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 Cosh

 http://functions.wolfram.com/01.20.21.4816.01

 Input Form

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

 Standard Form

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 MathML Form

 z n p z 2 sinh ( b z ) cosh v ( c z 2 ) z 2 - v - 2 - b 2 4 p p - n - 1 ( b 2 4 p ( s = 0 v - 1 2 ( - b 2 p 2 ( p + 2 c s - c v ) ( p - 2 c s + c v ) ( v s ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["v", Identity]], List[TagBox["s", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( b 2 4 p + 8 c s - 4 c v p + 2 c s - c v ( q = 0 n 2 q - n b n - q ( p - 2 c s + c v ) - n - 1 2 ( - ( b - 2 ( p - 2 c s + c v ) z ) 2 p - 2 c s + c v ) 1 2 ( - q - 1 ) ( 2 ( p - 2 c s + c v ) z - b ) 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 - 2 c s + c v ) z ) 2 4 ( p - 2 c s + c v ) ) - q = 0 n 2 q - n ( - b ) n - q ( p - 2 c s + c v ) - n - 1 2 ( b + 2 ( p - 2 c s + c v ) z ) q + 1 ( - ( b + 2 ( p - 2 c s + c v ) z ) 2 p - 2 c s + 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 ( p - 2 c s + c v ) z ) 2 4 ( p - 2 c s + c v ) ) ) - b 2 4 p - 8 c s + 4 c v p - 2 c s + c v q = 0 n 2 q - n ( - b ) n - q ( p + 2 c s - c v ) - n - 1 2 ( b + 2 ( p + 2 c s - c v ) z ) q + 1 ( - ( b + 2 ( p + 2 c s - c v ) z ) 2 p + 2 c s - 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 ( p + 2 c s - c v ) z ) 2 4 ( p + 2 c s - c v ) ) + b 2 4 p - 8 c s + 4 c v p - 2 c s + c v q = 0 n 2 q - n b n - q ( p + 2 c s - c v ) - n - 1 2 ( - ( b - 2 ( p + 2 c s - c v ) z ) 2 p + 2 c s - c v ) 1 2 ( - q - 1 ) ( 2 ( p + 2 c s - c v ) z - b ) 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 + 2 c s - c v ) z ) 2 4 ( p + 2 c s - c v ) ) ) ) / ( p + 2 c s - c v p - 2 c s + c v ) ) p 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]]]]] ( v mod 2 \$CellContext`v 2 - 1 ) q = 0 n 2 q - n ( - b ) n - q ( b + 2 p z ) q + 1 ( - ( b + 2 p z ) 2 p ) 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 p z ) 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 ) ( 2 p z - b ) 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 ) ) /; 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 -1 b 2 4 p -1 p -1 n -1 b 2 4 p -1 s 0 v -1 2 -1 -1 b 2 p 2 p 2 c s -1 c v p -1 2 c s c v -1 Binomial v s b 2 4 p 8 c s -1 4 c v -1 p 2 c s -1 c v 1 2 q 0 n 2 q -1 n b n -1 q p -1 2 c s c v -1 n -1 1 2 -1 b -1 2 p -1 2 c s c v z 2 p -1 2 c s c v -1 1 2 -1 q -1 2 p -1 2 c s c v z -1 b q 1 Binomial n q Gamma q 1 2 -1 -1 b -1 2 p -1 2 c s c v z 2 4 p -1 2 c s c v -1 -1 q 0 n 2 q -1 n -1 b n -1 q p -1 2 c s c v -1 n -1 1 2 b 2 p -1 2 c s c v z q 1 -1 b 2 p -1 2 c s c v z 2 p -1 2 c s c v -1 1 2 -1 q -1 Binomial n q Gamma q 1 2 -1 -1 b 2 p -1 2 c s c v z 2 4 p -1 2 c s c v -1 -1 b 2 4 p -1 8 c s 4 c v -1 p -1 2 c s c v 1 2 q 0 n 2 q -1 n -1 b n -1 q p 2 c s -1 c v -1 n -1 1 2 b 2 p 2 c s -1 c v z q 1 -1 b 2 p 2 c s -1 c v z 2 p 2 c s -1 c v -1 1 2 -1 q -1 Binomial n q Gamma q 1 2 -1 -1 b 2 p 2 c s -1 c v z 2 4 p 2 c s -1 c v -1 b 2 4 p -1 8 c s 4 c v -1 p -1 2 c s c v 1 2 q 0 n 2 q -1 n b n -1 q p 2 c s -1 c v -1 n -1 1 2 -1 b -1 2 p 2 c s -1 c v z 2 p 2 c s -1 c v -1 1 2 -1 q -1 2 p 2 c s -1 c v z -1 b q 1 Binomial n q Gamma q 1 2 -1 -1 b -1 2 p 2 c s -1 c v z 2 4 p 2 c s -1 c v -1 p 2 c s -1 c v 1 2 p -1 2 c s c v 1 2 -1 p n 1 Binomial v v 2 -1 \$CellContext`v 2 -1 q 0 n 2 q -1 n -1 b n -1 q b 2 p z q 1 -1 b 2 p z 2 p -1 1 2 -1 q -1 Binomial n q Gamma q 1 2 -1 -1 b 2 p z 2 4 p -1 -1 Binomial v v 2 -1 \$CellContext`v 2 -1 q 0 n 2 q -1 n b n -1 q -1 b -1 2 p z 2 p -1 1 2 -1 q -1 2 p z -1 b q 1 Binomial n q Gamma q 1 2 -1 -1 b -1 2 p z 2 4 p -1 v v 0 n [/itex]

 Rule Form

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

 2002-12-18