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 Cos

 http://functions.wolfram.com/01.07.21.2470.01

 Input Form

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

 Standard Form

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 MathML Form

 z n sin ( d z ) cos v ( z c + f z ) z 2 - v - 2 ( 4 - n 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 ( d + 2 f s - f v ) 2 ( - ( 2 c s - c v ) 2 4 d + 8 f s - 4 f v ( ( d + 2 f s - f v ) ) - 2 n k = 0 n h = 0 k ( - 1 ) k - h 4 k ( ( 2 c s - c v ) ) - h - k + 2 n ( ( c ( 2 s - v ) + 2 ( d + 2 f s - f v ) z ) ) h + k ( - ( c ( 2 s - v ) + 2 ( d + 2 f s - f v ) z ) 2 d + 2 f s - f 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]]]]] ( ( c v - 2 c s ) ( c ( 2 s - v ) + 2 ( d + 2 f s - f v ) z ) Γ ( 1 2 ( h + k + 1 ) , - ( c ( 2 s - v ) + 2 ( d + 2 f s - f v ) z ) 2 4 d + 8 f s - 4 f v ) + 2 - ( c ( 2 s - v ) + 2 ( d + 2 f s - f v ) z ) 2 d + 2 f s - f v ( d + 2 f s - f v ) Γ ( 1 2 ( h + k + 2 ) , - ( c ( 2 s - v ) + 2 ( d + 2 f s - f v ) z ) 2 4 d + 8 f s - 4 f v ) ) ) + - c 2 ( v - 2 s ) 2 4 ( d + f ( v - 2 s ) ) ( ( d + f ( v - 2 s ) ) 2 ) - 2 n - 1 ( ( - ( d + f ( v - 2 s ) ) ) 2 n k = 0 n h = 0 k ( - 1 ) k - h 4 k ( c ( v - 2 s ) ) - h - k + 2 n ( ( c ( v - 2 s ) + 2 ( d + f ( v - 2 s ) ) z ) ) h + k ( - ( c ( v - 2 s ) + 2 ( d + f ( v - 2 s ) ) z ) 2 d + f ( 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]]]]] ( 2 ( d + f ( v - 2 s ) ) - ( c ( v - 2 s ) + 2 ( d + f ( v - 2 s ) ) z ) 2 d + f ( v - 2 s ) Γ ( 1 2 ( h + k + 2 ) , - ( c ( v - 2 s ) + 2 ( d + f ( v - 2 s ) ) z ) 2 4 ( d + f ( v - 2 s ) ) ) - c ( v - 2 s ) ( c ( v - 2 s ) + 2 ( d + f ( v - 2 s ) ) z ) Γ ( 1 2 ( h + k + 1 ) , - ( c ( v - 2 s ) + 2 ( d + f ( v - 2 s ) ) z ) 2 4 ( d + f ( v - 2 s ) ) ) ) - c 2 ( v - 2 s ) 2 2 ( d + f ( v - 2 s ) ) ( ( d + f ( v - 2 s ) ) ) 2 n k = 0 n h = 0 k ( - 1 ) k - h 4 k ( - c ( v - 2 s ) ) - h - k + 2 n ( - ( c ( v - 2 s ) + 2 ( d + f ( v - 2 s ) ) z ) ) h + k ( ( c ( v - 2 s ) + 2 ( d + f ( v - 2 s ) ) z ) 2 d + f ( 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]]]]] ( - c ( v - 2 s ) ( c ( v - 2 s ) + 2 ( d + f ( v - 2 s ) ) z ) Γ ( 1 2 ( h + k + 1 ) , ( c ( v - 2 s ) + 2 ( d + f ( v - 2 s ) ) z ) 2 4 ( d + f ( v - 2 s ) ) ) - 2 ( d + f ( v - 2 s ) ) ( c ( v - 2 s ) + 2 ( d + f ( v - 2 s ) ) z ) 2 d + f ( v - 2 s ) Γ ( 1 2 ( h + k + 2 ) , ( c ( v - 2 s ) + 2 ( d + f ( v - 2 s ) ) z ) 2 4 ( d + f ( v - 2 s ) ) ) ) ) - 1 ( d + 2 f s - f v ) 2 ( ( 2 c s - c v ) 2 4 d + 8 f s - 4 f v ( - ( d + 2 f s - f v ) ) - 2 n k = 0 n h = 0 k ( - 1 ) k - h 4 k ( - ( 2 c s - c v ) ) - h - k + 2 n ( - ( c ( 2 s - v ) + 2 ( d + 2 f s - f v ) z ) ) h + k ( ( c ( 2 s - v ) + 2 ( d + 2 f s - f v ) z ) 2 d + 2 f s - f 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]]]]] ( ( c v - 2 c s ) ( c ( 2 s - v ) + 2 ( d + 2 f s - f v ) z ) Γ ( 1 2 ( h + k + 1 ) , ( c ( 2 s - v ) + 2 ( d + 2 f s - f v ) z ) 2 4 d + 8 f s - 4 f v ) - 2 ( d + 2 f s - f v ) ( c ( 2 s - v ) + 2 ( d + 2 f s - f v ) z ) 2 d + 2 f s - f v Γ ( 1 2 ( h + k + 2 ) , ( c ( 2 s - v ) + 2 ( d + 2 f s - f v ) z ) 2 4 d + 8 f s - 4 f v ) ) ) ) - 2 d - 2 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]]]]] ( Γ ( n + 1 , d z ) ( - d ) n + ( d ) n Γ ( n + 1 , - d z ) ) ( 1 - v mod 2 \$CellContext`v 2 ) ) /; n v + Condition z z n d z z 1 2 c f z v 2 -1 v -2 4 -1 n s 0 v -1 2 -1 Binomial v s 1 d 2 f s -1 f v 2 -1 -1 2 c s -1 c v 2 4 d 8 f s -1 4 f v -1 d 2 f s -1 f v -2 n h 0 k k 0 n -1 k -1 h 4 k 2 c s -1 c v -1 h -1 k 2 n c 2 s -1 v 2 d 2 f s -1 f v z 1 2 h k -1 c 2 s -1 v 2 d 2 f s -1 f v z 1 2 2 d 2 f s -1 f v -1 1 2 -1 h -1 k -1 Binomial k h Binomial n k c v -1 2 c s c 2 s -1 v 2 d 2 f s -1 f v z 1 2 Gamma 1 2 h k 1 -1 c 2 s -1 v 2 d 2 f s -1 f v z 1 2 2 4 d 8 f s -1 4 f v -1 2 -1 c 2 s -1 v 2 d 2 f s -1 f v z 1 2 2 d 2 f s -1 f v -1 1 2 d 2 f s -1 f v Gamma 1 2 h k 2 -1 c 2 s -1 v 2 d 2 f s -1 f v z 1 2 2 4 d 8 f s -1 4 f v -1 -1 c 2 v -1 2 s 2 4 d f v -1 2 s -1 d f v -1 2 s 2 -2 n -1 -1 d f v -1 2 s 2 n h 0 k k 0 n -1 k -1 h 4 k c v -1 2 s -1 h -1 k 2 n c v -1 2 s 2 d f v -1 2 s z 1 2 h k -1 c v -1 2 s 2 d f v -1 2 s z 1 2 2 d f v -1 2 s -1 1 2 -1 h -1 k -1 Binomial k h Binomial n k 2 d f v -1 2 s -1 c v -1 2 s 2 d f v -1 2 s z 1 2 2 d f v -1 2 s -1 1 2 Gamma 1 2 h k 2 -1 c v -1 2 s 2 d f v -1 2 s z 1 2 2 4 d f v -1 2 s -1 -1 c v -1 2 s c v -1 2 s 2 d f v -1 2 s z 1 2 Gamma 1 2 h k 1 -1 c v -1 2 s 2 d f v -1 2 s z 1 2 2 4 d f v -1 2 s -1 -1 c 2 v -1 2 s 2 2 d f v -1 2 s -1 d f v -1 2 s 2 n h 0 k k 0 n -1 k -1 h 4 k -1 c v -1 2 s -1 h -1 k 2 n -1 c v -1 2 s 2 d f v -1 2 s z 1 2 h k c v -1 2 s 2 d f v -1 2 s z 1 2 2 d f v -1 2 s -1 1 2 -1 h -1 k -1 Binomial k h Binomial n k -1 c v -1 2 s c v -1 2 s 2 d f v -1 2 s z 1 2 Gamma 1 2 h k 1 c v -1 2 s 2 d f v -1 2 s z 1 2 2 4 d f v -1 2 s -1 -1 2 d f v -1 2 s c v -1 2 s 2 d f v -1 2 s z 1 2 2 d f v -1 2 s -1 1 2 Gamma 1 2 h k 2 c v -1 2 s 2 d f v -1 2 s z 1 2 2 4 d f v -1 2 s -1 -1 1 d 2 f s -1 f v 2 -1 2 c s -1 c v 2 4 d 8 f s -1 4 f v -1 -1 d 2 f s -1 f v -2 n h 0 k k 0 n -1 k -1 h 4 k -1 2 c s -1 c v -1 h -1 k 2 n -1 c 2 s -1 v 2 d 2 f s -1 f v z 1 2 h k c 2 s -1 v 2 d 2 f s -1 f v z 1 2 2 d 2 f s -1 f v -1 1 2 -1 h -1 k -1 Binomial k h Binomial n k c v -1 2 c s c 2 s -1 v 2 d 2 f s -1 f v z 1 2 Gamma 1 2 h k 1 c 2 s -1 v 2 d 2 f s -1 f v z 1 2 2 4 d 8 f s -1 4 f v -1 -1 2 d 2 f s -1 f v c 2 s -1 v 2 d 2 f s -1 f v z 1 2 2 d 2 f s -1 f v -1 1 2 Gamma 1 2 h k 2 c 2 s -1 v 2 d 2 f s -1 f v z 1 2 2 4 d 8 f s -1 4 f v -1 -1 2 d -2 n -1 Binomial v v 2 -1 Gamma n 1 d z -1 d n d n Gamma n 1 -1 d z 1 -1 \$CellContext`v 2 n v SuperPlus [/itex]

 Rule Form

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

 2002-12-18