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 Cosh

 http://functions.wolfram.com/01.20.21.4010.01

 Input Form

 Integrate[z^n E^(p z) Cos[b z]^m Cosh[c z^2]^v, z] == 2^(-1 - m - v) ((1/p) ((2 Binomial[m, m/2] Binomial[v, v/2] Gamma[1 + n, (-p) z] (-1 + Mod[m, 2]) (-1 + Mod[v, 2]))/(-p)^n) + 2 z^(1 + n) Binomial[v, v/2] (-1 + Mod[v, 2]) Sum[Binomial[m, s] (ExpIntegralE[-n, (-(p - I b (m - 2 s))) z] + ExpIntegralE[-n, (-(p + I b (m - 2 s))) z]), {s, 0, Floor[(1/2) (-1 + m)]}] + Binomial[m, m/2] (-1 + Mod[m, 2]) Sum[(E^(p^2/(-8 c s + 4 c v)) Binomial[v, s] (Sqrt[c (-2 s + v)] Sum[2^(-n + q) (-p)^(n - q) (c (2 s - v))^ (-(1/2) - n) (p + 4 c s z - 2 c v z)^(1 + q) ((p + 4 c s z - 2 c v z)^2/(c (-2 s + v)))^((1/2) (-1 - q)) Binomial[n, q] Gamma[(1 + q)/2, (p + 4 c s z - 2 c v z)^2/ (4 c (-2 s + v))], {q, 0, n}] + E^(p^2/(4 c s - 2 c v)) Sqrt[c (2 s - v)] Sum[2^(-n + q) (-p)^(n - q) (c (-2 s + v))^ (-(1/2) - n) (p + 2 c (-2 s + v) z)^(1 + q) (-((p + 2 c (-2 s + v) z)^2/(c (-2 s + v))))^((1/2) (-1 - q)) Binomial[n, q] Gamma[(1 + q)/2, (p + 2 c (-2 s + v) z)^2/ (c (8 s - 4 v))], {q, 0, n}]))/Sqrt[(-c^2) (-2 s + v)^2], {s, 0, Floor[(1/2) (-1 + v)]}] + Sum[Binomial[v, k] Sum[(Binomial[m, s] (c (-2 k + v) Sum[2^(-n + q) (-p - I b (m - 2 s))^( n - q) (c (2 k - v))^(-(1/2) - n) (p + I b (m - 2 s) + 4 c k z - 2 c v z)^(1 + q) (-((p + I b (m - 2 s) + 4 c k z - 2 c v z)^2/(c (2 k - v))))^((1/2) (-1 - q)) Binomial[n, q] Gamma[(1 + q)/2, -((p + I b (m - 2 s) + 4 c k z - 2 c v z)^2/ (c (8 k - 4 v)))], {q, 0, n}] + Sqrt[c (2 k - v)] Sqrt[c (-2 k + v)] (E^((p^2 - b^2 (m - 2 s)^2)/(c (4 k - 2 v))) Sum[2^(-n + q) (-p + I b (m - 2 s))^(n - q) (c (-2 k + v))^ (-(1/2) - n) (p - I b (m - 2 s) + 2 c (-2 k + v) z)^(1 + q) ((p - I b (m - 2 s) + 2 c (-2 k + v) z)^2/(c (2 k - v)))^ ((1/2) (-1 - q)) Binomial[n, q] Gamma[(1 + q)/2, (p - I b (m - 2 s) + 2 c (-2 k + v) z)^2/(c (8 k - 4 v))], { q, 0, n}] + E^((p + I b (m - 2 s))^2/(c (4 k - 2 v))) Sum[2^(-n + q) (-p - I b (m - 2 s))^(n - q) (c (-2 k + v))^ (-(1/2) - n) (p + I b (m - 2 s) + 2 c (-2 k + v) z)^(1 + q) (-((p + I b (m - 2 s) + 2 c (-2 k + v) z)^2/(c (-2 k + v))))^ ((1/2) (-1 - q)) Binomial[n, q] Gamma[(1 + q)/2, (p + I b (m - 2 s) + 2 c (-2 k + v) z)^2/(c (8 k - 4 v))], { q, 0, n}]) - c E^((I b p (m - 2 s))/(c (2 k - v))) (2 k - v) Sum[2^(-n + q) (-p + I b (m - 2 s))^(n - q) (c (2 k - v))^( -(1/2) - n) (p - I b (m - 2 s) + 4 c k z - 2 c v z)^(1 + q) (-(((-I) b m + p + 2 I b s + 4 c k z - 2 c v z)^2/(2 c k - c v)))^((1/2) (-1 - q)) Binomial[n, q] Gamma[(1 + q)/2, (b (m - 2 s) + I (p + 4 c k z - 2 c v z))^2/(c (8 k - 4 v))], {q, 0, n}]))/E^((p + I b (m - 2 s))^2/(c (8 k - 