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 Cosh

 http://functions.wolfram.com/01.20.21.3921.01

 Input Form

 Integrate[z^n E^(p Sqrt[z]) Sin[b z] Cosh[c Sqrt[z]]^v, z] == I 2^(-2 - 2 n - v) (-1)^(n - 1) b^(-2 - 2 n) Binomial[v, v/2] (1 - Mod[v, 2]) (Sum[(-1)^(-h + j) 4^j p^(-h - j + 2 n) (p - 2 I b Sqrt[z])^(h + j) (-((I (p - 2 I b Sqrt[z])^2)/b))^ ((1/2) (-1 - h - j)) Binomial[j, h] Binomial[n, j] (p (p - 2 I b Sqrt[z]) Gamma[(1/2) (1 + h + j), -((I (p - 2 I b Sqrt[z])^2)/(4 b))] - 2 I b Sqrt[-((I (p - 2 I b Sqrt[z])^2)/b)] Gamma[(1/2) (2 + h + j), -((I (p - 2 I b Sqrt[z])^2)/(4 b))]), {j, 0, n}, {h, 0, j}]/ E^((I p^2)/(4 b)) - E^((I p^2)/(4 b)) Sum[(-1)^(-h + j) 4^j p^(-h - j + 2 n) (p + 2 I b Sqrt[z])^(h + j) ((I (p + 2 I b Sqrt[z])^2)/b)^((1/2) (-1 - h - j)) Binomial[j, h] Binomial[n, j] (p (p + 2 I b Sqrt[z]) Gamma[(1/2) (1 + h + j), (I (p + 2 I b Sqrt[z])^2)/(4 b)] + 2 I b Sqrt[(I (p + 2 I b Sqrt[z])^2)/b] Gamma[(1/2) (2 + h + j), (I (p + 2 I b Sqrt[z])^2)/(4 b)]), {j, 0, n}, {h, 0, j}]) + I (-1)^(n - 1) 2^(-2 - 2 n - v) b^(-2 - 2 n) Sum[Binomial[v, k] (Sum[(-1)^(-h + j) 4^j (p - c (-2 k + v))^ (-h - j + 2 n) (p - c (-2 k + v) - 2 I b Sqrt[z])^(h + j) (-((I (p - c (-2 k + v) - 2 I b Sqrt[z])^2)/b))^ ((1/2) (-1 - h - j)) Binomial[j, h] Binomial[n, j] ((p - c (-2 k + v)) (p - c (-2 k + v) - 2 I b Sqrt[z]) Gamma[(1/2) (1 + h + j), -((I (p - c (-2 k + v) - 2 I b Sqrt[z])^ 2)/(4 b))] - 2 I b Sqrt[-((I (p - c (-2 k + v) - 2 I b Sqrt[z])^2)/b)] Gamma[(1/2) (2 + h + j), -((I (p - c (-2 k + v) - 2 I b Sqrt[z])^2)/(4 b))]), {j, 0, n}, {h, 0, j}]/E^((I (p - c (-2 k + v))^2)/(4 b)) + Sum[(-1)^(-h + j) 4^j (p + c (-2 k + v))^(-h - j + 2 n) (p + c (-2 k + v) - 2 I b Sqrt[z])^(h + j) (-((I (p + c (-2 k + v) - 2 I b Sqrt[z])^2)/b))^ ((1/2) (-1 - h - j)) Binomial[j, h] Binomial[n, j] ((p + c (-2 k + v)) (p + c (-2 k + v) - 2 I b Sqrt[z]) Gamma[(1/2) (1 + h + j), -((I (p + c (-2 k + v) - 2 I b Sqrt[z])^ 2)/(4 b))] - 2 I b Sqrt[-((I (p + c (-2 k + v) - 2 I b Sqrt[z])^2)/b)] Gamma[(1/2) (2 + h + j), -((I (p + c (-2 k + v) - 2 I b Sqrt[z])^2)/(4 b))]), {j, 0, n}, {h, 0, j}]/E^((I (p + c (-2 k + v))^2)/(4 b)) - E^((I (p - c (-2 k + v))^2)/(4 b)) Sum[(-1)^(-h + j) 4^j (p - c (-2 k + v))^(-h - j + 2 n) (p - c (-2 k + v) + 2 I b Sqrt[z])^(h + j) ((I (p - c (-2 k + v) + 2 I b Sqrt[z])^2)/ b)^((1/2) (-1 - h - j)) Binomial[j, h] Binomial[n, j] ((p - c (-2 k + v)) (p - c (-2 k + v) + 2 I b Sqrt[z]) Gamma[(1/2) (1 + h + j), (I (p - c (-2 