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 Cosh

 http://functions.wolfram.com/01.20.21.4940.01

 Input Form

 Integrate[z^n Cos[b z^2 + d z]^m Cosh[c z^2 + f z], z] == (-2^(-2 - m)) c^(-1 - n) Binomial[m, m/2] (1 - Mod[m, 2]) (Sum[2^(-n + q) (-f)^(n - q) (f + 2 c z)^(1 + q) (-((f + 2 c z)^2/c))^ ((1/2) (-1 - q)) Binomial[n, q] Gamma[(1 + q)/2, -((f + 2 c z)^2/(4 c))], {q, 0, n}]/E^(f^2/(4 c)) + (-1)^(-1 + n) E^(f^2/(4 c)) Sum[2^(-n + q) f^(n - q) (-f - 2 c z)^(1 + q) ((f + 2 c z)^2/c)^((1/2) (-1 - q)) Binomial[n, q] Gamma[(1 + q)/2, (f + 2 c z)^2/(4 c)], {q, 0, n}]) - 2^(-2 - m) Sum[Binomial[m, k] (((-c + I b (2 k - m))^(-1 - n) Sum[2^(-n + q) (f - I d (2 k - m))^(n - q) (-f + I d (2 k - m) + 2 (-c + I b (2 k - m)) z)^(1 + q) (-((-f + I d (2 k - m) + 2 (-c + I b (2 k - m)) z)^2/ (-c + I b (2 k - m))))^((1/2) (-1 - q)) Binomial[n, q] Gamma[(1 + q)/2, -((-f + I d (2 k - m) + 2 (-c + I b (2 k - m)) z)^ 2/(4 (-c + I b (2 k - m))))], {q, 0, n}])/ E^((-f + I d (2 k - m))^2/(4 (-c + I b (2 k - m)))) + ((c + I b (2 k - m))^(-1 - n) Sum[2^(-n + q) (-f - I d (2 k - m))^ (n - q) (f + I d (2 k - m) + 2 (c + I b (2 k - m)) z)^(1 + q) (-((f + I d (2 k - m) + 2 (c + I b (2 k - m)) z)^2/ (c + I b (2 k - m))))^((1/2) (-1 - q)) Binomial[n, q] Gamma[(1 + q)/2, -((f + I d (2 k - m) + 2 (c + I b (2 k - m)) z)^2/ (4 (c + I b (2 k - m))))], {q, 0, n}])/ E^((f + I d (2 k - m))^2/(4 (c + I b (2 k - m)))) + ((-c + I b (-2 k + m))^(-1 - n) Sum[2^(-n + q) (f - I d (-2 k + m))^ (n - q) (-f + I d (-2 k + m) + 2 (-c + I b (-2 k + m)) z)^(1 + q) (-((-f + I d (-2 k + m) + 2 (-c + I b (-2 k + m)) z)^2/ (-c + I b (-2 k + m))))^((1/2) (-1 - q)) Binomial[n, q] Gamma[(1 + q)/2, -((-f + I d (-2 k + m) + 2 (-c + I b (-2 k + m)) z)^2/(4 (-c + I b (-2 k + m))))], {q, 0, n}])/ E^((-f + I d (-2 k + m))^2/(4 (-c + I b (-2 k + m)))) + ((c + I b (-2 k + m))^(-1 - n) Sum[2^(-n + q) (-f - I d (-2 k + m))^ (n - q) (f + I d (-2 k + m) + 2 (c + I b (-2 k + m)) z)^(1 + q) (-((f + I d (-2 k + m) + 2 (c + I b (-2 k + m)) z)^2/ (c + I b (-2 k + m))))^((1/2) (-1 - q)) Binomial[n, q] Gamma[(1 + q)/2, -((f + I d (-2 k + m) + 2 (c + I b (-2 k + m)) z)^ 2/(4 (c + I b (-2 k + m))))], {q, 0, n}])/ E^((f + I d (-2 k + m))^2/(4 (c + I b (-2 k + m))))), {k, 0, Floor[(1/2) (-1 + m)]}] /; Element[n, Integers] && n >= 0 && Element[m, Integers] && m > 0

