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 Cosh

 http://functions.wolfram.com/01.20.21.3980.01

 Input Form

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

 Rule Form

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

 2002-12-18