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 Cosh

 http://functions.wolfram.com/01.20.21.4020.01

 Input Form

 Integrate[z^n E^(b z^2 + e) Cos[a z^2 + q]^m Cosh[c z^2 + g]^v, z] == (-2^(-1 - m - v)) E^e z^(1 + n) ((-b) z^2)^((1/2) (-1 - n)) Binomial[m, m/2] Binomial[v, v/2] Gamma[(1 + n)/2, (-b) z^2] (1 - Mod[m, 2]) (1 - Mod[v, 2]) - 2^(-1 - m - v) z^(1 + n) Binomial[v, v/2] (1 - Mod[v, 2]) Sum[Binomial[m, s] (E^(e + I q (m - 2 s)) ((-b - I a (m - 2 s)) z^2)^ ((1/2) (-1 - n)) Gamma[(1 + n)/2, (-b - I a (m - 2 s)) z^2] + E^(e - I q (m - 2 s)) ((-b + I a (m - 2 s)) z^2)^((1/2) (-1 - n)) Gamma[(1 + n)/2, (-b + I a (m - 2 s)) z^2]), {s, 0, Floor[(1/2) (-1 + m)]}] - 2^(-1 - m - v) z^(1 + n) Binomial[m, m/2] (1 - Mod[m, 2]) Sum[Binomial[v, k] (E^(e + g (-2 k + v)) ((-b - c (-2 k + v)) z^2)^ ((1/2) (-1 - n)) Gamma[(1 + n)/2, (-b - c (-2 k + v)) z^2] + E^(e - g (-2 k + v)) ((-b + c (-2 k + v)) z^2)^((1/2) (-1 - n)) Gamma[(1 + n)/2, (-b + c (-2 k + v)) z^2]), {k, 0, Floor[(1/2) (-1 + v)]}] - 2^(-1 - m - v) z^(1 + n) Sum[Binomial[v, k] Sum[Binomial[m, s] (E^(e + I q (m - 2 s) + g (-2 k + v)) ((-b - I a (m - 2 s) - c (-2 k + v)) z^2)^((1/2) (-1 - n)) Gamma[(1 + n)/2, (-b - I a (m - 2 s) - c (-2 k + v)) z^2] + E^(e - I q (m - 2 s) + g (-2 k + v)) ((-b + I a (m - 2 s) - c (-2 k + v)) z^2)^((1/2) (-1 - n)) Gamma[(1 + n)/2, (-b + I a (m - 2 s) - c (-2 k + v)) z^2] + E^(e + I q (m - 2 s) - g (-2 k + v)) ((-b - I a (m - 2 s) + c (-2 k + v)) z^2)^((1/2) (-1 - n)) Gamma[(1 + n)/2, (-b - I a (m - 2 s) + c (-2 k + v)) z^2] + E^(e - I q (m - 2 s) - g (-2 k + v)) ((-b + I a (m - 2 s) + c (-2 k + v)) z^2)^((1/2) (-1 - n)) Gamma[(1 + n)/2, (-b + I a (m - 2 s) + c (-2 k + v)) z^2]), {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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 MathML Form

