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 Cosh

 http://functions.wolfram.com/01.20.21.3976.01

 Input Form

 Integrate[z^n E^(p z) Cos[b z^2] Cosh[c z]^v, z] == 2^(-2 - v) ((1/Sqrt[b^2]) (E^((I p^2)/(4 b)) Binomial[v, v/2] (-1 + Mod[v, 2]) ((Sqrt[I b] Sum[2^(-n + q) ((-I) b)^(-(1/2) - n) (-p)^(n - q) (p - 2 I b z)^(1 + q) (-((I (p - 2 I b z)^2)/b))^ ((1/2) (-1 - q)) Binomial[n, q] Gamma[(1 + q)/2, -((I (p - 2 I b z)^2)/(4 b))], {q, 0, n}])/E^((I p^2)/(2 b)) + Sqrt[(-I) b] Sum[2^(-n + q) (I b)^(-(1/2) - n) (-p)^(n - q) (p + 2 I b z)^(1 + q) ((I (p + 2 I b z)^2)/b)^((1/2) (-1 - q)) Binomial[n, q] Gamma[(1 + q)/2, (I (p + 2 I b z)^2)/(4 b)], {q, 0, n}])) + (1/((-I) b)^(3/2)) Sum[(Binomial[v, s] (I b E^((I c p (-2 s + v))/b) Sum[2^(-n + q) ((-I) b)^(-(1/2) - n) (-p + c (-2 s + v))^(n - q) (p + c (2 s - v) - 2 I b z)^(1 + q) (-((I (p + c (2 s - v) - 2 I b z)^2)/b))^((1/2) (-1 - q)) Binomial[n, q] Gamma[(1 + q)/2, -((I (p + c (2 s - v) - 2 I b z)^ 2)/(4 b))], {q, 0, n}] + I b Sum[2^(-n + q) ((-I) b)^(-(1/2) - n) (-p + 2 c s - c v)^(n - q) (p + c (-2 s + v) - 2 I b z)^(1 + q) (-((I (p + c (-2 s + v) - 2 I b z)^2)/b))^((1/2) (-1 - q)) Binomial[n, q] Gamma[(1 + q)/2, -((I (p + c (-2 s + v) - 2 I b z)^ 2)/(4 b))], {q, 0, n}] + Sqrt[(-I) b] Sqrt[I b] E^((I (p^2 + c^2 (-2 s + v)^2))/(2 b)) (Sum[2^(-n + q) (I b)^(-(1/2) - n) (-p + c (-2 s + v))^(n - q) (p + 2 c s - c v + 2 I b z)^(1 + q) ((I (p + 2 c s - c v + 2 I b z)^2)/b)^((1/2) (-1 - q)) Binomial[n, q] Gamma[(1 + q)/2, (I (p + 2 c s - c v + 2 I b z)^ 2)/(4 b)], {q, 0, n}] + Sum[2^(-n + q) (I b)^(-(1/2) - n) (-p + 2 c s - c v)^(n - q) (p + c (-2 s + v) + 2 I b z)^(1 + q) ((I (p + c (-2 s + v) + 2 I b z)^2)/b)^((1/2) (-1 - q)) Binomial[n, q] Gamma[(1 + q)/2, (I (p - 2 c s + c v + 2 I b z)^ 2)/(4 b)], {q, 0, n}]/E^((I c p (2 s - v))/b))))/ E^((I (p + c (-2 s + v))^2)/(4 b)), {s, 0, Floor[(1/2) (-1 + v)]}]) /; Element[v, Integers] && v > 0 && Element[n, Integers] && n >= 0

