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 Cosh

 http://functions.wolfram.com/01.20.21.3946.01

 Input Form

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

 Standard Form

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 MathML Form

 z n p z 2 sin m ( b z ) cosh v ( c z ) z - 2 - m - v - 1 z n + 1 ( - p z 2 ) 1 2 ( - n - 1 ) ( m m 2 ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["m", Identity]], List[TagBox[FractionBox["m", "2"], Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[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]]]]] Γ ( n + 1 2 , - p z 2 ) ( 1 - m mod 2 \$CellContext`m 2 ) ( 1 - v mod 2 \$CellContext`v 2 ) - - m 2 - m - v - 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]]]]] ( 1 - v mod 2 \$CellContext`v 2 ) k = 0 m - 1 2 ( - 1 ) k b 2 ( m - 2 k ) 2 4 p p - n - 1 ( m k ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["m", Identity]], List[TagBox["k", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( ( - 1 ) m q = 0 n 2 q - n ( b ( m - 2 k ) ) n - q ( 2 b k - b m + 2 p z ) q + 1 ( - ( 2 b k - b m + 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 , - ( 2 b k - b m + 2 p z ) 2 4 p ) + q = 0 n ( b ( k - m 2 ) ) n - q ( b ( m - 2 k ) + 2 p z ) q + 1 ( - ( b ( m - 2 k ) + 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 ( m - 2 k ) + 2 p z ) 2 4 p ) ) - 2 - m - v - 1 p - n - 1 ( m m 2 ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["m", Identity]], List[TagBox[FractionBox["m", "2"], Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( 1 - m mod 2 \$CellContext`m 2 ) s = 0 v - 1 2 - c 2 ( v - 2 s ) 2 4 p ( v s ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["v", Identity]], List[TagBox["s", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( q = 0 n 2 q - n ( c ( v - 2 s ) ) n - q ( 2 c s - c v + 2 p z ) q + 1 ( - ( 2 c s - c v + 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 , - ( 2 c s - c v + 2 p z ) 2 4 p ) + q = 0 n ( c ( s - v 2 ) ) n - q ( c ( v - 2 s ) + 2 p z ) q + 1 ( - ( c ( v - 2 s ) + 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 , - ( c ( v - 2 s ) + 2 p z ) 2 4 p ) ) - 1 p ( 2 - m - v - 1 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]]]]] k = 0 m - 1 2 ( - 1 ) k - - b 2 ( m - 2 k ) 2 + c 2 ( v - 2 s ) 2 + m p π 2 p ( m k ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["m", Identity]], List[TagBox["k", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( ( 2 b k - b m - 2 c s + c v ) 2 4 p q = 0 n p - n - 1 2 ( ( s - v 2 ) c + ( k - m 2 ) b ) n - q ( - 2 b k + b m - 2 c s + c v + 2 p z ) q + 1 ( - ( - 2 b k + b m - 2 c s + c v + 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 , - ( - 2 b k + b m - 2 c s + c v + 2 p z ) 2 4 p ) + ( - 2 b k + b m - 2 c s + c v ) 2 4 p + m π q = 0 n 2 q - n p - n - 1 2 ( - 2 b k + b m + 2 c s - c v ) n - q ( 2 b k - b m - 2 c s + c v + 2 p z ) q + 1 ( - ( 2 b k - b m - 2 c s + c v + 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 , - ( 2 b k - b m - 2 c s + c v + 2 p z ) 2 4 p ) + ( - 