 z n b z 2 + e cos m ( a z 2 + q ) cosh v ( c z 2 + g ) z - 2 - m - v - 1 e 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]]]]] Γ ( n + 1 2 , - b z 2 ) ( 1 - m mod 2 \$CellContext`m 2 ) ( 1 - v mod 2 \$CellContext`v 2 ) ( - b z 2 ) 1 2 ( - n - 1 ) - 2 - m - v - 1 z n + 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]]]]] ( 1 - v mod 2 \$CellContext`v 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]]]]] ( e - q ( m - 2 s ) Γ ( n + 1 2 , ( a ( m - 2 s ) - b ) z 2 ) ( ( a ( m - 2 s ) - b ) z 2 ) 1 2 ( - n - 1 ) + e + q ( m - 2 s ) ( ( - b - a ( m - 2 s ) ) z 2 ) 1 2 ( - n - 1 ) Γ ( n + 1 2 , ( - b - a ( m - 2 s ) ) z 2 ) ) - 2 - m - v - 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]]]]] ( 1 - m mod 2 \$CellContext`m 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]]]]] ( e - g ( v - 2 k ) Γ ( n + 1 2 , ( c ( v - 2 k ) - b ) z 2 ) ( ( c ( v - 2 k ) - b ) z 2 ) 1 2 ( - n - 1 ) + e + g ( v - 2 k ) ( ( - b - c ( v - 2 k ) ) z 2 ) 1 2 ( - n - 1 ) Γ ( n + 1 2 , ( - b - c ( v - 2 k ) ) z 2 ) ) - 2 - m - v - 1 z n + 1 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]]]]] ( e - q ( m - 2 s ) - g ( v - 2 k ) Γ ( n + 1 2 , ( - b + a ( m - 2 s ) + c ( v - 2 k ) ) z 2 ) ( ( - b + a ( m - 2 s ) + c ( v - 2 k ) ) z 2 ) 1 2 ( - n - 1 ) + e + q ( m - 2 s ) - g ( v - 2 k ) ( ( - b - a ( m - 2 s ) + c ( v - 2 k ) ) z 2 ) 1 2 ( - n - 1 ) Γ ( n + 1 2 , ( - b - a ( m - 2 s ) + c ( v - 2 k ) ) z 2 ) + e - q ( m - 2 s ) + g ( v - 2 k ) ( ( - b + a ( m - 2 s ) - c ( v - 2 k ) ) z 2 ) 1 2 ( - n - 1 ) Γ ( n + 1 2 , ( - b + a ( m - 2 s ) - c ( v - 2 k ) ) z 2 ) + e + q ( m - 2 s ) + g ( v - 2 k ) ( ( - b - a ( m - 2 s ) - c ( v - 2 k ) ) z 2 ) 1 2 ( - n - 1 ) Γ ( n + 1 2 , ( - b - a ( m - 2 s ) - c ( v - 2 k ) ) z 2 ) ) /; m + v + n Condition z z n b z 2 e a z 2 q m c z 2 g v -1 2 -1 m -1 v -1 e z n 1 Binomial m m 2 -1 Binomial v v 2 -1 Gamma n 1 2 -1 -1 b z 2 1 -1 \$CellContext`m 2 1 -1 \$CellContext`v 2 -1 b z 2 1 2 -1 n -1 -1 2 -1 m -1 v -1 z n 1 Binomial v v 2 -1 1 -1 \$CellContext`v 2 s 0 m -1 2 -1 Binomial m s e -1 q m -1 2 s Gamma n 1 2 -1 a m -1 2 s -1 b z 2 a m -1 2 s -1 b z 2 1 2 -1 n -1 e q m -1 2 s -1 b -1 a m -1 2 s z 2 1 2 -1 n -1 Gamma n 1 2 -1 -1 b -1 a m -1 2 s z 2 -1 2 -1 m -1 v -1 z n 1 Binomial m m 2 -1 1 -1 \$CellContext`m 2 k 0 v -1 2 -1 Binomial v k e -1 g v -1 2 k Gamma n 1 2 -1 c v -1 2 k -1 b z 2 c v -1 2 k -1 b z 2 1 2 -1 n -1 e g v -1 2 k -1 b -1 c v -1 2 k z 2 1 2 -1 n -1 Gamma n 1 2 -1 -1 b -1 c v -1 2 k z 2 -1 2 -1 m -1 v -1 z n 1 k 0 v -1 2 -1 Binomial v k s 0 m -1 2 -1 Binomial m s e -1 q m -1 2 s -1 g v -1 2 k Gamma n 1 2 -1 -1 b a m -1 2 s c v -1 2 k z 2 -1 b a m -1 2 s c v -1 2 k z 2 1 2 -1 n -1 e q m -1 2 s -1 g v -1 2 k -1 b -1 a m -1 2 s c v -1 2 k z 2 1 2 -1 n -1 Gamma n 1 2 -1 -1 b -1 a m -1 2 s c v -1 2 k z 2 e -1 q m -1 2 s g v -1 2 k -1 b a m -1 2 s -1 c v -1 2 k z 2 1 2 -1 n -1 Gamma n 1 2 -1 -1 b a m -1 2 s -1 c v -1 2 k z 2 e q m -1 2 s g v -1 2 k -1 b -1 a m -1 2 s -1 c v -1 2 k z 2 1 2 -1 n -1 Gamma n 1 2 -1 -1 b -1 a m -1 2 s -1 c v -1 2 k z 2 m SuperPlus v SuperPlus n [/itex]

 Rule Form

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

 2002-12-18