - 3 c 2 ( v - 2 s ) 2 - 4 b π ) 4 b q = 0 n 2 q - n ( ( d + 2 c s - c v ) ) n - q ( ( - d - 2 c s + c v - 2 b z ) ) q + 1 ( ( - d - 2 c s + c v - 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 c s + c v - 2 b z ) 2 4 b ) - ( - b ) n ( d + c ( v - 2 s ) ) 2 4 b q = 0 n 2 q - n ( ( - d - 2 c s + c v ) ) n - q ( - ( - d - 2 c s + c v - 2 b z ) ) q + 1 ( - ( - d - 2 c s + c v - 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 c s + c v - 2 b z ) 2 4 b ) ) ) /; n v + Condition z z n b z 2 d z c z v 2 -1 v -2 b -2 n -1 d 2 4 b -1 Binomial v v 2 -1 \$CellContext`v 2 -1 b n q 0 n 2 q -1 n d n -1 q -1 d 2 b z q d 2 b z 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 1 d 2 b z -1 -1 d 2 2 b -1 b -1 b n q 0 n 2 q -1 n -1 d n -1 q d 2 b z q -1 d 2 b z 2 b -1 1 -1 q 2 -1 Binomial n q Gamma q 1 2 -1 -1 d 2 b z 2 4 b -1 -1 b -2 n -1 s 0 v -1 2 -1 -1 d 2 -1 c 2 v -1 2 s 2 -1 b 2 b -1 Binomial v s -1 -1 d -1 2 c s c v 2 4 b -1 q 0 n 2 q -1 n -1 d 2 c s -1 c v n -1 q d c v -1 2 s 2 b z q 1 -1 d c v -1 2 s 2 b z 2 b -1 1 2 -1 q -1 Binomial n q Gamma q 1 2 -1 -1 d c v -1 2 s 2 b z 2 4 b -1 -1 b n b n -1 -3 d 2 2 c 2 s -1 v d -1 3 c 2 v -1 2 s 2 -1 4 b 4 b -1 q 0 n 2 q -1 n -1 -1 d 2 c s -1 c v n -1 q -1 d 2 c s -1 c v -1 2 b z q 1 d c v -1 2 s 2 b z 2 b -1 1 2 -1 q -1 Binomial n q Gamma q 1 2 -1 d c v -1 2 s 2 b z 2 4 b -1 b n -1 -3 d 2 -1 2 c 2 s -1 v d -1 3 c 2 v -1 2 s 2 -1 4 b 4 b -1 q 0 n 2 q -1 n d 2 c s -1 c v n -1 q -1 d -1 2 c s c v -1 2 b z q 1 -1 d -1 2 c s c v -1 2 b z 2 b -1 1 2 -1 q -1 Binomial n q Gamma q 1 2 -1 -1 d -1 2 c s c v -1 2 b z 2 4 b -1 -1 -1 b n d c v -1 2 s 2 4 b -1 q 0 n 2 q -1 n -1 d -1 2 c s c v n -1 q -1 -1 d -1 2 c s c v -1 2 b z q 1 -1 -1 d -1 2 c s c v -1 2 b z 2 b -1 1 2 -1 q -1 Binomial n q Gamma q 1 2 -1 -1 -1 d -1 2 c s c v -1 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