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 Cosh

 http://functions.wolfram.com/01.20.21.4008.01

 Input Form

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