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 Cosh

 http://functions.wolfram.com/01.20.21.1444.01

 Input Form

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

 Standard Form

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 MathML Form

 z n p z cos m ( b z 2 ) cosh ( c z 2 ) z 2 - m - 2 k = 0 m - 1 2 ( m k ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["m", Identity, Rule[Editable, True]]], List[TagBox["k", Identity, Rule[Editable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False]] ( - 1 - c + 2 b k - b m ( p 2 4 c - 8 b k + 4 b m q = 0 n 2 q - n ( - c + 2 b k - b m ) - n - 1 2 ( - p ) n - q ( p + 2 ( - c + 2 b k - b m ) z ) q + 1 ( - ( p + 2 ( - c + 2 b k - b m ) z ) 2 - c + 2 b k - b m ) 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 + 2 b k - b m ) z ) 2 - 4 c + 8 b k - 4 b m ) ) - 1 - c - 2 b k + b m ( p 2 4 c + 8 b k - 4 b m q = 0 n 2 q - n ( - c - 2 b k + b m ) - n - 1 2 ( - p ) n - q ( p + 2 ( - c - 2 b k + b m ) z ) q + 1 ( ( p + 2 ( - c - 2 b k + b m ) z ) 2 c + 2 b k - b m ) 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 - 2 b k + b m ) z ) 2 4 c + 8 b k - 4 b m ) ) - 1 c - 2 b k + b m ( p 2 - 4 c + 8 b k - 4 b m q = 0 n 2 q - n ( c - 2 b k + b m ) - n - 1 2 ( - p ) n - q ( p + 2 ( c - 2 b k + b m ) z ) q + 1 ( - ( p + 2 ( c - 2 b k + b m ) z ) 2 c - 2 b k + b m ) 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 - 2 b k + b m ) z ) 2 - 4 c + 8 b k - 4 b m ) ) - 1 c + 2 b k - b m ( p 2 - 4 c - 8 b k + 4 b m q = 0 n 2 q - n ( c + 2 b k - b m ) - n - 1 2 ( - p ) n - q ( p + 2 ( c + 2 b k - b m ) z ) q + 1 ( - ( p + 2 ( c + 2 b k - b m ) z ) 2 c + 2 b k - b m ) 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 + 2 b k - b m ) z ) 2 4 c + 8 b k - 4 b m ) ) ) - 2 - m - 2 ( m m 2 ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["m", Identity, Rule[Editable, True]]], List[TagBox[FractionBox["m", "2"], Identity, Rule[Editable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False]] ( 1 - m mod 2 \$CellContext`m 2 ) ( 1 - c ( p 2 4 c q = 0 n 2 q - n ( - c ) - n - 1 2 ( - p ) n - q ( p - 2 c z ) q + 1 ( ( p - 2 c z ) 2 c ) 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 z ) 2 4 c ) ) + 1 c ( - p 2 4 c q = 0 n 2 q - n c - n - 1 2 ( - p ) n - q ( p + 2 c z ) q + 1 ( - ( p + 2 c z ) 2 c ) 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 z ) 2 4 c ) ) ) /; m + n Condition z z n p z b z 2 m c z 2 2 -1 m -2 k 0 m -1 2 -1 Binomial m k -1 1 -1 c 2 b k -1 b m 1 2 -1 p 2 4 c -1 8 b k 4 b m -1 q 0 n 2 q -1 n -1 c 2 b k -1 b m -1 n -1 1 2 -1 p n -1 q p 2 -1 c 2 b k -1 b m z q 1 -1 p 2 -1 c 2 b k -1 b m z 2 -1 c 2 b k -1 b m -1 1 2 -1 q -1 Binomial n q Gamma q 1 2 -1 -1 p 2 -1 c 2 b k -1 b m z 2 -4 c 8 b k -1 4 b m -1 -1 1 -1 c -1 2 b k b m 1 2 -1 p 2 4 c 8 b k -1 4 b m -1 q 0 n 2 q -1 n -1 c -1 2 b k b m -1 n -1 1 2 -1 p n -1 q p 2 -1 c -1 2 b k b m z q 1 p 2 -1 c -1 2 b k b m z 2 c 2 b k -1 b m -1 1 2 -1 q -1 Binomial n q Gamma q 1 2 -1 p 2 -1 c -1 2 b k b m z 2 4 c 8 b k -1 4 b m -1 -1 1 c -1 2 b k b m 1 2 -1 p 2 -4 c 8 b k -1 4 b m -1 q 0 n 2 q -1 n c -1 2 b k b m -1 n -1 1 2 -1 p n -1 q p 2 c -1 2 b k b m z q 1 -1 p 2 c -1 2 b k b m z 2 c -1 2 b k b m -1 1 2 -1 q -1 Binomial n q Gamma q 1 2 -1 p 2 c -1 2 b k b m z 2 -4 c 8 b k -1 4 b m -1 -1 1 c 2 b k -1 b m 1 2 -1 p 2 -4 c -1 8 b k 4 b m -1 q 0 n 2 q -1 n c 2 b k -1 b m -1 n -1 1 2 -1 p n -1 q p 2 c 2 b k -1 b m z q 1 -1 p 2 c 2 b k -1 b m z 2 c 2 b k -1 b m -1 1 2 -1 q -1 Binomial n q Gamma q 1 2 -1 -1 p 2 c 2 b k -1 b m z 2 4 c 8 b k -1 4 b m -1 -1 2 -1 m -2 Binomial m m 2 -1 1 -1 \$CellContext`m 2 1 -1 c 1 2 -1 p 2 4 c -1 q 0 n 2 q -1 n -1 c -1 n -1 1 2 -1 p n -1 q p -1 2 c z q 1 p -1 2 c z 2 c -1 1 2 -1 q -1 Binomial n q Gamma q 1 2 -1 p -1 2 c z 2 4 c -1 1 c 1 2 -1 -1 p 2 4 c -1 q 0 n 2 q -1 n c -1 n -1 1 2 -1 p n -1 q p 2 c z q 1 -1 p 2 c z 2 c -1 1 2 -1 q -1 Binomial n q Gamma q 1 2 -1 -1 p 2 c z 2 4 c -1 m SuperPlus n [/itex]

 Rule Form

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

 2002-12-18