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 Cosh

 http://functions.wolfram.com/01.20.21.1434.01

 Input Form

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

 Rule Form

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

 2002-12-18