 Standard Form

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 MathML Form

 z n p z cos ( b z 2 ) cosh v ( c z ) z 2 - v - 2 ( 1 b 2 ( p 2 4 b ( v v 2 ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["v", Identity, Rule[Editable, True]]], List[TagBox[FractionBox["v", "2"], Identity, Rule[Editable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False]] ( v mod 2 \$CellContext`v 2 - 1 ) ( - b q = 0 n 2 q - n ( b ) - n - 1 2 ( - p ) n - q ( p + 2 b z ) q + 1 ( ( p + 2 b z ) 2 b ) 1 2 ( - q - 1 ) ( n q ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["n", Identity, Rule[Editable, True]]], List[TagBox["q", Identity, Rule[Editable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False]] Γ ( q + 1 2 , ( p + 2 b z ) 2 4 b ) + b - p 2 2 b q = 0 n 2 q - n ( - b ) - n - 1 2 ( - p ) n - q ( p - 2 b z ) q + 1 ( - ( p - 2 b z ) 2 b ) 1 2 ( - q - 1 ) ( n q ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["n", Identity, Rule[Editable, True]]], List[TagBox["q", Identity, Rule[Editable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False]] Γ ( q + 1 2 , - ( p - 2 b z ) 2 4 b ) ) ) + 1 ( - b ) 3 / 2 ( s = 0 v - 1 2 - ( p + c ( v - 2 s ) ) 2 4 b ( v s ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["v", Identity, Rule[Editable, True]]], List[TagBox["s", Identity, Rule[Editable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False]] ( - b b ( p 2 + c 2 ( v - 2 s ) 2 ) 2 b ( - c p ( 2 s - v ) b q = 0 n 2 q - n ( b ) - n - 1 2 ( - p + 2 c s - c v ) n - q ( p + c ( v - 2 s ) + 2 b z ) q + 1 ( ( p + c ( v - 2 s ) + 2 b z ) 2 b ) 1 2 ( - q - 1 ) ( n q ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["n", Identity, Rule[Editable, True]]], List[TagBox["q", Identity, Rule[Editable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False]] Γ ( q + 1 2 , ( p - 2 c s + c v + 2 b z ) 2 4 b ) + q = 0 n 2 q - n ( b ) - n - 1 2 ( c ( v - 2 s ) - p ) n - q ( p + 2 c s - c v + 2 b z ) q + 1 ( ( p + 2 c s - c v + 2 b z ) 2 b ) 1 2 ( - q - 1 ) ( n q ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["n", Identity, Rule[Editable, True]]], List[TagBox["q", Identity, Rule[Editable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False]] Γ ( q + 1 2 , ( p + 2 c s - c v + 2 b z ) 2 4 b ) ) + b c p ( v - 2 s ) b q = 0 n 2 q - n ( - b ) - n - 1 2 ( c ( v - 2 s ) - p ) n - q ( p + c ( 2 s - v ) - 2 b z ) q + 1 ( - ( p + c ( 2 s - v ) - 2 b z ) 2 b ) 1 2 ( - q - 1 ) ( n q ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["n", Identity, Rule[Editable, True]]], List[TagBox["q", Identity, Rule[Editable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False]] Γ ( q + 1 2 , - ( p + c ( 2 s - v ) - 2 b z ) 2 4 b ) + b q = 0 n 2 q - n ( - b ) - n - 1 2 ( - p + 2 c s - c v ) n - q ( p + c ( v - 2 s ) - 2 b z ) q + 1 ( - ( p + c ( v - 2 s ) - 2 b z ) 2 b ) 1 2 ( - q - 1 ) ( n q ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["n", Identity, Rule[Editable, True]]], List[TagBox["q", Identity, Rule[Editable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False]] Γ ( q + 1 2 , - ( p + c ( v - 2 s ) - 2 b z ) 2 4 b ) ) ) ) /; v + n Condition z z n p z b z 2 c z v 2 -1 v -2 1 b 2 1 2 -1 p 2 4 b -1 Binomial v v 2 -1 \$CellContext`v 2 -1 -1 b 1 2 q 0 n 2 q -1 n b -1 n -1 1 2 -1 p n -1 q p 2 b z q 1 p 2 b z 2 b -1 1 2 -1 q -1 Binomial n q Gamma q 1 2 -1 p 2 b z 2 4 b -1 b 1 2 -1 p 2 2 b -1 q 0 n 2 q -1 n -1 b -1 n -1 1 2 -1 p n -1 q p -1 2 b z q 1 -1 p -1 2 b z 2 b -1 1 2 -1 q -1 Binomial n q Gamma q 1 2 -1 -1 p -1 2 b z 2 4 b -1 1 -1 b 3 2 -1 s 0 v -1 2 -1 -1 p c v -1 2 s 2 4 b -1 Binomial v s -1 b 1 2 b 1 2 p 2 c 2 v -1 2 s 2 2 b -1 -1 c p 2 s -1 v b -1 q 0 n 2 q -1 n b -1 n -1 1 2 -1 p 2 c s -1 c v n -1 q p c v -1 2 s 2 b z q 1 p c v -1 2 s 2 b z 2 b -1 1 2 -1 q -1 Binomial n q Gamma q 1 2 -1 p -1 2 c s c v 2 b z 2 4 b -1 q 0 n 2 q -1 n b -1 n -1 1 2 c v -1 2 s -1 p n -1 q p 2 c s -1 c v 2 b z q 1 p 2 c s -1 c v 2 b z 2 b -1 1 2 -1 q -1 Binomial n q Gamma q 1 2 -1 p 2 c s -1 c v 2 b z 2 4 b -1 b c p v -1 2 s b -1 q 0 n 2 q -1 n -1 b -1 n -1 1 2 c v -1 2 s -1 p n -1 q p c 2 s -1 v -1 2 b z q 1 -1 p c 2 s -1 v -1 2 b z 2 b -1 1 2 -1 q -1 Binomial n q Gamma q 1 2 -1 -1 p c 2 s -1 v -1 2 b z 2 4 b -1 b q 0 n 2 q -1 n -1 b -1 n -1 1 2 -1 p 2 c s -1 c v n -1 q p c v -1 2 s -1 2 b z q 1 -1 p c v -1 2 s -1 2 b z 2 b -1 1 2 -1 q -1 Binomial n q Gamma q 1 2 -1 -1 p c v -1 2 s -1 2 b z 2 4 b -1 v SuperPlus n [/itex]

 Rule Form

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

 2002-12-18