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 Cos

 http://functions.wolfram.com/01.07.21.2430.01

 Input Form

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

 Standard Form

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 MathML Form

 z n sin ( b z ) cos v ( c z ) z - 2 - 2 n - v - 2 ( s = 0 v - 1 2 b 2 8 c s - 4 c v ( c 2 ( v - 2 s ) 2 ) - 2 n - 1 ( v s ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["v", Identity]], List[TagBox["s", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( - k = 0 n h = 0 k ( - 1 ) k - h 4 k ( - b ) - h - k + 2 n ( ( b + 2 c ( 2 s - v ) z ) 2 c ( 2 s - v ) ) 1 2 ( - h - k - 1 ) ( - ( b - 2 c ( v - 2 s ) z ) ) h + k ( 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 ( 2 s - v ) z ) Γ ( 1 2 ( h + k + 1 ) , ( b + 2 c ( 2 s - v ) z ) 2 c ( 8 s - 4 v ) ) + 2 c ( 2 s - v ) ( b + 2 c ( 2 s - v ) z ) 2 c ( 2 s - v ) Γ ( 1 2 ( h + k + 2 ) , ( b + 2 c ( 2 s - v ) z ) 2 c ( 8 s - 4 v ) ) ) ( c ( 2 s - v ) ) 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 ( 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 ( b + 2 c ( v - 2 s ) z ) Γ ( 1 2 ( h + k + 1 ) , ( b + 2 c ( v - 2 s ) z ) 2 c ( 8 s - 4 v ) ) + 2 c ( 2 s - v ) ( b + 2 c ( v - 2 s ) z ) 2 c ( 2 s - v ) Γ ( 1 2 ( h + k + 2 ) , ( b + 2 c ( v - 2 s ) z ) 2 c ( 8 s - 4 v ) ) ) ) ( c ( 2 s - v ) ) 2 n + b 2 2 c v - 4 c 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 ( 2 s - v ) z ) ) h + k ( - ( b + 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 ( b + 2 c ( 2 s - v ) z ) Γ ( 1 2 ( h + k + 1 ) , - ( b + 2 c ( 2 s - v ) z ) 2 c ( 8 s - 4 v ) ) + 2 c - ( b + 2 c ( 2 s - v ) z ) 2 c ( 2 s - v ) ( v - 2 s ) Γ ( 1 2 ( h + k + 2 ) , - ( b + 2 c ( 2 s - v ) z ) 2 c ( 8 s - 4 v ) ) ) - b 2 2 c v - 4 c 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 ( 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 ( b + 2 c ( v - 2 s ) z ) Γ ( 1 2 ( h + k + 1 ) , - ( b + 2 c ( v - 2 s ) z ) 2 c ( 8 s - 4 v ) ) - 2 c ( 2 s - v ) - ( b + 2 c ( v - 2 s ) z ) 2 c ( 2 s - v ) Γ ( 1 2 ( h + k + 2 ) , - ( b + 2 c ( v - 2 s ) z ) 2 c ( 8 s - 4 v ) ) ) ) - 4 n + 1 b - 4 n - 2 ( 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 ) 2 n Γ ( 2 ( n + 1 ) , - b z ) - ( - b ) 2 n Γ ( 2 ( n + 1 ) , b z ) ) ( v mod 2 \$CellContext`v 2 - 1 ) ) /; n v + Condition z z n b z 1 2 c z v -1 2 -2 n -1 v -2 s 0 v -1 2 -1 b 2 8 c s -1 4 c v -1 c 2 v -1 2 s 2 -2 n -1 Binomial v s -1 h 0 k k 0 n -1 k -1 h 4 k -1 b -1 h -1 k 2 n b 2 c 2 s -1 v z 1 2 2 c 2 s -1 v -1 1 2 -1 h -1 k -1 -1 b -1 2 c v -1 2 s z 1 2 h k Binomial k h Binomial n k b b 2 c 2 s -1 v z 1 2 Gamma 1 2 h k 1 b 2 c 2 s -1 v z 1 2 2 c 8 s -1 4 v -1 2 c 2 s -1 v b 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 2 c 2 s -1 v z 1 2 2 c 8 s -1 4 v -1 c 2 s -1 v 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 b 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 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 c 8 s -1 4 v -1 2 c 2 s -1 v b 2 c v -1 2 s z 1 2 2 c 2 s -1 v -1 1 2 Gamma 1 2 h k 2 b 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 2 2 c v -1 4 c s -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 2 s -1 v z 1 2 h k -1 b 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 b 2 c 2 s -1 v z 1 2 Gamma 1 2 h k 1 -1 b 2 c 2 s -1 v z 1 2 2 c 8 s -1 4 v -1 2 c -1 b 2 c 2 s -1 v z 1 2 2 c 2 s -1 v -1 1 2 v -1 2 s Gamma 1 2 h k 2 -1 b 2 c 2 s -1 v z 1 2 2 c 8 s -1 4 v -1 -1 b 2 2 c v -1 4 c 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 -1 b 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 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 c 8 s -1 4 v -1 -1 2 c 2 s -1 v -1 b 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 2 c v -1 2 s z 1 2 2 c 8 s -1 4 v -1 -1 4 n 1 b -4 n -2 Binomial v v 2 -1 b 2 n Gamma 2 n 1 -1 b z 1 2 -1 -1 b 2 n Gamma 2 n 1 b z 1 2 \$CellContext`v 2 -1 n v SuperPlus [/itex]

 Rule Form

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

 2002-12-18