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 Cosh

 http://functions.wolfram.com/01.20.21.4012.01

 Input Form

 Integrate[z^n E^(p z^2) Cos[b z]^m Cosh[c z^2]^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[m, m/2] (-1 + Mod[m, 2]) Sum[Binomial[v, s] ((-((-(p + 2 c s - c v)) z^2)^((1/2) (-1 - n))) Gamma[(1 + n)/2, (-(p + 2 c s - c v)) z^2] - ((-(p + c (-2 s + v))) z^2)^((1/2) (-1 - n)) Gamma[(1 + n)/2, (-(p + c (-2 s + v))) z^2]), {s, 0, Floor[(1/2) (-1 + v)]}] + p^(-1 - n) Binomial[v, v/2] (-1 + Mod[v, 2]) Sum[E^((b^2 (m - 2 s)^2)/(4 p)) Binomial[m, s] (Sum[2^(-n + q) ((-I) b (m - 2 s))^(n - q) ((b (m - 2 s) - 2 I p z)^2/p)^((1/2) (-1 - q)) (I b (m - 2 s) + 2 p z)^(1 + q) Binomial[n, q] Gamma[(1 + q)/2, (b (m - 2 s) - 2 I p z)^2/(4 p)], {q, 0, n}] + Sum[2^(-n + q) (I b (m - 2 s))^(n - q) ((b (m - 2 s) + 2 I p z)^2/p)^ ((1/2) (-1 - q)) ((-I) b (m - 2 s) + 2 p z)^(1 + q) Binomial[n, q] Gamma[(1 + q)/2, (b (m - 2 s) + 2 I p z)^2/(4 p)], {q, 0, n}]), {s, 0, Floor[(1/2) (-1 + m)]}] + Sum[Binomial[v, k] Sum[-(Binomial[m, s] (E^((b^2 (m - 2 s)^2)/(4 (2 c k + p - c v))) Sqrt[p + c (-2 k + v)] Sum[(I b (-(m/2) + s))^(n - q) (2 c k + p - c v)^(-(1/2) - n) ((b (m - 2 s) - 2 I (2 c k + p - c v) z)^2/(2 c k + p - c v))^((1/2) (-1 - q)) (I b (m - 2 s) + 2 (2 c k + p - c v) z)^(1 + q) Binomial[n, q] Gamma[(1 + q)/2, (b (m - 2 s) - 2 I (2 c k + p - c v) z)^2/ (4 (2 c k + p - c v))], {q, 0, n}] + E^((b^2 (m - 2 s)^2)/(4 (2 c k + p - c v))) Sqrt[-2 c k + p + c v] Sum[2^(-n + q) (I b (m - 2 s))^(n - q) (2 c k + p - c v)^( -(1/2) - n) ((b (m - 2 s) + 2 I (2 c k + p - c v) z)^2/ (2 c k + p - c v))^((1/2) (-1 - q)) ((-I) b (m - 2 s) + 2 (2 c k + p - c v) z)^(1 + q) Binomial[n, q] Gamma[(1 + q)/2, (b (m - 2 s) + 2 I (2 c k + p - c v) z)^2/ (4 (2 c k + p - c v))], {q, 0, n}] + E^((b^2 (m - 2 s)^2)/(4 (-2 c k + p + c v))) Sqrt[2 c k + p - c v] (Sum[2^(-n + q) (I b (m - 2 s))^(n - q) (-2 c k + p + c v)^ (-(1/2) - n) (-(I b (m - 2 s) - 2 (-2 c k + p + c v) z)^2/ (-2 c k + p + c v))^((1/2) (-1 - q)) ((-I) b (m - 2 s) + 2 (-2 c k + p + c v) z)^(1 + q) Binomial[n, q] Gamma[ (1 + q)/2, -((I b (m - 2 s) - 2 (-2 c k + p + c v) z)^2/ (4 (-2 c k + p + c v)))], {q, 0, n}] + Sum[(I b (-(m/2) + s))^(n - q) (-2 c k + p + c v)^(-(1/2) - n) (I b (m - 2 s) + 2 (-2 c k + p + c v) z)^(1 + q) (-(I b (m - 2 s) + 2 (-2 c k + p + c v) z)^2/(-2 c k + p + c v))^((1/2) (-1 - q)) Binomial[n, q] Gamma[(1 + q)/2, -((I b (m - 2 s) + 2 (-2 c k + p + c v) z)^2/(4 (-2 c k + p + c v)))], {q, 0, n}])))/(Sqrt[2 c k + p - c v] Sqrt[p + c (-2 k + v)]), {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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RowBox[List[RowBox[List["m", "\[Element]", "Integers"]], "\[And]", RowBox[List["m", ">", "0"]], "\[And]", RowBox[List["v", "\[Element]", "Integers"]], "\[And]", RowBox[List["v", ">", "0"]], "\[And]", RowBox[List["n", "\[Element]", "Integers"]], "\[And]", RowBox[List["n", "\[GreaterEqual]", "0"]]]]]]]]

 MathML Form

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