k + v) + 2 I b Sqrt[z])^2)/ (4 b)] + 2 I b Sqrt[(I (p - c (-2 k + v) + 2 I b Sqrt[z])^2)/b] Gamma[(1/2) (2 + h + j), (I (p - c (-2 k + v) + 2 I b Sqrt[z])^2)/ (4 b)]), {j, 0, n}, {h, 0, j}] - E^((I (p + c (-2 k + v))^2)/(4 b)) Sum[(-1)^(-h + j) 4^j (p + c (-2 k + v))^(-h - j + 2 n) (p + c (-2 k + v) + 2 I b Sqrt[z])^(h + j) ((I (p + c (-2 k + v) + 2 I b Sqrt[z])^2)/ b)^((1/2) (-1 - h - j)) Binomial[j, h] Binomial[n, j] ((p + c (-2 k + v)) (p + c (-2 k + v) + 2 I b Sqrt[z]) Gamma[(1/2) (1 + h + j), (I (p + c (-2 k + v) + 2 I b Sqrt[z])^2)/ (4 b)] + 2 I b Sqrt[(I (p + c (-2 k + v) + 2 I b Sqrt[z])^2)/b] Gamma[(1/2) (2 + h + j), (I (p + c (-2 k + v) + 2 I b Sqrt[z])^2)/ (4 b)]), {j, 0, n}, {h, 0, j}]), {k, 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 p z sin ( b z ) cosh v ( c z ) z ( - 1 ) n - 1 2 - 2 n - v - 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 - v mod 2 \$CellContext`v 2 ) b - 2 n - 2 ( - p 2 4 b j = 0 n h = 0 j ( - 1 ) j - h 4 j p - h - j + 2 n ( p - 2 b z ) h + j ( - ( p - 2 b z ) 2 b ) 1 2 ( - h - j - 1 ) ( j h ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["j", Identity]], List[TagBox["h", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( n j ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["n", Identity]], List[TagBox["j", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( p ( p - 2 b z ) Γ ( 1 2 ( h + j + 1 ) , - ( p - 2 b z ) 2 4 b ) - 2 b - ( p - 2 b z ) 2 b Γ ( 1 2 ( h + j + 2 ) , - ( p - 2 b z ) 2 4 b ) ) - p 2 4 b j = 0 n h = 0 j ( - 1 ) j - h 4 j p - h - j + 2 n ( p + 2 b z ) h + j ( ( p + 2 b z ) 2 b ) 1 2 ( - h - j - 1 ) ( j h ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["j", Identity]], List[TagBox["h", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( n j ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["n", Identity]], List[TagBox["j", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( p ( p + 2 b z ) Γ ( 1 2 ( h + j + 1 ) , ( p + 2 b z ) 2 4 b ) + 2 ( p + 2 b z ) 2 b b Γ ( 1 2 ( h + j + 2 ) , ( p + 2 b z ) 2 4 b ) ) ) + ( - 1 ) n - 1 2 - 2 n - v - 2 b - 2 n - 2 k = 0 v - 1 2 ( v k ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["v", Identity]], List[TagBox["k", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( - ( p + c ( v - 2 k ) ) 2 4 b j = 0 n h = 0 j ( - 1 ) j - h 4 j ( p + c ( v - 2 k ) ) - h - j + 2 n ( p + c ( v - 2 k ) + 2 b z ) h + j ( ( p + c ( v - 2 k ) + 2 b z ) 2 b ) 1 2 ( - h - j - 1 ) ( j h ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["j", Identity]], List[TagBox["h", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( n j ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["n", Identity]], List[TagBox["j", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( ( p + c ( v - 2 k ) ) ( p + c ( v - 2 k ) + 2 b z ) Γ ( 1 2 ( h + j + 1 ) , ( p + c ( v - 2 k ) + 2 b z ) 2 4 b ) + 2 ( p + c ( v - 2 k ) + 2 b z ) 2 b b Γ ( 1 2 ( h + j + 2 ) , ( p + c ( v - 2 k ) + 2 b z ) 2 4 b ) ) - ( p - c ( v - 2 k ) ) 2 4 b j = 0 n h = 0 j ( - 1 ) j - h 4 j ( p - c ( v - 2 k ) ) - h - j + 2 n ( p - c ( v - 2 k ) + 2 b z ) h + j ( ( p - c ( v - 2 k ) + 2 b z ) 2 b ) 1 2 ( - h - j - 1 ) ( j h ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["j", Identity]], List[TagBox["h", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( n j ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["n", Identity]], List[TagBox["j", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( ( p - c ( v - 2 k ) ) ( p - c ( v - 2 k ) + 2 b z ) Γ ( 1 2 ( h + j + 1 ) , ( p - c ( v - 2 k ) + 2 b z ) 2 4 b ) + 2 ( p - c ( v - 2 k ) + 2 b z ) 2 b b Γ ( 1 2 ( h + j + 2 ) , ( p - c ( v - 2 k ) + 2 b z ) 2 4 b ) ) + - ( p + c ( v - 2 k ) ) 2 4 b j = 0 n h = 0 j ( - 1 ) j - h 4 j ( p + c ( v - 2 k ) ) - h - j + 2 n ( p + c ( v - 2 k ) - 2 b z ) h + j ( - ( p + c ( v - 2 k ) - 2 b z ) 2 b ) 1 2 ( - h - j - 1 ) ( j h ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["j", Identity]], List[TagBox["h", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( n j ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["n", Identity]], List[TagBox["j", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( ( p + c ( v - 2 k ) ) ( p + c ( v - 2 k ) - 2 b z ) Γ ( 1 2 ( h + j + 1 ) , - ( p + c ( v - 2 k ) - 2 b z ) 2 4 b ) - 2 b - ( p + c ( v - 2 k ) - 2 b z ) 2 b Γ ( 1 2 ( h + j + 2 ) , - ( p + c ( v - 2 k ) - 2 b z ) 2 4 b ) ) + - ( p - c ( v - 2 k ) ) 2 4 b j = 0 n h = 0 j ( - 1 ) j - h 4 j ( p - c ( v - 2 k ) ) - h - j + 2 n ( p - c ( v - 2 k ) - 2 b z ) h + j ( - ( p - c ( v - 2 k ) - 2 b z ) 2 b ) 1 2 ( - h - j - 1 ) ( j h ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["j", Identity]], List[TagBox["h", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( n j ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["n", Identity]], List[TagBox["j", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( ( p - c ( v - 2 k ) ) ( p - c ( v - 2 k ) - 2 b z ) Γ ( 1 2 ( h + j + 1 ) , - ( p - c ( v - 2 k ) - 2 b z ) 2 4 b ) - 2 b - ( p - c ( v - 2 k ) - 2 b z ) 2 b Γ ( 1 2 ( h + j + 2 ) , - ( p - c ( v - 2 k ) - 2 b z ) 2 4 b ) ) ) /; n v + Condition z z n p z 1 2 b z c z 1 2 v -1 n -1 2 -2 n -1 v -2 Binomial v v 2 -1 1 -1 \$CellContext`v 2 b -2 n -2 -1 p 2 4 b -1 h 0 j j 0 n -1 j -1 h 4 j p -1 h -1 j 2 n p -1 2 b z 1 2 h j -1 p -1 2 b z 1 2 2 b -1 1 2 -1 h -1 j -1 Binomial j h Binomial n j p p -1 2 b z 1 2 Gamma 1 2 h j 1 -1 p -1 2 b z 1 2 2 4 b -1 -1 2 b -1 p -1 2 b z 1 2 2 b -1 1 2 Gamma 1 2 h j 2 -1 p -1 2 b z 1 2 2 4 b -1 -1 p 2 4 b -1 h 0 j j 0 n -1 j -1 h 4 j p -1 h -1 j 2 n p 2 b z 1 2 h j p 2 b z 1 2 2 b -1 1 2 -1 h -1 j -1 Binomial j h Binomial n j p p 2 b z 1 2 Gamma 1 2 h j 1 p 2 b z 1 2 2 4 b -1 2 p 2 b z 1 2 2 b -1 1 2 b Gamma 1 2 h j 2 p 2 b z 1 2 2 4 b -1 -1 n -1 2 -2 n -1 v -2 b -2 n -2 k 0 v -1 2 -1 Binomial v k -1 p c v -1 2 k 2 4 b -1 h 0 j j 0 n -1 j -1 h 4 j p c v -1 2 k -1 h -1 j 2 n p c v -1 2 k 2 b z 1 2 h j p c v -1 2 k 2 b z 1 2 2 b -1 1 2 -1 h -1 j -1 Binomial j h Binomial n j p c v -1 2 k p c v -1 2 k 2 b z 1 2 Gamma 1 2 h j 1 p c v -1 2 k 2 b z 1 2 2 4 b -1 2 p c v -1 2 k 2 b z 1 2 2 b -1 1 2 b Gamma 1 2 h j 2 p c v -1 2 k 2 b z 1 2 2 4 b -1 -1 p -1 c v -1 2 k 2 4 b -1 h 0 j j 0 n -1 j -1 h 4 j p -1 c v -1 2 k -1 h -1 j 2 n p -1 c v -1 2 k 2 b z 1 2 h j p -1 c v -1 2 k 2 b z 1 2 2 b -1 1 2 -1 h -1 j -1 Binomial j h Binomial n j p -1 c v -1 2 k p -1 c v -1 2 k 2 b z 1 2 Gamma 1 2 h j 1 p -1 c v -1 2 k 2 b z 1 2 2 4 b -1 2 p -1 c v -1 2 k 2 b z 1 2 2 b -1 1 2 b Gamma 1 2 h j 2 p -1 c v -1 2 k 2 b z 1 2 2 4 b -1 -1 p c v -1 2 k 2 4 b -1 h 0 j j 0 n -1 j -1 h 4 j p c v -1 2 k -1 h -1 j 2 n p c v -1 2 k -1 2 b z 1 2 h j -1 p c v -1 2 k -1 2 b z 1 2 2 b -1 1 2 -1 h -1 j -1 Binomial j h Binomial n j p c v -1 2 k p c v -1 2 k -1 2 b z 1 2 Gamma 1 2 h j 1 -1 p c v -1 2 k -1 2 b z 1 2 2 4 b -1 -1 2 b -1 p c v -1 2 k -1 2 b z 1 2 2 b -1 1 2 Gamma 1 2 h j 2 -1 p c v -1 2 k -1 2 b z 1 2 2 4 b -1 -1 p -1 c v -1 2 k 2 4 b -1 h 0 j j 0 n -1 j -1 h 4 j p -1 c v -1 2 k -1 h -1 j 2 n p -1 c v -1 2 k -1 2 b z 1 2 h j -1 p -1 c v -1 2 k -1 2 b z 1 2 2 b -1 1 2 -1 h -1 j -1 Binomial j h Binomial n j p -1 c v -1 2 k p -1 c v -1 2 k -1 2 b z 1 2 Gamma 1 2 h j 1 -1 p -1 c v -1 2 k -1 2 b z 1 2 2 4 b -1 -1 2 b -1 p -1 c v -1 2 k -1 2 b z 1 2 2 b -1 1 2 Gamma 1 2 h j 2 -1 p -1 c v -1 2 k -1 2 b z 1 2 2 4 b -1 n v SuperPlus [/itex]

 Rule Form

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

 2002-12-18