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 Cos

 http://functions.wolfram.com/01.07.21.2488.01

 Input Form

 Integrate[z^n Sin[b Sqrt[z] + e] Cos[c Sqrt[z] + f z + g]^v, z] == 2^(-2 - v) ((4 (-1)^n 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^((1/2) I (2 e + Pi))) - (I (-(1/4))^n Sum[(Binomial[v, s] (((-E^(2 I e)) Sum[(-1)^(-h + k) 4^k (I (b + c (-2 s + v)))^ (-h - k + 2 n) (I (b + c (-2 s + v) + 2 f (-2 s + v) Sqrt[z]))^(h + k) (-((I (b + c (-2 s + v) + 2 f (-2 s + v) Sqrt[z])^2)/(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 f (-2 s + v) Sqrt[z]) Gamma[ (1/2) (1 + h + k), -((I (b + c (-2 s + v) + 2 f (-2 s + v) Sqrt[z])^2)/(4 f (-2 s + v)))] + 2 I f (-2 s + v) Sqrt[-((I (b + c (-2 s + v) + 2 f (-2 s + v) Sqrt[z])^2)/ (f (-2 s + v)))] Gamma[(1/2) (2 + h + k), -((I (b + c (-2 s + v) + 2 f (-2 s + v) Sqrt[z])^2)/ (4 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/(f (-2 s + v)))) Sum[(-1)^(-h + k) 4^k ((-I) (b + c (-2 s + v)))^(-h - k + 2 n) ((-I) (b + c (-2 s + v) + 2 f (-2 s + v) Sqrt[z]))^(h + k) ((I (b + c (-2 s + v) + 2 f (-2 s + v) Sqrt[z])^2)/ (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 f (-2 s + v) Sqrt[z]) Gamma[(1/2) (1 + h + k), (I (b + c (-2 s + v) + 2 f (-2 s + v) Sqrt[z])^2)/(4 f (-2 s + v))] - 2 I f (-2 s + v) Sqrt[(I (b + c (-2 s + v) + 2 f (-2 s + v) Sqrt[z])^2)/(f (-2 s + v))] Gamma[(1/2) (2 + h + k), (I (b + c (-2 s + v) + 2 f (-2 s + v) Sqrt[z])^2)/ (4 f (-2 s + v))]), {k, 0, n}, {h, 0, k}])/ E^((I (b + c (-2 s + v))^2)/(4 f (-2 s + v))) - E^(I (2 e + g (4 s - 2 v) - (b + 2 c s - c v)^2/(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 (2 f s - f v) Sqrt[z]))^(h + k) (-((I (b + c (2 s - v) + 2 (2 f s - f v) Sqrt[z])^2)/ (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 (2 f s - f v) Sqrt[z]) Gamma[(1/2) (1 + h + k), -((I (b + c (2 s - v) + 2 (2 f s - f v) Sqrt[z])^2)/ (8 f s - 4 f v))] + 2 I (2 f s - f v) Sqrt[ -((I (b + c (2 s - v) + 2 (2 f s - f v) Sqrt[z])^2)/ (2 f s - f v))] Gamma[(1/2) (2 + h + k), -((I (b + c (2 s - v) + 2 (2 f s - f v) Sqrt[z])^2)/ (8 f s - 4 f v))]), {k, 0, n}, {h, 0, k}] + E^((I (b + 2 c s - c v)^2)/(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 (2 f s - f v) Sqrt[z]))^(h + k) ((I (b + c (2 s - v) + 2 (2 f s - f v) Sqrt[z])^2)/(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 (2 f s - f v) Sqrt[z]) Gamma[(1/2) (1 + h + k), (I (b + c (2 s - v) + 2 (2 f s - f v) Sqrt[z])^2)/(8 f s - 4 f v)] - 2 I (2 f s - f v) Sqrt[ (I (b + c (2 s - v) + 2 (2 f s - f v) Sqrt[z])^2)/ (2 f s - f v)] Gamma[(1/2) (2 + h + k), (I (b + c (2 s - v) + 2 (2 f s - f v) Sqrt[z])^2)/ (8 f s - 4 f v)]), {k, 0, n}, {h, 0, k}]))/ (E^(I (e + 2 g s - g v)) (-2 s + v)^(2 (1 + n))), {s, 0, Floor[(1/2) (-1 + v)]}])/f^(2 (1 + n))) /; 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 ( z c + f z + g ) z 2 - v - 2 ( 4 ( - 1 ) n b - 2 ( n + 1 ) - 1 2 ( 2 e + π ) ( 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 ) - ( - 1 4 ) n f - 2 ( n + 1 ) s = 0 v - 1 2 - ( e + 2 g s - g v ) ( v - 2 s ) - 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]]]]] ( - ( b + c ( v - 2 s ) ) 2 4 f ( v - 2 s ) ( 1 2 ( ( b + c ( v - 2 s ) ) 2 f ( v - 2 s ) + 8 g s - 4 g v ) 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 f ( v - 2 s ) z ) ) h + k ( ( b + c ( v - 2 s ) + 2 f ( v - 2 s ) z ) 2 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 f ( v - 2 s ) z ) Γ ( 1 2 ( h + k + 1 ) , ( b + c ( v - 2 s ) + 2 f ( v - 2 s ) z ) 2 4 f ( v - 2 s ) ) - 2 f ( v - 2 s ) ( b + c ( v - 2 s ) + 2 f ( v - 2 s ) z ) 2 f ( v - 2 s ) Γ ( 1 2 ( h + k + 2 ) , ( b + c ( v - 2 s ) + 2 f ( v - 