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 Cosh

 http://functions.wolfram.com/01.20.21.4515.01

 Input Form

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