 Standard Form

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"+", "m"]], ")"]]]]]]]]], ")"]], RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "-", "q"]], ")"]]]]], " ", RowBox[List["Binomial", "[", RowBox[List["n", ",", "q"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox[RowBox[List["1", "+", "q"]], "2"], ",", RowBox[List["-", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List["f", "+", RowBox[List["\[ImaginaryI]", " ", "d", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "m"]], ")"]]]], "+", RowBox[List["2", " ", RowBox[List["(", RowBox[List["c", "+", RowBox[List["\[ImaginaryI]", " ", "b", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "m"]], ")"]]]]]], ")"]], " ", "z"]]]], ")"]], "2"], RowBox[List["4", " ", RowBox[List["(", RowBox[List["c", "+", RowBox[List["\[ImaginaryI]", " ", "b", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "m"]], ")"]]]]]], ")"]]]]]]]]], "]"]]]]]]]]]], ")"]]]]]]]]]]]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "\[And]", RowBox[List["n", "\[GreaterEqual]", "0"]], "\[And]", RowBox[List["m", "\[Element]", "Integers"]], "\[And]", RowBox[List["m", ">", "0"]]]]]]]]

 MathML Form

 z n cos m ( b z 2 + d z ) cosh ( c z 2 + f z ) z - 2 - m - 2 ( 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]]]]] ( 1 - m mod 2 \$CellContext`m 2 ) ( ( - 1 ) n - 1 f 2 4 c q = 0 n 2 q - n f n - q ( - f - 2 c z ) q + 1 ( ( f + 2 c z ) 2 c ) 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 , ( f + 2 c z ) 2 4 c ) + - f 2 4 c q = 0 n 2 q - n ( - f ) n - q ( f + 2 c z ) q + 1 ( - ( f + 2 c z ) 2 c ) 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 , - ( f + 2 c z ) 2 4 c ) ) c - n - 1 - 2 - m - 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]]]]] ( - ( d ( 2 k - m ) - f ) 2 4 ( b ( 2 k - m ) - c ) ( q = 0 n 2 q - n ( f - d ( 2 k - m ) ) n - q ( - f + d ( 2 k - m ) + 2 ( b ( 2 k - m ) - c ) z ) q + 1 ( - ( - f + d ( 2 k - m ) + 2 ( b ( 2 k - m ) - c ) z ) 2 b ( 2 k - m ) - c ) 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 , - ( - f + d ( 2 k - m ) + 2 ( b ( 2 k - m ) - c ) z ) 2 4 ( b ( 2 k - m ) - c ) ) ) ( b ( 2 k - m ) - c ) - n - 1 + - ( f + d ( 2 k - m ) ) 2 4 ( c + b ( 2 k - m ) ) ( c + b ( 2 k - m ) ) - n - 1 q = 0 n 2 q - n ( - f - d ( 2 k - m ) ) n - q ( f + d ( 2 k - m ) + 2 ( c + b ( 2 k - m ) ) z ) q + 1 ( - ( f + d ( 2 k - m ) + 2 ( c + b ( 2 k - m ) ) z ) 2 c + 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 , - ( f + d ( 2 k - m ) + 2 ( c + b ( 2 k - m ) ) z ) 2 4 ( c + b ( 2 k - m ) ) ) + - ( d ( m - 2 k ) - f ) 2 4 ( b ( m - 2 k ) - c ) ( b ( m - 2 k ) - c ) - n - 1 q = 0 n 2 q - n ( f - d ( m - 2 k ) ) n - q ( - f + d ( m - 2 k ) + 2 ( b ( m - 2 k ) - c ) z ) q + 1 ( - ( - f + d ( m - 2 k ) + 2 ( b ( m - 2 k ) - c ) z ) 2 b ( m - 2 k ) - c ) 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 , - ( - f + d ( m - 2 k ) + 2 ( b ( m - 2 k ) - c ) z ) 2 4 ( b ( m - 2 k ) - c ) ) + - ( f + d ( m - 2 k ) ) 2 4 ( c + b ( m - 2 k ) ) ( c + b ( m - 2 k ) ) - n - 1 q = 0 n 2 q - n ( - f - d ( m - 2 k ) ) n - q ( f + d ( m - 2 k ) + 2 ( c + b ( m - 2 k ) ) z ) q + 1 ( - ( f + d ( m - 2 k ) + 2 ( c + b ( m - 2 k ) ) z ) 2 c + 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 , - ( f + d ( m - 2 k ) + 2 ( c + b ( m - 2 k ) ) z ) 2 4 ( c + b ( m - 2 k ) ) ) ) /; n m + Condition z z n b z 2 d z m c z 2 f z -1 2 -1 m -2 Binomial m m 2 -1 1 -1 \$CellContext`m 2 -1 n -1 f 2 4 c -1 q 0 n 2 q -1 n f n -1 q -1 f -1 2 c z q 1 f 2 c z 2 c -1 1 2 -1 q -1 Binomial n q Gamma q 1 2 -1 f 2 c z 2 4 c -1 -1 f 2 4 c -1 q 0 n 2 q -1 n -1 f n -1 q f 2 c z q 1 -1 f 2 c z 2 c -1 1 2 -1 q -1 Binomial n q Gamma q 1 2 -1 -1 f 2 c z 2 4 c -1 c -1 n -1 -1 2 -1 m -2 k 0 m -1 2 -1 Binomial m k -1 d 2 k -1 m -1 f 2 4 b 2 k -1 m -1 c -1 q 0 n 2 q -1 n f -1 d 2 k -1 m n -1 q -1 f d 2 k -1 m 2 b 2 k -1 m -1 c z q 1 -1 -1 f d 2 k -1 m 2 b 2 k -1 m -1 c z 2 b 2 k -1 m -1 c -1 1 2 -1 q -1 Binomial n q Gamma q 1 2 -1 -1 -1 f d 2 k -1 m 2 b 2 k -1 m -1 c z 2 4 b 2 k -1 m -1 c -1 b 2 k -1 m -1 c -1 n -1 -1 f d 2 k -1 m 2 4 c b 2 k -1 m -1 c b 2 k -1 m -1 n -1 q 0 n 2 q -1 n -1 f -1 d 2 k -1 m n -1 q f d 2 k -1 m 2 c b 2 k -1 m z q 1 -1 f d 2 k -1 m 2 c b 2 k -1 m z 2 c b 2 k -1 m -1 1 2 -1 q -1 Binomial n q Gamma q 1 2 -1 -1 f d 2 k -1 m 2 c b 2 k -1 m z 2 4 c b 2 k -1 m -1 -1 d m -1 2 k -1 f 2 4 b m -1 2 k -1 c -1 b m -1 2 k -1 c -1 n -1 q 0 n 2 q -1 n f -1 d m -1 2 k n -1 q -1 f d m -1 2 k 2 b m -1 2 k -1 c z q 1 -1 -1 f d m -1 2 k 2 b m -1 2 k -1 c z 2 b m -1 2 k -1 c -1 1 2 -1 q -1 Binomial n q Gamma q 1 2 -1 -1 -1 f d m -1 2 k 2 b m -1 2 k -1 c z 2 4 b m -1 2 k -1 c -1 -1 f d m -1 2 k 2 4 c b m -1 2 k -1 c b m -1 2 k -1 n -1 q 0 n 2 q -1 n -1 f -1 d m -1 2 k n -1 q f d m -1 2 k 2 c b m -1 2 k z q 1 -1 f d m -1 2 k 2 c b m -1 2 k z 2 c b m -1 2 k -1 1 2 -1 q -1 Binomial n q Gamma q 1 2 -1 -1 f d m -1 2 k 2 c b m -1 2 k z 2 4 c b m -1 2 k -1 n m SuperPlus [/itex]

 Rule Form

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

 2003-08-21