2 b k + b m - 2 c s + c v ) 2 4 p q = 0 n 2 q - n p - n - 1 2 ( 2 b k - b m - 2 c s + c v ) n - q ( - 2 b k + b m + 2 c s - c v + 2 p z ) q + 1 ( - ( - 2 b k + b m + 2 c s - c v + 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 , - ( - 2 b k + b m + 2 c s - c v + 2 p z ) 2 4 p ) + ( 2 b k - b m - 2 c s + c v ) 2 4 p + m π q = 0 n 2 q - n p - n - 1 2 ( b ( m - 2 k ) + c ( v - 2 s ) ) n - q ( - ( b ( m - 2 k ) - 2 c s + c v - 2 p z ) 2 p ) 1 2 ( - q - 1 ) ( 2 b k - b m + 2 c s - c v + 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 ( m - 2 k ) - 2 c s + c v - 2 p z ) 2 4 p ) ) ) /; m TagBox["\[DoubleStruckCapitalZ]", Function[Integers]] m > 0 v TagBox["\[DoubleStruckCapitalZ]", Function[Integers]] v > 0 n Condition z z n p z 2 b z m c z v -1 2 -1 m -1 v -1 z n 1 -1 p z 2 1 2 -1 n -1 Binomial m m 2 -1 Binomial v v 2 -1 Gamma n 1 2 -1 -1 p z 2 1 -1 \$CellContext`m 2 1 -1 \$CellContext`v 2 -1 -1 m 2 -1 m -1 v -1 Binomial v v 2 -1 1 -1 \$CellContext`v 2 k 0 m -1 2 -1 -1 k b 2 m -1 2 k 2 4 p -1 p -1 n -1 Binomial m k -1 m q 0 n 2 q -1 n b m -1 2 k n -1 q 2 b k -1 b m 2 p z q 1 -1 2 b k -1 b m 2 p z 2 p -1 1 2 -1 q -1 Binomial n q Gamma q 1 2 -1 -1 2 b k -1 b m 2 p z 2 4 p -1 q 0 n b k -1 m 2 -1 n -1 q b m -1 2 k 2 p z q 1 -1 b m -1 2 k 2 p z 2 p -1 1 2 -1 q -1 Binomial n q Gamma q 1 2 -1 -1 b m -1 2 k 2 p z 2 4 p -1 -1 2 -1 m -1 v -1 p -1 n -1 Binomial m m 2 -1 1 -1 \$CellContext`m 2 s 0 v -1 2 -1 -1 c 2 v -1 2 s 2 4 p -1 Binomial v s q 0 n 2 q -1 n c v -1 2 s n -1 q 2 c s -1 c v 2 p z q 1 -1 2 c s -1 c v 2 p z 2 p -1 1 2 -1 q -1 Binomial n q Gamma q 1 2 -1 -1 2 c s -1 c v 2 p z 2 4 p -1 q 0 n c s -1 v 2 -1 n -1 q c v -1 2 s 2 p z q 1 -1 c v -1 2 s 2 p z 2 p -1 1 2 -1 q -1 Binomial n q Gamma q 1 2 -1 -1 c v -1 2 s 2 p z 2 4 p -1 -1 1 p 1 2 -1 2 -1 m -1 v -1 s 0 v -1 2 -1 Binomial v s k 0 m -1 2 -1 -1 k -1 -1 b 2 m -1 2 k 2 c 2 v -1 2 s 2 m p 2 p -1 Binomial m k 2 b k -1 b m -1 2 c s c v 2 4 p -1 q 0 n p -1 n -1 1 2 s -1 v 2 -1 c k -1 m 2 -1 b n -1 q -2 b k b m -1 2 c s c v 2 p z q 1 -1 -2 b k b m -1 2 c s c v 2 p z 2 p -1 1 2 -1 q -1 Binomial n q Gamma q 1 2 -1 -1 -2 b k b m -1 2 c s c v 2 p z 2 4 p -1 -2 b k b m -1 2 c s c v 2 4 p -1 m q 0 n 2 q -1 n p -1 n -1 1 2 -2 b k b m 2 c s -1 c v n -1 q 2 b k -1 b m -1 2 c s c v 2 p z q 1 -1 2 b k -1 b m -1 2 c s c v 2 p z 2 p -1 1 2 -1 q -1 Binomial n q Gamma q 1 2 -1 -1 2 b k -1 b m -1 2 c s c v 2 p z 2 4 p -1 -2 b k b m -1 2 c s c v 2 4 p -1 q 0 n 2 q -1 n p -1 n -1 1 2 2 b k -1 b m -1 2 c s c v n -1 q -2 b k b m 2 c s -1 c v 2 p z q 1 -1 -2 b k b m 2 c s -1 c v 2 p z 2 p -1 1 2 -1 q -1 Binomial n q Gamma q 1 2 -1 -1 -2 b k b m 2 c s -1 c v 2 p z 2 4 p -1 2 b k -1 b m -1 2 c s c v 2 4 p -1 m q 0 n 2 q -1 n p -1 n -1 1 2 b m -1 2 k c v -1 2 s n -1 q -1 b m -1 2 k -1 2 c s c v -1 2 p z 2 p -1 1 2 -1 q -1 2 b k -1 b m 2 c s -1 c v 2 p z q 1 Binomial n q Gamma q 1 2 -1 -1 b m -1 2 k -1 2 c s c v -1 2 p z 2 4 p -1 m m 0 v v 0 n [/itex]

 Rule Form

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

 2002-12-18