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 Cos

 http://functions.wolfram.com/01.07.21.2508.01

 Input Form

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

 Standard Form

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 MathML Form

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

 Rule Form

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

 2002-12-18