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 Cos

 http://functions.wolfram.com/01.07.21.2490.01

 Input Form

 Integrate[z^n Sin[b Sqrt[z] + d z] Cos[c Sqrt[z] + f z + g]^v, z] == 2^(-2 - v) (((-I) (-(1/4))^n Binomial[v, v/2] (1 - Mod[v, 2]) (Sum[(-1)^(-h + k) 4^k (I b)^(-h - k + 2 n) (I (b + 2 d Sqrt[z]))^ (h + k) (-((I (b + 2 d Sqrt[z])^2)/d))^((1/2) (-1 - h - k)) Binomial[k, h] Binomial[n, k] (b (b + 2 d Sqrt[z]) Gamma[(1/2) (1 + h + k), -((I (b + 2 d Sqrt[z])^2)/(4 d))] - 2 I d Sqrt[-((I (b + 2 d Sqrt[z])^2)/d)] Gamma[(1/2) (2 + h + k), -((I (b + 2 d Sqrt[z])^2)/(4 d))]), {k, 0, n}, {h, 0, k}] - E^((I b^2)/(2 d)) Sum[(-1)^(-h + k) 4^k ((-I) b)^(-h - k + 2 n) ((-I) (b + 2 d Sqrt[z]))^(h + k) ((I (b + 2 d Sqrt[z])^2)/d)^ ((1/2) (-1 - h - k)) Binomial[k, h] Binomial[n, k] (b (b + 2 d Sqrt[z]) Gamma[(1/2) (1 + h + k), (I (b + 2 d Sqrt[z])^2)/(4 d)] + 2 I d Sqrt[(I (b + 2 d Sqrt[z])^2)/d] Gamma[(1/2) (2 + h + k), (I (b + 2 d Sqrt[z])^2)/(4 d)]), {k, 0, n}, {h, 0, k}]))/ (d^(2 (1 + n)) E^((I b^2)/(4 d))) + (I Sum[(Binomial[v, s] ((E^(I (g (4 s - 2 v) - (b + 2 c s - c v)^2/ (4 d + 8 f s - 4 f v))) Sum[(-1)^(-h + k) 4^k (I (b + 2 c s - c v))^(-h - k + 2 n) (I (b + c (2 s - v) + 2 (d + 2 f s - f v) Sqrt[z]))^(h + k) (-((I (b + 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] ((-b - 2 c s + c v) (b + c (2 s - v) + 2 (d + 2 f s - f v) Sqrt[z]) Gamma[(1/2) (1 + h + k), -((I (b + 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 (b + 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 (b + 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 - (E^((I (b + 2 c s - c v)^2)/(4 d + 8 f s - 4 f v)) Sum[(-1)^(-h + k) 4^k ((-I) (b + 2 c s - c v))^(-h - k + 2 n) ((-I) (b + c (2 s - v) + 2 (d + 2 f s - f v) Sqrt[z]))^(h + k) ((I (b + 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] ((-b - 2 c s + c v) (b + c (2 s - v) + 2 (d + 2 f s - f v) Sqrt[z]) Gamma[(1/2) (1 + h + k), (I (b + 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 (b + 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 (b + 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 (b + c (-2 s + v)))^(-h - k + 2 n) (I (b + c (-2 s + v) + 2 (d + f (-2 s + v)) Sqrt[z]))^(h + k) (-((I (b + 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] ((-b - c (-2 s + v)) (b + c (-2 s + v) + 2 (d + f (-2 s + v)) Sqrt[z]) Gamma[(1/2) (1 + h + k), -((I (b + 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 (b + 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 (b + 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^((1/2) I (8 g s - 4 g v + (b + c (-2 s + v))^2/ (d + f (-2 s + v)))) (I (d + f (-2 s + v)))^(2 n) Sum[(-1)^(-h + k) 4^k ((-I) (b + c (-2 s + v)))^(-h - k + 2 n) ((-I) (b + c (-2 s + v) + 2 (d + f (-2 s + v)) Sqrt[z]))^ (h + k) ((I (b + 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] ((-b - c (-2 s + v)) (b + c (-2 s + v) + 2 (d + f (-2 s + v)) Sqrt[z]) Gamma[(1/2) (1 + h + k), (I (b + 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 (b + 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 (b + 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 (b + c (-2 s + v))^2)/(4 (d + f (-2 s + v))))))/ E^(I (2 g s - g 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 ( z b + d z ) cos v ( z c + f z + g ) z 2 - v - 2 ( 4 - n s = 0 v - 1 2 - ( 2 g s - g v ) ( 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 ( ( g ( 4 s - 2 v ) - ( b + 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 ( ( b + 2 c s - c v ) ) - h - k + 2 n ( ( b + c ( 2 s - v ) + 2 ( d + 2 f s - f v ) z ) ) h + k ( - ( b + 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]]]]] ( ( - b - 2 c s + c v ) ( b + c ( 2 s - v ) + 2 ( d + 2 f s - f v ) z ) Γ ( 1 2 ( h + k + 1 ) , - ( b + c ( 2 s - v ) + 2 ( d + 2 f s - f v ) z ) 2 4 d + 8 f s - 4 f v ) + 2 - ( b + 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 ) , - ( b + c ( 2 s - v ) + 2 ( d + 2 f s - f v ) z ) 2 4 d + 8 f s - 4 f v ) ) ) + - ( b + c ( 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 ( ( b + c ( v - 2 s ) ) ) - h - k + 2 n ( ( b + c ( v - 2 s ) + 2 ( d + f ( v - 2 s ) ) z ) ) h + k ( - ( b + 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]]]]] ( ( - b - c ( v - 2 s ) ) ( b + c ( v - 2 s ) + 2 ( d + f ( v - 2 s ) ) z ) Γ ( 1 2 ( h + k + 1 ) , - ( b + c ( v - 2 s ) + 2 ( d + f ( v - 2 s ) ) z ) 2 4 ( d + f ( v - 2 s ) ) ) + 2 ( d + f ( v - 2 s ) ) Γ ( 1 2 ( h + k + 2 ) , - ( b + c ( v - 2 s ) + 2 ( d + f ( v - 2 s ) ) z ) 2 4 ( d + f ( v - 2 s ) ) ) - ( b + c ( v - 2 s ) + 2 ( d + f ( v - 2 s ) ) z ) 2 d + f ( v - 2 s ) ) - 1 2 ( ( b + c ( v - 2 s ) ) 2 d + f ( v - 2 s ) + 8 g s - 4 g v ) ( ( d + f ( v - 2 s ) ) ) 2 n k = 0 n h = 0 k ( - 1 ) k - h 4 k ( - ( b + c ( v - 2 s ) ) ) - h - k + 2 n ( - ( b + c ( v - 2 s ) + 2 ( d + f ( v - 2 s ) ) z ) ) h + k ( ( b + 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]]]]] ( ( - b - c ( v - 2 s ) ) ( b + c ( v - 2 s ) + 2 ( d + f ( v - 2 s ) ) z ) Γ ( 1 2 ( h + k + 1 ) , ( b + c ( v - 2 s ) + 2 ( d + f ( v - 2 s ) ) z ) 2 4 ( d + f ( v - 2 s ) ) ) - 2 ( d + f ( v - 2 s ) ) ( b + c ( v - 2 s ) + 2 ( d + f ( v - 2 s ) ) z ) 2 d + f ( v - 2 s ) Γ ( 1 2 ( h + k + 2 ) , ( b + 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 ( ( b + 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 ( - ( b + 2 c s - c v ) ) - h - k + 2 n ( - ( b + c ( 2 s - v ) + 2 ( d + 2 f s - f v ) z ) ) h + k ( ( b + 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]]]]] ( ( - b - 2 c s + c v ) ( b + c ( 2 s - v ) + 2 ( d + 2 f s - f v ) z ) Γ ( 1 2 ( h + k + 1 ) , ( b + 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 ) ( b + 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 ) , ( b + c ( 2 s - v ) + 2 ( d + 2 f s - f v ) z ) 2 4 d + 8 f s - 4 f v ) ) ) ) - ( - 1 4 ) n d - 2 ( n + 1 ) - b 2 4 d ( 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]]]]] ( 1 - v mod 2 \$CellContext`v 2 ) ( k = 0 n h = 0 k ( - 1 ) k - h 4 k ( b ) - h - k + 2 n ( ( b + 2 d z ) ) h + k ( - ( b + 2 d z ) 2 d ) 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 ( b + 2 d z ) Γ ( 1 2 ( h + k + 1 ) , - ( b + 2 d z ) 2 4 d ) - 2 d - ( b + 2 d z ) 2 d Γ ( 1 2 ( h + k + 2 ) , - ( b + 2 d z ) 2 4 d ) ) - b 2 2 d k = 0 n h = 0 k ( - 1 ) k - h 4 k ( - b ) - h - k + 2 n ( - ( b + 2 d z ) ) h + k ( ( b + 2 d z ) 2 d ) 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 ( b + 2 d z ) Γ ( 1 2 ( h + k + 1 ) , ( b + 2 d z ) 2 4 d ) + 2 ( b + 2 d z ) 2 d d Γ ( 1 2 ( h + k + 2 ) , ( b + 2 d z ) 2 4 d ) ) ) ) /; n v + Condition z z n z 1 2 b d z z 1 2 c f z g v 2 -1 v -2 4 -1 n s 0 v -1 2 -1 -1 2 g s -1 g v Binomial v s 1 d 2 f s -1 f v 2 -1 g 4 s -1 2 v -1 b 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 b 2 c s -1 c v -1 h -1 k 2 n b c 2 s -1 v 2 d 2 f s -1 f v z 1 2 h k -1 b 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 -1 b -1 2 c s c v b c 2 s -1 v 2 d 2 f s -1 f v z 1 2 Gamma 1 2 h k 1 -1 b 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 b 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 b 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 b c 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 b c v -1 2 s -1 h -1 k 2 n b c v -1 2 s 2 d f v -1 2 s z 1 2 h k -1 b 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 b -1 c v -1 2 s b c v -1 2 s 2 d f v -1 2 s z 1 2 Gamma 1 2 h k 1 -1 b c v -1 2 s 2 d f v -1 2 s z 1 2 2 4 d f v -1 2 s -1 2 d f v -1 2 s Gamma 1 2 h k 2 -1 b 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 b 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 1 2 b c v -1 2 s 2 d f v -1 2 s -1 8 g s -1 4 g v d f v -1 2 s 2 n h 0 k k 0 n -1 k -1 h 4 k -1 b c v -1 2 s -1 h -1 k 2 n -1 b c v -1 2 s 2 d f v -1 2 s z 1 2 h k b 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 b -1 c v -1 2 s b c v -1 2 s 2 d f v -1 2 s z 1 2 Gamma 1 2 h k 1 b 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 b 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 b 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 b 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 b 2 c s -1 c v -1 h -1 k 2 n -1 b c 2 s -1 v 2 d 2 f s -1 f v z 1 2 h k b 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 -1 b -1 2 c s c v b c 2 s -1 v 2 d 2 f s -1 f v z 1 2 Gamma 1 2 h k 1 b 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 b 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 b 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 -1 1 4 n d -2 n 1 -1 b 2 4 d -1 Binomial v v 2 -1 1 -1 \$CellContext`v 2 h 0 k k 0 n -1 k -1 h 4 k b -1 h -1 k 2 n b 2 d z 1 2 h k -1 b 2 d z 1 2 2 d -1 1 2 -1 h -1 k -1 Binomial k h Binomial n k b b 2 d z 1 2 Gamma 1 2 h k 1 -1 b 2 d z 1 2 2 4 d -1 -1 2 d -1 b 2 d z 1 2 2 d -1 1 2 Gamma 1 2 h k 2 -1 b 2 d z 1 2 2 4 d -1 -1 b 2 2 d -1 h 0 k k 0 n -1 k -1 h 4 k -1 b -1 h -1 k 2 n -1 b 2 d z 1 2 h k b 2 d z 1 2 2 d -1 1 2 -1 h -1 k -1 Binomial k h Binomial n k b b 2 d z 1 2 Gamma 1 2 h k 1 b 2 d z 1 2 2 4 d -1 2 b 2 d z 1 2 2 d -1 1 2 d Gamma 1 2 h k 2 b 2 d z 1 2 2 4 d -1 n v SuperPlus [/itex]

 Rule Form

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

 2002-12-18