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 Cos

 http://functions.wolfram.com/01.07.21.2432.01

 Input Form

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

 Standard Form

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 MathML Form

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

 Rule Form

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

 2002-12-18