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 Cos

 http://functions.wolfram.com/01.07.21.2435.01

 Input Form

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