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 Cos

 http://functions.wolfram.com/01.07.21.2720.01

 Input Form

 Integrate[z^n E^(p Sqrt[z]) Sin[b Sqrt[z]] Cos[c z]^v, z] == (I z^(1 + n) Binomial[v, v/2] (Gamma[2 (1 + n), ((-I) b - p) Sqrt[z]]/ (((-I) b - p) Sqrt[z])^(2 (1 + n)) - Gamma[2 (1 + n), I (b + I p) Sqrt[z]]/(I (b + I p) Sqrt[z])^ (2 (1 + n))) (1 - Mod[v, 2]))/2^v + (1/c^2) (I 2^(-2 - 2 n - v) Sum[(1/(-2 s + v)^2) (Binomial[v, s] (Sum[(-1)^(-h + k) 4^k (I b + p)^(-h - k + 2 n) (I b + p + 2 I c (2 s - v) Sqrt[z])^(h + k) ((I (I b + p + 2 I c (2 s - v) Sqrt[z])^2)/(c (2 s - v)))^ ((1/2) (-1 - h - k)) Binomial[k, h] Binomial[n, k] ((I b + p) (I b + p + 2 I c (2 s - v) Sqrt[z]) Gamma[(1/2) (1 + h + k), (I (I b + p + 2 I c (2 s - v) Sqrt[z])^2)/ (c (8 s - 4 v))] + 2 I c (2 s - v) Sqrt[(I (I b + p + 2 I c (2 s - v) Sqrt[z])^2)/(c (2 s - v))] Gamma[(1/2) (2 + h + k), (I (I b + p + 2 I c (2 s - v) Sqrt[z])^ 2)/(c (8 s - 4 v))]), {k, 0, n}, {h, 0, k}]/ (E^((I b + p)^2/(8 I c s - 4 I c v)) (I c (2 s - v))^(2 n)) + (E^((I (b + I p)^2)/(4 (2 c s - c v))) (-Sum[(-1)^(-h + k) 4^k ((-I) b + p)^(-h - k + 2 n) ((-I) b + p + 2 I c (-2 s + v) Sqrt[z])^(h + k) ((I ((-I) b + p + 2 I c (-2 s + v) Sqrt[z])^2)/(c (-2 s + v)))^ ((1/2) (-1 - h - k)) Binomial[k, h] Binomial[n, k] ( ((-I) b + p) ((-I) b + p + 2 I c (-2 s + v) Sqrt[z]) Gamma[(1/2) (1 + h + k), (I (b + I p + 2 c (2 s - v) Sqrt[z])^2)/(c (8 s - 4 v))] + 2 I c (-2 s + v) Sqrt[(I ((-I) b + p + 2 I c (-2 s + v) Sqrt[z])^2)/ (c (-2 s + v))] Gamma[(1/2) (2 + h + k), (I (b + I p + 2 c (2 s - v) Sqrt[z])^2)/(c (8 s - 4 v))]), {k, 0, n}, {h, 0, k}] + E^((b p)/(2 c s - c v)) Sum[(-1)^(-h + k) 4^k (I b + p)^(-h - k + 2 n) (I b + p + 2 I c (-2 s + v) Sqrt[z])^(h + k) ((I (I b + p + 2 I c (-2 s + v) Sqrt[z])^2)/(c (-2 s + v)))^ ((1/2) (-1 - h - k)) Binomial[k, h] Binomial[n, k] ( (I b + p) (I b + p + 2 I c (-2 s + v) Sqrt[z]) Gamma[(1/2) (1 + h + k), (I (b - I p + 2 c (-2 s + v) Sqrt[z])^2)/(c (8 s - 4 v))] + 2 I c (-2 s + v) Sqrt[(I (I b + p + 2 I c (-2 s + v) Sqrt[z])^2)/ (c (-2 s + v))] Gamma[(1/2) (2 + h + k), (I (b - I p + 2 c (-2 s + v) Sqrt[z])^2)/(c (8 s - 4 v))]), {k, 0, n}, {h, 0, k}]))/(I c (-2 s + v))^(2 n) - Sum[(-1)^(-h + k) 4^k ((-I) b + p)^(-h - k + 2 n) ((-I) b + p + 2 I c (2 s - v) Sqrt[z])^(h + k) (-((I (b + I p + 2 c (-2 s + v) Sqrt[z])^2)/(c (2 s - v))))^ ((1/2) (-1 - h - k)) Binomial[k, h] Binomial[n, k] (((-I) b + p) ((-I) b + p + 2 I c (2 s - v) Sqrt[z]) Gamma[(1/2) (1 + h + k), -((I (b + I p + 2 c (-2 s + v) Sqrt[z])^2)/(c (8 s - 4 v)))] + 2 I c (2 s - v) Sqrt[-((I (b + I p + 2 c (-2 s + v) Sqrt[z])^2)/(c (2 s - v)))] Gamma[(1/2) (2 + h + k), -((I (b + I p + 2 c (-2 s + v) Sqrt[z])^2)/(c (8 s - 4 v)))]), {k, 0, n}, {h, 0, k}]/ (E^((I (b + I p)^2)/(4 (2 c s - c v))) (I c (2 