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 Cos

 http://functions.wolfram.com/01.07.21.2727.01

 Input Form

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

 Standard Form

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 MathML Form

 z n p z 2 sin ( b z ) cos v ( c z 2 ) z - 2 - v - 2 p - n - 1 ( ( 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]]]]] ( 1 p - 2 c s + c v ( b 2 4 p - 8 c s + 4 c v q = 0 n 2 q - n ( b ) n - q ( p - 2 c s + c v ) - n - 1 2 ( - ( b + 2 p z + 4 c s z - 2 c v z ) ) q + 1 ( ( b + 2 p z + 4 c s z - 2 c v z ) 2 p - 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 , ( b + 2 p z + 4 c s z - 2 c v z ) 2 4 ( p - 2 c s + c v ) ) ) + 1 p + 2 c s - c v ( b 2 4 p + 8 c s - 4 c v q = 0 n 2 q - n ( b ) n - q ( p + 2 c s - c v ) - n - 1 2 ( - ( b + 2 p z - 4 c s z + 2 c v z ) ) q + 1 ( ( b + 2 p z - 4 c s z + 2 c v z ) 2 p + 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 , ( b + 2 p z - 4 c s z + 2 c v z ) 2 4 ( p + 2 c s - c v ) ) ) - 1 p + 2 c s - c v ( b 2 4 p + 8 c s - 4 c v q = 0 n 2 q - n ( - b ) n - q ( p + 2 c s - c v ) - n - 1 2 ( ( b - 2 p z + 4 c s z - 2 c v z ) 2 p + 2 c s - c v ) 1 2 ( - q - 1 ) ( b + 2 ( p + 2 c s - c v ) z ) 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 , ( b - 2 p z + 4 c s z - 2 c v z ) 2 4 ( p + 2 c s - c v ) ) ) - 1 p - 2 c s + c v ( b 2 4 p - 8 c s + 4 c v q = 0 n 2 q - n ( - b ) n - q ( p - 2 c s + c v ) - n - 1 2 ( ( b - 2 p z - 4 c s z + 2 c v z ) 2 p - 2 c s + c v ) 1 2 ( - q - 1 ) ( b + 2 ( p - 2 c s + c v ) z ) 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 , ( b - 2 p z - 4 c s z + 2 c v z ) 2 4 ( p - 2 c s + c v ) ) ) ) ) p n + 1 - b 2 4 p ( 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 ( b ) n - q ( ( b + 2 p z ) 2 p ) 1 2 ( - q - 1 ) ( - b + 2 p z ) 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 , ( b + 2 p z ) 2 4 p ) + b 2 4 p ( 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 ( - b ) n - q ( ( b - 2 p z ) 2 p ) 1 2 ( - q - 1 ) ( b + 2 p z ) 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 , ( b - 2 p z ) 2 4 p ) ) /; v + n Condition z z n p z 2 b z c z 2 v -1 2 -1 v -2 p -1 n -1 s 0 v -1 2 -1 Binomial v s 1 p -1 2 c s c v 1 2 -1 b 2 4 p -1 8 c s 4 c v -1 q 0 n 2 q -1 n b n -1 q p -1 2 c s c v -1 n -1 1 2 -1 b 2 p z 4 c s z -1 2 c v z q 1 b 2 p z 4 c s z -1 2 c v z 2 p -1 2 c s c v -1 1 2 -1 q -1 Binomial n q Gamma q 1 2 -1 b 2 p z 4 c s z -1 2 c v z 2 4 p -1 2 c s c v -1 1 p 2 c s -1 c v 1 2 -1 b 2 4 p 8 c s -1 4 c v -1 q 0 n 2 q -1 n b n -1 q p 2 c s -1 c v -1 n -1 1 2 -1 b 2 p z -1 4 c s z 2 c v z q 1 b 2 p z -1 4 c s z 2 c v z 2 p 2 c s -1 c v -1 1 2 -1 q -1 Binomial n q Gamma q 1 2 -1 b 2 p z -1 4 c s z 2 c v z 2 4 p 2 c s -1 c v -1 -1 1 p 2 c s -1 c v 1 2 -1 b 2 4 p 8 c s -1 4 c v -1 q 0 n 2 q -1 n -1 b n -1 q p 2 c s -1 c v -1 n -1 1 2 b -1 2 p z 4 c s z -1 2 c v z 2 p 2 c s -1 c v -1 1 2 -1 q -1 b 2 p 2 c s -1 c v z q 1 Binomial n q Gamma q 1 2 -1 b -1 2 p z 4 c s z -1 2 c v z 2 4 p 2 c s -1 c v -1 -1 1 p -1 2 c s c v 1 2 -1 b 2 4 p -1 8 c s 4 c v -1 q 0 n 2 q -1 n -1 b n -1 q p -1 2 c s c v -1 n -1 1 2 b -1 2 p z -1 4 c s z 2 c v z 2 p -1 2 c s c v -1 1 2 -1 q -1 b 2 p -1 2 c s c v z q 1 Binomial n q Gamma q 1 2 -1 b -1 2 p z -1 4 c s z 2 c v z 2 4 p -1 2 c s c v -1 p n 1 -1 b 2 4 p -1 Binomial v v 2 -1 \$CellContext`v 2 -1 q 0 n 2 q -1 n b n -1 q b 2 p z 2 p -1 1 2 -1 q -1 -1 b 2 p z q 1 Binomial n q Gamma q 1 2 -1 b 2 p z 2 4 p -1 b 2 4 p -1 Binomial v v 2 -1 \$CellContext`v 2 -1 q 0 n 2 q -1 n -1 b n -1 q b -1 2 p z 2 p -1 1 2 -1 q -1 b 2 p z q 1 Binomial n q Gamma q 1 2 -1 b -1 2 p z 2 4 p -1 v SuperPlus n [/itex]

 Rule Form

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

 2002-12-18