2 s ) z ) 2 4 f ( v - 2 s ) ) ) - 2 e 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 f ( v - 2 s ) z ) ) h + k ( - ( b + c ( v - 2 s ) + 2 f ( v - 2 s ) z ) 2 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 f ( v - 2 s ) z ) Γ ( 1 2 ( h + k + 1 ) , - ( b + c ( v - 2 s ) + 2 f ( v - 2 s ) z ) 2 4 f ( v - 2 s ) ) + 2 f ( v - 2 s ) - ( b + c ( v - 2 s ) + 2 f ( v - 2 s ) z ) 2 f ( v - 2 s ) Γ ( 1 2 ( h + k + 2 ) , - ( b + c ( v - 2 s ) + 2 f ( v - 2 s ) z ) 2 4 f ( v - 2 s ) ) ) ) + ( b + 2 c s - c v ) 2 8 f s - 4 f v 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 ( 2 f s - f v ) z ) ) h + k ( ( b + c ( 2 s - v ) + 2 ( 2 f s - f v ) z ) 2 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 ( 2 f s - f v ) z ) Γ ( 1 2 ( h + k + 1 ) , ( b + c ( 2 s - v ) + 2 ( 2 f s - f v ) z ) 2 8 f s - 4 f v ) - 2 ( 2 f s - f v ) ( b + c ( 2 s - v ) + 2 ( 2 f s - f v ) z ) 2 2 f s - f v Γ ( 1 2 ( h + k + 2 ) , ( b + c ( 2 s - v ) + 2 ( 2 f s - f v ) z ) 2 8 f s - 4 f v ) ) - ( - ( b + 2 c s - c v ) 2 8 f s - 4 f v + 2 e + g ( 4 s - 2 v ) ) 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 ( 2 f s - f v ) z ) ) h + k ( - ( b + c ( 2 s - v ) + 2 ( 2 f s - f v ) z ) 2 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 ( 2 f s - f v ) z ) Γ ( 1 2 ( h + k + 1 ) , - ( b + c ( 2 s - v ) + 2 ( 2 f s - f v ) z ) 2 8 f s - 4 f v ) + 2 - ( b + c ( 2 s - v ) + 2 ( 2 f s - f v ) z ) 2 2 f s - f v ( 2 f s - f v ) Γ ( 1 2 ( h + k + 2 ) , - ( b + c ( 2 s - v ) + 2 ( 2 f s - f v ) z ) 2 8 f s - 4 f v ) ) ) ) /; n v + Condition z z n z 1 2 b e z 1 2 c f z g v 2 -1 v -2 4 -1 n b -2 n 1 -1 1 2 2 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 -1 -1 1 4 n f -2 n 1 s 0 v -1 2 -1 -1 e 2 g s -1 g v v -1 2 s -2 n 1 Binomial v s -1 b c v -1 2 s 2 4 f v -1 2 s -1 1 2 b c v -1 2 s 2 f v -1 2 s -1 8 g s -1 4 g v 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 f v -1 2 s z 1 2 h k b c v -1 2 s 2 f v -1 2 s z 1 2 2 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 f v -1 2 s z 1 2 Gamma 1 2 h k 1 b c v -1 2 s 2 f v -1 2 s z 1 2 2 4 f v -1 2 s -1 -1 2 f v -1 2 s b c v -1 2 s 2 f v -1 2 s z 1 2 2 f v -1 2 s -1 1 2 Gamma 1 2 h k 2 b c v -1 2 s 2 f v -1 2 s z 1 2 2 4 f v -1 2 s -1 -1 2 e 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 f v -1 2 s z 1 2 h k -1 b c v -1 2 s 2 f v -1 2 s z 1 2 2 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 f v -1 2 s z 1 2 Gamma 1 2 h k 1 -1 b c v -1 2 s 2 f v -1 2 s z 1 2 2 4 f v -1 2 s -1 2 f v -1 2 s -1 b c v -1 2 s 2 f v -1 2 s z 1 2 2 f v -1 2 s -1 1 2 Gamma 1 2 h k 2 -1 b c v -1 2 s 2 f v -1 2 s z 1 2 2 4 f v -1 2 s -1 b 2 c s -1 c v 2 8 f s -1 4 f v -1 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 2 f s -1 f v z 1 2 h k b c 2 s -1 v 2 2 f s -1 f v z 1 2 2 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 2 f s -1 f v z 1 2 Gamma 1 2 h k 1 b c 2 s -1 v 2 2 f s -1 f v z 1 2 2 8 f s -1 4 f v -1 -1 2 2 f s -1 f v b c 2 s -1 v 2 2 f s -1 f v z 1 2 2 2 f s -1 f v -1 1 2 Gamma 1 2 h k 2 b c 2 s -1 v 2 2 f s -1 f v z 1 2 2 8 f s -1 4 f v -1 -1 -1 b 2 c s -1 c v 2 8 f s -1 4 f v -1 2 e g 4 s -1 2 v 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 2 f s -1 f v z 1 2 h k -1 b c 2 s -1 v 2 2 f s -1 f v z 1 2 2 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 2 f s -1 f v z 1 2 Gamma 1 2 h k 1 -1 b c 2 s -1 v 2 2 f s -1 f v z 1 2 2 8 f s -1 4 f v -1 2 -1 b c 2 s -1 v 2 2 f s -1 f v z 1 2 2 2 f s -1 f v -1 1 2 2 f s -1 f v Gamma 1 2 h k 2 -1 b c 2 s -1 v 2 2 f s -1 f v z 1 2 2 8 f s -1 4 f v -1 n v SuperPlus [/itex]

 Rule Form

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

 2002-12-18