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 Cosh

 http://functions.wolfram.com/01.20.21.4491.01

 Input Form

 Integrate[z^n Sinh[b z^2 + d z] Cosh[c z^2 + g]^v, z] == 2^(-2 - v) ((b^(-1 - n) Binomial[v, v/2] (-1 + Mod[v, 2]) (Sum[2^(-n + q) (-d)^(n - q) (d + 2 b z)^(1 + q) (-((d + 2 b z)^2/b))^ ((1/2) (-1 - q)) Binomial[n, q] Gamma[(1 + q)/2, -((d + 2 b z)^2/(4 b))], {q, 0, n}] - E^(d^2/(2 b)) Sum[2^(-n + q) (-d)^(n - q) (d + 2 b z)^(1 + q) ((d + 2 b z)^2/b)^ ((1/2) (-1 - q)) Binomial[n, q] Gamma[(1 + q)/2, (d + 2 b z)^2/(4 b)], {q, 0, n}]))/E^(d^2/(4 b)) + Sum[E^(2 g s - g v) Binomial[v, s] (E^(2 g (-2 s + v) - d^2/(-4 b - 8 c s + 4 c v)) (-b - 2 c s + c v)^(-1 - n) Sum[2^(-n + q) d^(n - q) (-((d + 2 (b + 2 c s - c v) z)^2/(-b - 2 c s + c v)))^ ((1/2) (-1 - q)) (-d + 2 (-b - 2 c s + c v) z)^(1 + q) Binomial[n, q] Gamma[(1 + q)/2, -((d + 2 (b + 2 c s - c v) z)^2/ (-4 b - 8 c s + 4 c v))], {q, 0, n}] - E^(d^2/(-4 b - 8 c s + 4 c v)) (b + 2 c s - c v)^(-1 - n) Sum[2^(-n + q) (-d)^(n - q) (d + 2 (b + 2 c s - c v) z)^(1 + q) ((d + 2 (b + 2 c s - c v) z)^2/(-b - 2 c s + c v))^((1/2) (-1 - q)) Binomial[n, q] Gamma[(1 + q)/2, (d + 2 (b + 2 c s - c v) z)^2/ (-4 b - 8 c s + 4 c v)], {q, 0, n}] + ((-b + 2 c s - c v)^(-1 - n) Sum[2^(-n + q) d^(n - q) (-d - 2 b z + 4 c s z - 2 c v z)^(1 + q) (-((d + 2 (b + c (-2 s + v)) z)^2/(-b + 2 c s - c v)))^ ((1/2) (-1 - q)) Binomial[n, q] Gamma[(1 + q)/2, -((d + 2 (b + c (-2 s + v)) z)^2/(-4 b + 8 c s - 4 c v))], {q, 0, n}])/E^(d^2/(-4 b + 8 c s - 4 c v)) - E^(2 g (-2 s + v) + d^2/(-4 b + 8 c s - 4 c v)) (b + c (-2 s + v))^(-1 - n) Sum[2^(-n + q) (-d)^(n - q) (d + 2 (b + c (-2 s + v)) z)^(1 + q) ((d + 2 (b + c (-2 s + v)) z)^2/(-b + 2 c s - c v))^ ((1/2) (-1 - q)) Binomial[n, q] Gamma[(1 + q)/2, (d + 2 (b + c (-2 s + v)) z)^2/(-4 b + 8 c s - 4 c v)], {q, 0, n}]), {s, 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 sinh ( b z 2 + d z ) cosh v ( c z 2 + g ) z 2 - v - 2 ( - d 2 4 b ( 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 ( - d ) n - q ( d + 2 b z ) q + 1 ( - ( d + 2 b z ) 2 b ) 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 , - ( d + 2 b z ) 2 4 b ) - d 2 2 b q = 0 n 2 q - n ( - d ) n - q ( d + 2 b z ) q + 1 ( ( d + 2 b z ) 2 b ) 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 , ( d + 2 b z ) 2 4 b ) ) b - n - 1 + s = 0 v - 1 2 2 g s - g v ( v s ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["v", Identity]], List[TagBox["s", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( 2 g ( v - 2 s ) - d 2 - 4 b - 8 c s + 4 c v ( q = 0 n 2 q - n d n - q ( - ( d + 2 ( b + 2 c s - c v ) z ) 2 - b - 2 c s + c v ) 1 2 ( - q - 1 ) ( 2 ( - b - 2 c s + c v ) z - d ) 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 , - ( d + 2 ( b + 2 c s - c v ) z ) 2 - 4 b - 8 c s + 4 c v ) ) ( - b - 2 c s + c v ) - n - 1 - d 2 - 4 b - 8 c s + 4 c v ( b + 2 c s - c v ) - n - 1 q = 0 n 2 q - n ( - d ) n - q ( d + 2 ( b + 2 c s - c v ) z ) q + 1 ( ( d + 2 ( b + 2 c s - c v ) z ) 2 - b - 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 , ( d + 2 ( b + 2 c s - c v ) z ) 2 - 4 b - 8 c s + 4 c v ) - d 2 - 4 b + 8 c s - 4 c v + 2 g ( v - 2 s ) ( b + c ( v - 2 s ) ) - n - 1 q = 0 n 2 q - n ( - d ) n - q ( d + 2 ( b + c ( v - 2 s ) ) z ) q + 1 ( ( d + 2 ( b + c ( v - 2 s ) ) z ) 2 - b + 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 , ( d + 2 ( b + c ( v - 2 s ) ) z ) 2 - 4 b + 8 c s - 4 c v ) + - d 2 - 4 b + 8 c s - 4 c v ( - b + 2 c s - c v ) - n - 1 q = 0 n 2 q - n d n - q ( - d - 2 b z + 4 c s z - 2 c v z ) q + 1 ( - ( d + 2 ( b + c ( v - 2 s ) ) z ) 2 - b + 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 , - ( d + 2 ( b + c ( v - 2 s ) ) z ) 2 - 4 b + 8 c s - 4 c v ) ) ) /; n v + Condition z z n b z 2 d z c z 2 g v 2 -1 v -2 -1 d 2 4 b -1 Binomial v v 2 -1 \$CellContext`v 2 -1 q 0 n 2 q -1 n -1 d n -1 q d 2 b z q 1 -1 d 2 b z 2 b -1 1 2 -1 q -1 Binomial n q Gamma q 1 2 -1 -1 d 2 b z 2 4 b -1 -1 d 2 2 b -1 q 0 n 2 q -1 n -1 d n -1 q d 2 b z q 1 d 2 b z 2 b -1 1 2 -1 q -1 Binomial n q Gamma q 1 2 -1 d 2 b z 2 4 b -1 b -1 n -1 s 0 v -1 2 -1 2 g s -1 g v Binomial v s 2 g v -1 2 s -1 d 2 -4 b -1 8 c s 4 c v -1 q 0 n 2 q -1 n d n -1 q -1 d 2 b 2 c s -1 c v z 2 -1 b -1 2 c s c v -1 1 2 -1 q -1 2 -1 b -1 2 c s c v z -1 d q 1 Binomial n q Gamma q 1 2 -1 -1 d 2 b 2 c s -1 c v z 2 -4 b -1 8 c s 4 c v -1 -1 b -1 2 c s c v -1 n -1 -1 d 2 -4 b -1 8 c s 4 c v -1 b 2 c s -1 c v -1 n -1 q 0 n 2 q -1 n -1 d n -1 q d 2 b 2 c s -1 c v z q 1 d 2 b 2 c s -1 c v z 2 -1 b -1 2 c s c v -1 1 2 -1 q -1 Binomial n q Gamma q 1 2 -1 d 2 b 2 c s -1 c v z 2 -4 b -1 8 c s 4 c v -1 -1 d 2 -4 b 8 c s -1 4 c v -1 2 g v -1 2 s b c v -1 2 s -1 n -1 q 0 n 2 q -1 n -1 d n -1 q d 2 b c v -1 2 s z q 1 d 2 b c v -1 2 s z 2 -1 b 2 c s -1 c v -1 1 2 -1 q -1 Binomial n q Gamma q 1 2 -1 d 2 b c v -1 2 s z 2 -4 b 8 c s -1 4 c v -1 -1 d 2 -4 b 8 c s -1 4 c v -1 -1 b 2 c s -1 c v -1 n -1 q 0 n 2 q -1 n d n -1 q -1 d -1 2 b z 4 c s z -1 2 c v z q 1 -1 d 2 b c v -1 2 s z 2 -1 b 2 c s -1 c v -1 1 2 -1 q -1 Binomial n q Gamma q 1 2 -1 -1 d 2 b c v -1 2 s z 2 -4 b 8 c s -1 4 c v -1 n v SuperPlus [/itex]

 Rule Form

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

 2002-12-18