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 Cosh

 http://functions.wolfram.com/01.20.21.1451.01

 Input Form

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

 Standard Form

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 MathML Form

 z n b z 2 + d z + e cos m ( a z 2 + p z + q ) cosh ( c z 2 + f z + g ) 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 ) ( ( b - c ) - n - 1 - ( d - f ) 2 4 ( b - c ) + e - g j = 0 n 2 j - n ( f - d ) n - j ( d - f + 2 ( b - c ) z ) j + 1 ( - ( d - f + 2 ( b - c ) z ) 2 b - c ) 1 2 ( - j - 1 ) ( n j ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["n", Identity]], List[TagBox["j", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] Γ ( j + 1 2 , - ( d - f + 2 ( b - c ) z ) 2 4 ( b - c ) ) + ( b + c ) - n - 1 - ( d + f ) 2 4 ( b + c ) + e + g j = 0 n 2 j - n ( - d - f ) n - j ( d + f + 2 ( b + c ) z ) j + 1 ( - ( d + f + 2 ( b + c ) z ) 2 b + c ) 1 2 ( - j - 1 ) ( n j ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["n", Identity]], List[TagBox["j", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] Γ ( j + 1 2 , - ( d + f + 2 ( b + c ) z ) 2 4 ( b + c ) ) ) - 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 - f + ( 2 k - m ) p ) 2 4 ( b - c + a ( 2 k - m ) ) + e - g + ( 2 k - m ) q ( b - c + a ( 2 k - m ) ) - n - 1 j = 0 n 2 j - n ( - d + f - ( 2 k - m ) p ) n - j ( d - f + ( 2 k - m ) p + 2 ( b - c + a ( 2 k - m ) ) z ) j + 1 ( - ( d - f + ( 2 k - m ) p + 2 ( b - c + a ( 2 k - m ) ) z ) 2 b - c + a ( 2 k - m ) ) 1 2 ( - j - 1 ) ( n j ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["n", Identity]], List[TagBox["j", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] Γ ( j + 1 2 , - ( d - f + ( 2 k - m ) p + 2 ( b - c + a ( 2 k - m ) ) z ) 2 4 ( b - c + a ( 2 k - m ) ) ) + - ( d + f + ( 2 k - m ) p ) 2 4 ( b + c + a ( 2 k - m ) ) + e + g + ( 2 k - m ) q ( b + c + a ( 2 k - m ) ) - n - 1 j = 0 n 2 j - n ( - d - f - ( 2 k - m ) p ) n - j ( d + f + ( 2 k - m ) p + 2 ( b + c + a ( 2 k - m ) ) z ) j + 1 ( - ( d + f + ( 2 k - m ) p + 2 ( b + c + a ( 2 k - m ) ) z ) 2 b + c + a ( 2 k - m ) ) 1 2 ( - j - 1 ) ( n j ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["n", Identity]], List[TagBox["j", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] Γ ( j + 1 2 , - ( d + f + ( 2 k - m ) p + 2 ( b + c + a ( 2 k - m ) ) z ) 2 4 ( b + c + a ( 2 k - m ) ) ) + - ( d - f + ( m - 2 k ) p ) 2 4 ( b - c + a ( m - 2 k ) ) + e - g + ( m - 2 k ) q ( b - c + a ( m - 2 k ) ) - n - 1 j = 0 n 2 j - n ( - d + f - ( m - 2 k ) p ) n - j ( d - f + ( m - 2 k ) p + 2 ( b - c + a ( m - 2 k ) ) z ) j + 1 ( - ( d - f + ( m - 2 k ) p + 2 ( b - c + a ( m - 2 k ) ) z ) 2 b - c + a ( m - 2 k ) ) 1 2 ( - j - 1 ) ( n j ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["n", Identity]], List[TagBox["j", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] Γ ( j + 1 2 , - ( d - f + ( m - 2 k ) p + 2 ( b - c + a ( m - 2 k ) ) z ) 2 4 ( b - c + a ( m - 2 k ) ) ) + - ( d + f + ( m - 2 k ) p ) 2 4 ( b + c + a ( m - 2 k ) ) + e + g + ( m - 2 k ) q ( b + c + a ( m - 2 k ) ) - n - 1 j = 0 n 2 j - n ( - d - f - ( m - 2 k ) p ) n - j ( d + f + ( m - 2 k ) p + 2 ( b + c + a ( m - 2 k ) ) z ) j + 1 ( - ( d + f + ( m - 2 k ) p + 2 ( b + c + a ( m - 2 k ) ) z ) 2 b + c + a ( m - 2 k ) ) 1 2 ( - j - 1 ) ( n j ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["n", Identity]], List[TagBox["j", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] Γ ( j + 1 2 , - ( d + f + ( m - 2 k ) p + 2 ( b + c + a ( m - 2 k ) ) z ) 2 4 ( b + c + a ( m - 2 k ) ) ) ) /; n m + Condition z z n b z 2 d z e a z 2 p z q m c z 2 f z g -1 2 -1 m -2 Binomial m m 2 -1 1 -1 \$CellContext`m 2 b -1 c -1 n -1 -1 d -1 f 2 4 b -1 c -1 e -1 g j 0 n 2 j -1 n f -1 d n -1 j d -1 f 2 b -1 c z j 1 -1 d -1 f 2 b -1 c z 2 b -1 c -1 1 2 -1 j -1 Binomial n j Gamma j 1 2 -1 -1 d -1 f 2 b -1 c z 2 4 b -1 c -1 b c -1 n -1 -1 d f 2 4 b c -1 e g j 0 n 2 j -1 n -1 d -1 f n -1 j d f 2 b c z j 1 -1 d f 2 b c z 2 b c -1 1 2 -1 j -1 Binomial n j Gamma j 1 2 -1 -1 d f 2 b c z 2 4 b c -1 -1 2 -1 m -2 k 0 m -1 2 -1 Binomial m k -1 d -1 f 2 k -1 m p 2 4 b -1 c a 2 k -1 m -1 e -1 g 2 k -1 m q b -1 c a 2 k -1 m -1 n -1 j 0 n 2 j -1 n -1 d f -1 2 k -1 m p n -1 j d -1 f 2 k -1 m p 2 b -1 c a 2 k -1 m z j 1 -1 d -1 f 2 k -1 m p 2 b -1 c a 2 k -1 m z 2 b -1 c a 2 k -1 m -1 1 2 -1 j -1 Binomial n j Gamma j 1 2 -1 -1 d -1 f 2 k -1 m p 2 b -1 c a 2 k -1 m z 2 4 b -1 c a 2 k -1 m -1 -1 d f 2 k -1 m p 2 4 b c a 2 k -1 m -1 e g 2 k -1 m q b c a 2 k -1 m -1 n -1 j 0 n 2 j -1 n -1 d -1 f -1 2 k -1 m p n -1 j d f 2 k -1 m p 2 b c a 2 k -1 m z j 1 -1 d f 2 k -1 m p 2 b c a 2 k -1 m z 2 b c a 2 k -1 m -1 1 2 -1 j -1 Binomial n j Gamma j 1 2 -1 -1 d f 2 k -1 m p 2 b c a 2 k -1 m z 2 4 b c a 2 k -1 m -1 -1 d -1 f m -1 2 k p 2 4 b -1 c a m -1 2 k -1 e -1 g m -1 2 k q b -1 c a m -1 2 k -1 n -1 j 0 n 2 j -1 n -1 d f -1 m -1 2 k p n -1 j d -1 f m -1 2 k p 2 b -1 c a m -1 2 k z j 1 -1 d -1 f m -1 2 k p 2 b -1 c a m -1 2 k z 2 b -1 c a m -1 2 k -1 1 2 -1 j -1 Binomial n j Gamma j 1 2 -1 -1 d -1 f m -1 2 k p 2 b -1 c a m -1 2 k z 2 4 b -1 c a m -1 2 k -1 -1 d f m -1 2 k p 2 4 b c a m -1 2 k -1 e g m -1 2 k q b c a m -1 2 k -1 n -1 j 0 n 2 j -1 n -1 d -1 f -1 m -1 2 k p n -1 j d f m -1 2 k p 2 b c a m -1 2 k z j 1 -1 d f m -1 2 k p 2 b c a m -1 2 k z 2 b c a m -1 2 k -1 1 2 -1 j -1 Binomial n j Gamma j 1 2 -1 -1 d f m -1 2 k p 2 b c a m -1 2 k z 2 4 b c a m -1 2 k -1 n m SuperPlus [/itex]

 Rule Form

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

 2002-12-18