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 Cosh

 http://functions.wolfram.com/01.20.21.4004.01

 Input Form

 Integrate[z^n E^(p z^2) Cos[b z^2]^m Cosh[c z]^v, z] == 2^(-1 - m - v) ((-z^(1 + n)) Binomial[m, m/2] Binomial[v, v/2] ExpIntegralE[1/2 - n/2, (-p) z^2] (-1 + Mod[m, 2]) (-1 + Mod[v, 2]) + z^(1 + n) Binomial[v, v/2] (-1 + Mod[v, 2]) Sum[Binomial[m, s] (((-(p - I b (m - 2 s))) z^2)^((1/2) (-1 - n)) Gamma[(1 + n)/2, (-(p - I b (m - 2 s))) z^2] + ((-(p + I b (m - 2 s))) z^2)^((1/2) (-1 - n)) Gamma[(1 + n)/2, (-(p + I b (m - 2 s))) z^2]), {s, 0, Floor[(1/2) (-1 + m)]}] + 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)]}] - Sum[Binomial[m, k] Sum[(Binomial[v, s] ((E^((c^2 (-2 s + v)^2)/(4 (I b (-2 k + m) + p))) Sqrt[I b (-2 k + m) + p] Sum[2^(-n + q) (I b (2 k - m) + p)^( -(1/2) - n) (c (-2 s + v))^(n - q) (c (2 s - v) + 2 (I b (2 k - m) + p) z)^(1 + q) (-((c (2 s - v) + 2 (I b (2 k - m) + p) z)^2/(I b (2 k - m) + p)))^((1/2) (-1 - q)) Binomial[n, q] Gamma[(1 + q)/2, -((c (2 s - v) + 2 (I b (2 k - m) + p) z)^2/ (4 (I b (2 k - m) + p)))], {q, 0, n}] + E^((c^2 (-2 s + v)^2)/(4 (I b (-2 k + m) + p))) Sqrt[-2 I b k + I b m + p] Sum[(I b (2 k - m) + p)^(-(1/2) - n) (c (s - v/2))^(n - q) (c (-2 s + v) + 2 (I b (2 k - m) + p) z)^( 1 + q) (-((c (-2 s + v) + 2 (I b (2 k - m) + p) z)^2/ (I b (2 k - m) + p)))^((1/2) (-1 - q)) Binomial[n, q] Gamma[(1 + q)/2, -((c (-2 s + v) + 2 (I b (2 k - m) + p) z)^2/ (4 (I b (2 k - m) + p)))], {q, 0, n}] + E^((c^2 (-2 s + v)^2)/(4 (I b (2 k - m) + p))) Sqrt[I b (2 k - m) + p] (Sum[2^(-n + q) (-2 I b k + I b m + p)^ (-(1/2) - n) (c (-2 s + v))^(n - q) (c (2 s - v) + 2 (-2 I b k + I b m + p) z)^(1 + q) (-((c (2 s - v) + 2 (-2 I b k + I b m + p) z)^2/(-2 I b k + I b m + p)))^ ((1/2) (-1 - q)) Binomial[n, q] Gamma[(1 + q)/2, -((c (2 s - v) + 2 (-2 I b k + I b m + p) z)^2/ (4 (-2 I b k + I b m + p)))], {q, 0, n}] + Sum[(-2 I b k + I b m + p)^(-(1/2) - n) (c (s - v/2))^(n - q) (c (-2 s + v) + 2 (-2 I b k + I b m + p) z)^(1 + q) (-(c (-2 s + v) + 2 (-2 I b k + I b m + p) z)^2/(-2 I b k + I b m + p))^((1/2) (-1 - q)) Binomial[n, q] Gamma[ (1 + q)/2, -((c (-2 s + v) + 2 (-2 I b k + I b m + p) z)^2/ (4 (-2 I b k + I b m + p)))], {q, 0, n}]))/ (Sqrt[I b (2 k - m) + p] Sqrt[I b (-2 k + m) + p])))/ E^((c^2 p (-2 s + v)^2)/(2 (b^2 (-2 k + m)^2 + p^2))), {s, 0, Floor[(1/2) (-1 + v)]}], {k, 0, Floor[(1/2) (-1 + m)]}]) /; 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 cos m ( b z 2 ) cosh v ( c z ) z 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]]]]] ( m mod 2 \$CellContext`m 2 - 1 ) 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 ) ) - z 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]]]]] E TagBox["E", ExpIntegralE] 1 2 - n 2 ( - p z 2 ) ( m mod 2 \$CellContext`m 2 - 1 ) ( v mod 2 \$CellContext`v 2 - 1 ) + z 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 ) s = 0 m - 1 2 ( m s ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["m", Identity]], List[TagBox["s", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( ( - ( p + b ( m - 2 s ) ) z 2 ) 1 2 ( - n - 1 ) Γ ( n + 1 2 , - ( p + b ( m - 2 s ) ) z 2 ) + ( - ( p - b ( m - 2 s ) ) z 2 ) 1 2 ( - n - 1 ) Γ ( n + 1 2 , - ( p - b ( m - 2 s ) ) z 2 ) ) - k = 0 m - 1 2 ( m k ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["m", Identity]], List[TagBox["k", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] s = 0 v - 1 2 - c 2 p ( v - 2 s ) 2 2 ( b 2 ( m - 2 k ) 2 + p 2 ) ( v s ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["v", Identity]], List[TagBox["s", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( c 2 ( v - 2 s ) 2 4 ( b ( m - 2 k ) + p ) b ( m - 2 k ) + p q = 0 n 2 q - n ( b ( 2 k - m ) + p ) - n - 1 2 ( c ( v - 2 s ) ) n - q ( c ( 2 s - v ) + 2 ( b ( 2 k - m ) + p ) z ) q + 1 ( - ( c ( 2 s - v ) + 2 ( b ( 2 k - m ) + p ) z ) 2 / ( b ( 2 k - m ) + 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 ( 2 s - v ) + 2 ( b ( 2 k - m ) + p ) z ) 2 4 ( b ( 2 k - m ) + p ) ) + c 2 ( v - 2 s ) 2 4 ( b ( m - 2 k ) + p ) - 2 b k + b m + p q = 0 n ( b ( 2 k - m ) + p ) - n - 1 2 ( c ( s - v 2 ) ) n - q ( c ( v - 2 s ) + 2 ( b ( 2 k - m ) + p ) z ) q + 1 ( - ( c ( v - 2 s ) + 2 ( b ( 2 k - m ) + p ) z ) 2 / ( b ( 2 k - m ) + 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 ( b ( 2 k - m ) + p ) z ) 2 4 ( b ( 2 k - m ) + p ) ) + c 2 ( v - 2 s ) 2 4 ( b ( 2 k - m ) + p ) b ( 2 k - m ) + p ( q = 0 n 2 q - n ( - 2 b k + b m + p ) - n - 1 2 ( c ( v - 2 s ) ) n - q ( c ( 2 s - v ) + 2 ( - 2 b k + b m + p ) z ) q + 1 ( - ( c ( 2 s - v ) + 2 ( - 2 b k + b m + p ) z ) 2 / ( - 2 b k + b m + 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 ( 2 s - v ) + 2 ( - 2 b k + b m + p ) z ) 2 / ( 4 ( - 2 b k + b m + p ) ) ) + q = 0 n ( - 2 b k + b m + p ) - n - 1 2 ( c ( s - v 2 ) ) n - q ( c ( v - 2 s ) + 2 ( - 2 b k + b m + p ) z ) q + 1 ( - ( c ( v - 2 s ) + 2 ( - 2 b k + b m + p ) z ) 2 / ( - 2 b k + b m + 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 ( - 2 b k + b m + p ) z ) 2 / ( 4 ( - 2 b k + b m + p ) ) ) ) ) / ( b ( 2 k - m ) + p b ( m - 2 k ) + p ) ) /; m + v + n Condition z z n p z 2 b z 2 m c z v 2 -1 m -1 v -1 p -1 n -1 Binomial m m 2 -1 \$CellContext`m 2 -1 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 z n 1 Binomial m m 2 -1 Binomial v v 2 -1 ExpIntegralE 1 2 -1 n 2 -1 -1 p z 2 \$CellContext`m 2 -1 \$CellContext`v 2 -1 z n 1 Binomial v v 2 -1 \$CellContext`v 2 -1 s 0 m -1 2 -1 Binomial m s -1 p b m -1 2 s z 2 1 2 -1 n -1 Gamma n 1 2 -1 -1 p b m -1 2 s z 2 -1 p -1 b m -1 2 s z 2 1 2 -1 n -1 Gamma n 1 2 -1 -1 p -1 b m -1 2 s z 2 -1 k 0 m -1 2 -1 Binomial m k s 0 v -1 2 -1 -1 c 2 p v -1 2 s 2 2 b 2 m -1 2 k 2 p 2 -1 Binomial v s c 2 v -1 2 s 2 4 b m -1 2 k p -1 b m -1 2 k p 1 2 q 0 n 2 q -1 n b 2 k -1 m p -1 n -1 1 2 c v -1 2 s n -1 q c 2 s -1 v 2 b 2 k -1 m p z q 1 -1 c 2 s -1 v 2 b 2 k -1 m p z 2 b 2 k -1 m p -1 1 2 -1 q -1 Binomial n q Gamma q 1 2 -1 -1 c 2 s -1 v 2 b 2 k -1 m p z 2 4 b 2 k -1 m p -1 c 2 v -1 2 s 2 4 b m -1 2 k p -1 -2 b k b m p 1 2 q 0 n b 2 k -1 m p -1 n -1 1 2 c s -1 v 2 -1 n -1 q c v -1 2 s 2 b 2 k -1 m p z q 1 -1 c v -1 2 s 2 b 2 k -1 m p z 2 b 2 k -1 m p -1 1 2 -1 q -1 Binomial n q Gamma q 1 2 -1 -1 c v -1 2 s 2 b 2 k -1 m p z 2 4 b 2 k -1 m p -1 c 2 v -1 2 s 2 4 b 2 k -1 m p -1 b 2 k -1 m p 1 2 q 0 n 2 q -1 n -2 b k b m p -1 n -1 1 2 c v -1 2 s n -1 q c 2 s -1 v 2 -2 b k b m p z q 1 -1 c 2 s -1 v 2 -2 b k b m p z 2 -2 b k b m p -1 1 2 -1 q -1 Binomial n q Gamma q 1 2 -1 -1 c 2 s -1 v 2 -2 b k b m p z 2 4 -2 b k b m p -1 q 0 n -2 b k b m p -1 n -1 1 2 c s -1 v 2 -1 n -1 q c v -1 2 s 2 -2 b k b m p z q 1 -1 c v -1 2 s 2 -2 b k b m p z 2 -2 b k b m p -1 1 2 -1 q -1 Binomial n q Gamma q 1 2 -1 -1 c v -1 2 s 2 -2 b k b m p z 2 4 -2 b k b m p -1 b 2 k -1 m p 1 2 b m -1 2 k p 1 2 -1 m SuperPlus v SuperPlus n [/itex]

 Rule Form

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

 2002-12-18