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 Cosh

 http://functions.wolfram.com/01.20.21.4499.01

 Input Form

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