4 v)))/ (c (2 k - v))^(3/2), {s, 0, Floor[(1/2) (-1 + m)]}], {k, 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 cos m ( b z ) cosh v ( c z 2 ) z 2 - m - v - 1 ( 2 ( - p ) - n ( 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 , - p z ) ( m mod 2 \$CellContext`m 2 - 1 ) ( v mod 2 \$CellContext`v 2 - 1 ) p + ( 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 1 - c 2 ( v - 2 s ) 2 ( p 2 4 c v - 8 c s ( v s ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["v", Identity]], List[TagBox["s", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( c ( v - 2 s ) q = 0 n 2 q - n ( - p ) n - q ( c ( 2 s - v ) ) - n - 1 2 ( p + 4 c s z - 2 c v z ) q + 1 ( ( p + 4 c s z - 2 c v z ) 2 c ( v - 2 s ) ) 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 , ( p + 4 c s z - 2 c v z ) 2 4 c ( v - 2 s ) ) + p 2 4 c s - 2 c v c ( 2 s - v ) q = 0 n 2 q - n ( - p ) n - q ( c ( v - 2 s ) ) - n - 1 2 ( p + 2 c ( v - 2 s ) z ) q + 1 ( - ( p + 2 c ( v - 2 s ) z ) 2 c ( v - 2 s ) ) 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 , ( p + 2 c ( v - 2 s ) z ) 2 c ( 8 s - 4 v ) ) ) ) + 2 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]]]]] ( E TagBox["E", ExpIntegralE] - n ( - ( p + b ( m - 2 s ) ) z ) + E TagBox["E", ExpIntegralE] - n ( - ( p - b ( m - 2 s ) ) z ) ) + k = 0 v - 1 2 ( v k ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["v", Identity]], List[TagBox["k", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] s = 0 m - 1 2 1 ( c ( 2 k - v ) ) 3 / 2 ( - ( p + b ( m - 2 s ) ) 2 c ( 8 k - 4 v ) ( m s ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["m", Identity]], List[TagBox["s", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( c ( 2 k - v ) c ( v - 2 k ) ( ( p + b ( m - 2 s ) ) 2 c ( 4 k - 2 v ) q = 0 n 2 q - n ( - p - b ( m - 2 s ) ) n - q ( c ( v - 2 k ) ) - n - 1 2 ( p + b ( m - 2 s ) + 2 c ( v - 2 k ) z ) q + 1 ( - ( p + b ( m - 2 s ) + 2 c ( v - 2 k ) z ) 2 c ( v - 2 k ) ) 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 , ( p + b ( m - 2 s ) + 2 c ( v - 2 k ) z ) 2 c ( 8 k - 4 v ) ) + p 2 - b 2 ( m - 2 s ) 2 c ( 4 k - 2 v ) q = 0 n 2 q - n ( b ( m - 2 s ) - p ) n - q ( c ( v - 2 k ) ) - n - 1 2 ( p - b ( m - 2 s ) + 2 c ( v - 2 k ) z ) q + 1 ( ( p - b ( m - 2 s ) + 2 c ( v - 2 k ) z ) 2 c ( 2 k - 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 , ( p - b ( m - 2 s ) + 2 c ( v - 2 k ) z ) 2 c ( 8 k - 4 v ) ) ) - c b p ( m - 2 s ) c ( 2 k - v ) ( 2 k - v ) q = 0 n 2 q - n ( b ( m - 2 s ) - p ) n - q ( c ( 2 k - v ) ) - n - 1 2 ( p - b ( m - 2 s ) + 4 c k z - 2 c v z ) q + 1 ( - ( - b m + p + 2 b s + 4 c k z - 2 c v z ) 2 2 c k - 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 , ( b ( m - 2 s ) + ( p + 4 c k z - 2 c v z ) ) 2 c ( 8 k - 4 v ) ) + c ( v - 2 k ) q = 0 n 2 q - n ( - p - b ( m - 2 s ) ) n - q ( c ( 2 k - v ) ) - n - 1 2 ( p + b ( m - 2 s ) + 4 c k z - 2 c v z ) q + 1 ( - ( p + b ( m - 2 s ) + 4 c k z - 2 c v z ) 2 c ( 2 k - 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 , - ( p + b ( m - 2 s ) + 4 c k z - 2 c v z ) 2 c ( 8 k - 4 v ) ) ) ) ) /; m TagBox["\[DoubleStruckCapitalZ]", Function[Integers]] m > 0 v TagBox["\[DoubleStruckCapitalZ]", Function[Integers]] v > 0 n Condition z z n p z b z m c z 2 v 2 -1 m -1 v -1 2 -1 p -1 n Binomial m m 2 -1 Binomial v v 2 -1 Gamma n 1 -1 p z \$CellContext`m 2 -1 \$CellContext`v 2 -1 p -1 Binomial m m 2 -1 \$CellContext`m 2 -1 s 0 v -1 2 -1 1 -1 c 2 v -1 2 s 2 1 2 -1 p 2 4 c v -1 8 c s -1 Binomial v s c v -1 2 s 1 2 q 0 n 2 q -1 n -1 p n -1 q c 2 s -1 v -1 n -1 1 2 p 4 c s z -1 2 c v z q 1 p 4 c s z -1 2 c v z 2 c v -1 2 s -1 1 2 -1 q -1 Binomial n q Gamma q 1 2 -1 p 4 c s z -1 2 c v z 2 4 c v -1 2 s -1 p 2 4 c s -1 2 c v -1 c 2 s -1 v 1 2 q 0 n 2 q -1 n -1 p n -1 q c v -1 2 s -1 n -1 1 2 p 2 c v -1 2 s z q 1 -1 p 2 c v -1 2 s z 2 c v -1 2 s -1 1 2 -1 q -1 Binomial n q Gamma q 1 2 -1 p 2 c v -1 2 s z 2 c 8 s -1 4 v -1 2 z n 1 Binomial v v 2 -1 \$CellContext`v 2 -1 s 0 m -1 2 -1 Binomial m s ExpIntegralE -1 n -1 p b m -1 2 s z ExpIntegralE -1 n -1 p -1 b m -1 2 s z k 0 v -1 2 -1 Binomial v k s 0 m -1 2 -1 1 c 2 k -1 v 3 2 -1 -1 p b m -1 2 s 2 c 8 k -1 4 v -1 Binomial m s c 2 k -1 v 1 2 c v -1 2 k 1 2 p b m -1 2 s 2 c 4 k -1 2 v -1 q 0 n 2 q -1 n -1 p -1 b m -1 2 s n -1 q c v -1 2 k -1 n -1 1 2 p b m -1 2 s 2 c v -1 2 k z q 1 -1 p b m -1 2 s 2 c v -1 2 k z 2 c v -1 2 k -1 1 2 -1 q -1 Binomial n q Gamma q 1 2 -1 p b m -1 2 s 2 c v -1 2 k z 2 c 8 k -1 4 v -1 p 2 -1 b 2 m -1 2 s 2 c 4 k -1 2 v -1 q 0 n 2 q -1 n b m -1 2 s -1 p n -1 q c v -1 2 k -1 n -1 1 2 p -1 b m -1 2 s 2 c v -1 2 k z q 1 p -1 b m -1 2 s 2 c v -1 2 k z 2 c 2 k -1 v -1 1 2 -1 q -1 Binomial n q Gamma q 1 2 -1 p -1 b m -1 2 s 2 c v -1 2 k z 2 c 8 k -1 4 v -1 -1 c b p m -1 2 s c 2 k -1 v -1 2 k -1 v q 0 n 2 q -1 n b m -1 2 s -1 p n -1 q c 2 k -1 v -1 n -1 1 2 p -1 b m -1 2 s 4 c k z -1 2 c v z q 1 -1 -1 b m p 2 b s 4 c k z -1 2 c v z 2 2 c k -1 c v -1 1 2 -1 q -1 Binomial n q Gamma q 1 2 -1 b m -1 2 s p 4 c k z -1 2 c v z 2 c 8 k -1 4 v -1 c v -1 2 k q 0 n 2 q -1 n -1 p -1 b m -1 2 s n -1 q c 2 k -1 v -1 n -1 1 2 p b m -1 2 s 4 c k z -1 2 c v z q 1 -1 p b m -1 2 s 4 c k z -1 2 c v z 2 c 2 k -1 v -1 1 2 -1 q -1 Binomial n q Gamma q 1 2 -1 -1 p b m -1 2 s 4 c k z -1 2 c v z 2 c 8 k -1 4 v -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