s - v))^(2 n)))), {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 sin ( b z ) cos v ( c z ) z 2 - v ( 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]]]]] ( ( ( - b - p ) z ) - 2 ( n + 1 ) Γ ( 2 ( n + 1 ) , ( - b - p ) z ) - ( ( b + p ) z ) - 2 ( n + 1 ) Γ ( 2 ( n + 1 ) , ( b + p ) z ) ) ( 1 - v mod 2 \$CellContext`v 2 ) z n + 1 + 2 - 2 n - v - 2 c 2 s = 0 v - 1 2 1 ( v - 2 s ) 2 ( ( v s ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["v", Identity]], List[TagBox["s", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( - ( b + p ) 2 8 c s - 4 c v ( k = 0 n h = 0 k ( - 1 ) k - h 4 k ( b + p ) - h - k + 2 n ( b + p + 2 c ( 2 s - v ) z ) h + k ( ( b + p + 2 c ( 2 s - v ) z ) 2 c ( 2 s - v ) ) 1 2 ( - h - k - 1 ) ( k h ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["k", Identity]], List[TagBox["h", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( n k ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["n", Identity]], List[TagBox["k", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( ( b + p ) ( b + p + 2 c ( 2 s - v ) z ) Γ ( 1 2 ( h + k + 1 ) , ( b + p + 2 c ( 2 s - v ) z ) 2 c ( 8 s - 4 v ) ) + 2 c ( 2 s - v ) ( b + p + 2 c ( 2 s - v ) z ) 2 c ( 2 s - v ) Γ ( 1 2 ( h + k + 2 ) , ( b + p + 2 c ( 2 s - v ) z ) 2 c ( 8 s - 4 v ) ) ) ) ( c ( 2 s - v ) ) - 2 n - - ( b + p ) 2 4 ( 2 c s - c v ) ( k = 0 n h = 0 k ( - 1 ) k - h 4 k ( - b + p ) - h - k + 2 n ( - b + p + 2 c ( 2 s - v ) z ) h + k ( - ( b + p + 2 c ( v - 2 s ) z ) 2 c ( 2 s - v ) ) 1 2 ( - h - k - 1 ) ( k h ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["k", Identity]], List[TagBox["h", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( n k ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["n", Identity]], List[TagBox["k", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( ( - b + p ) ( - b + p + 2 c ( 2 s - v ) z ) Γ ( 1 2 ( h + k + 1 ) , - ( b + p + 2 c ( v - 2 s ) z ) 2 c ( 8 s - 4 v ) ) + 2 c ( 2 s - v ) - ( b + p + 2 c ( v - 2 s ) z ) 2 c ( 2 s - v ) Γ ( 1 2 ( h + k + 2 ) , - ( b + p + 2 c ( v - 2 s ) z ) 2 c ( 8 s - 4 v ) ) ) ) ( c ( 2 s - v ) ) - 2 n + ( b + p ) 2 4 ( 2 c s - c v ) ( c ( v - 2 s ) ) - 2 n ( b p 2 c s - c v k = 0 n h = 0 k ( - 1 ) k - h 4 k ( b + p ) - h - k + 2 n ( b + p + 2 c ( v - 2 s ) z ) h + k ( ( b + p + 2 c ( v - 2 s ) z ) 2 c ( v - 2 s ) ) 1 2 ( - h - k - 1 ) ( k h ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["k", Identity]], List[TagBox["h", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( n k ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["n", Identity]], List[TagBox["k", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( ( b + p ) ( b + p + 2 c ( v - 2 s ) z ) Γ ( 1 2 ( h + k + 1 ) , ( b - p + 2 c ( v - 2 s ) z ) 2 c ( 8 s - 4 v ) ) + 2 c ( v - 2 s ) ( b + p + 2 c ( v - 2 s ) z ) 2 c ( v - 2 s ) Γ ( 1 2 ( h + k + 2 ) , ( b - p + 2 c ( v - 2 s ) z ) 2 c ( 8 s - 4 v ) ) ) - k = 0 n h = 0 k ( - 1 ) k - h 4 k ( - b + p ) - h - k + 2 n ( - b + p + 2 c ( v - 2 s ) z ) h + k ( ( - b + p + 2 c ( v - 2 s ) z ) 2 c ( v - 2 s ) ) 1 2 ( - h - k - 1 ) ( k h ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["k", Identity]], List[TagBox["h", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( n k ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["n", Identity]], List[TagBox["k", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( ( - b + p ) ( - b + p + 2 c ( v - 2 s ) z ) Γ ( 1 2 ( h + k + 1 ) , ( b + p + 2 c ( 2 s - v ) z ) 2 c ( 8 s - 4 v ) ) + 2 c ( v - 2 s ) ( - b + p + 2 c ( v - 2 s ) z ) 2 c ( v - 2 s ) Γ ( 1 2 ( h + k + 2 ) , ( b + p + 2 c ( 2 s - v ) z ) 2 c ( 8 s - 4 v ) ) ) ) ) ) /; v + n Condition z z n p z 1 2 b z 1 2 c z v 2 -1 v Binomial v v 2 -1 -1 b -1 p z 1 2 -2 n 1 Gamma 2 n 1 -1 b -1 p z 1 2 -1 b p z 1 2 -2 n 1 Gamma 2 n 1 b p z 1 2 1 -1 \$CellContext`v 2 z n 1 2 -2 n -1 v -2 c 2 -1 s 0 v -1 2 -1 1 v -1 2 s 2 -1 Binomial v s -1 b p 2 8 c s -1 4 c v -1 h 0 k k 0 n -1 k -1 h 4 k b p -1 h -1 k 2 n b p 2 c 2 s -1 v z 1 2 h k b p 2 c 2 s -1 v z 1 2 2 c 2 s -1 v -1 1 2 -1 h -1 k -1 Binomial k h Binomial n k b p b p 2 c 2 s -1 v z 1 2 Gamma 1 2 h k 1 b p 2 c 2 s -1 v z 1 2 2 c 8 s -1 4 v -1 2 c 2 s -1 v b p 2 c 2 s -1 v z 1 2 2 c 2 s -1 v -1 1 2 Gamma 1 2 h k 2 b p 2 c 2 s -1 v z 1 2 2 c 8 s -1 4 v -1 c 2 s -1 v -2 n -1 -1 b p 2 4 2 c s -1 c v -1 h 0 k k 0 n -1 k -1 h 4 k -1 b p -1 h -1 k 2 n -1 b p 2 c 2 s -1 v z 1 2 h k -1 b p 2 c v -1 2 s z 1 2 2 c 2 s -1 v -1 1 2 -1 h -1 k -1 Binomial k h Binomial n k -1 b p -1 b p 2 c 2 s -1 v z 1 2 Gamma 1 2 h k 1 -1 b p 2 c v -1 2 s z 1 2 2 c 8 s -1 4 v -1 2 c 2 s -1 v -1 b p 2 c v -1 2 s z 1 2 2 c 2 s -1 v -1 1 2 Gamma 1 2 h k 2 -1 b p 2 c v -1 2 s z 1 2 2 c 8 s -1 4 v -1 c 2 s -1 v -2 n b p 2 4 2 c s -1 c v -1 c v -1 2 s -2 n b p 2 c s -1 c v -1 h 0 k k 0 n -1 k -1 h 4 k b p -1 h -1 k 2 n b p 2 c v -1 2 s z 1 2 h k b p 2 c v -1 2 s z 1 2 2 c v -1 2 s -1 1 2 -1 h -1 k -1 Binomial k h Binomial n k b p b p 2 c v -1 2 s z 1 2 Gamma 1 2 h k 1 b -1 p 2 c v -1 2 s z 1 2 2 c 8 s -1 4 v -1 2 c v -1 2 s b p 2 c v -1 2 s z 1 2 2 c v -1 2 s -1 1 2 Gamma 1 2 h k 2 b -1 p 2 c v -1 2 s z 1 2 2 c 8 s -1 4 v -1 -1 h 0 k k 0 n -1 k -1 h 4 k -1 b p -1 h -1 k 2 n -1 b p 2 c v -1 2 s z 1 2 h k -1 b p 2 c v -1 2 s z 1 2 2 c v -1 2 s -1 1 2 -1 h -1 k -1 Binomial k h Binomial n k -1 b p -1 b p 2 c v -1 2 s z 1 2 Gamma 1 2 h k 1 b p 2 c 2 s -1 v z 1 2 2 c 8 s -1 4 v -1 2 c v -1 2 s -1 b p 2 c v -1 2 s z 1 2 2 c v -1 2 s -1 1 2 Gamma 1 2 h k 2 b p 2 c 2 s -1 v z 1 2 2 c 8 s -1 4 v -1 v SuperPlus n [/itex]

 Rule Form

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

 2002-12-18