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 Cosh

 http://functions.wolfram.com/01.20.21.4856.01

 Input Form

 Integrate[z^n E^(b z^2 + e) Sinh[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]))/I^m - 2^(-1 - m - v) z^(1 + n) Binomial[v, v/2] (1 - Mod[v, 2]) Sum[(-1)^k Binomial[m, k] (E^(e + (-2 k + m) q) ((-b - a (-2 k + m)) z^2)^((1/2) (-1 - n)) Gamma[(1 + n)/2, (-b - a (-2 k + m)) z^2] + (-1)^m E^(e - (-2 k + m) q) ((-b + a (-2 k + m)) z^2)^((1/2) (-1 - n)) Gamma[(1 + n)/2, (-b + a (-2 k + m)) z^2]), {k, 0, Floor[(1/2) (-1 + m)]}] - (2^(-1 - m - v) z^(1 + n) Binomial[m, m/2] (1 - Mod[m, 2]) Sum[Binomial[v, s] (E^(e + g (-2 s + v)) ((-b - c (-2 s + v)) z^2)^ ((1/2) (-1 - n)) Gamma[(1 + n)/2, (-b - c (-2 s + v)) z^2] + E^(e - g (-2 s + v)) ((-b + c (-2 s + v)) z^2)^((1/2) (-1 - n)) Gamma[(1 + n)/2, (-b + c (-2 s + v)) z^2]), {s, 0, Floor[(1/2) (-1 + v)]}])/I^m - (2^(-1 - m - v) z^(1 + n) Sum[(-1)^k Binomial[m, k] Sum[Binomial[v, s] (E^(e + (I m Pi)/2 + (-2 k + m) q + g (-2 s + v)) ((-b - a (-2 k + m) - c (-2 s + v)) z^2)^((1/2) (-1 - n)) Gamma[(1 + n)/2, (-b - a (-2 k + m) - c (-2 s + v)) z^2] + E^(e - (I m Pi)/2 - (-2 k + m) q + g (-2 s + v)) ((-b + a (-2 k + m) - c (-2 s + v)) z^2)^((1/2) (-1 - n)) Gamma[(1 + n)/2, (-b + a (-2 k + m) - c (-2 s + v)) z^2] + E^(e + (I m Pi)/2 + (-2 k + m) q - g (-2 s + v)) ((-b - a (-2 k + m) + c (-2 s + v)) z^2)^((1/2) (-1 - n)) Gamma[(1 + n)/2, (-b - a (-2 k + m) + c (-2 s + v)) z^2] + E^(e - (I m Pi)/2 - (-2 k + m) q - g (-2 s + v)) ((-b + a (-2 k + m) + c (-2 s + v)) z^2)^((1/2) (-1 - n)) Gamma[(1 + n)/2, (-b + a (-2 k + m) + c (-2 s + v)) z^2]), {s, 0, Floor[(1/2) (-1 + v)]}], {k, 0, Floor[(1/2) (-1 + m)]}])/ I^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 b z 2 + e sinh m ( a z 2 + q ) cosh v ( c z 2 + g ) z - 2 - m - v - 1 - m e z n + 1 ( - b z 2 ) 1 2 ( - 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 ) - 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 ) k = 0 m - 1 2 ( - 1 ) k ( m k ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["m", Identity]], List[TagBox["k", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( ( - 1 ) m e - ( m - 2 k ) q Γ ( n + 1 2 , ( a ( m - 2 k ) - b ) z 2 ) ( ( a ( m - 2 k ) - b ) z 2 ) 1 2 ( - n - 1 ) + e + ( m - 2 k ) q ( ( - b - a ( m - 2 k ) ) z 2 ) 1 2 ( - n - 1 ) Γ ( n + 1 2 , ( - b - a ( m - 2 k ) ) z 2 ) ) - 2 - m - v - 1 - m 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 ) 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 - g ( v - 2 s ) Γ ( n + 1 2 , ( c ( v - 2 s ) - b ) z 2 ) ( ( c ( v - 2 s ) - b ) z 2 ) 1 2 ( - n - 1 ) + e + g ( v - 2 s ) ( ( - b - c ( v - 2 s ) ) z 2 ) 1 2 ( - n - 1 ) Γ ( n + 1 2 , ( - b - c ( v - 2 s ) ) z 2 ) ) - 2 - m - v - 1 - m z n + 1 k = 0 m - 1 2 ( - 1 ) k ( 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 ( v s ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["v", Identity]], List[TagBox["s", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( e - ( m - 2 k ) q - g ( v - 2 s ) - m π 2 Γ ( n + 1 2 , ( - b + a ( m - 2 k ) + c ( v - 2 s ) ) z 2 ) ( ( - b + a ( m - 2 k ) + c ( v - 2 s ) ) z 2 ) 1 2 ( - n - 1 ) + e + ( m - 2 k ) q - g ( v - 2 s ) + m π 2 ( ( - b - a ( m - 2 k ) + c ( v - 2 s ) ) z 2 ) 1 2 ( - n - 1 ) Γ ( n + 1 2 , ( - b - a ( m - 2 k ) + c ( v - 2 s ) ) z 2 ) + e - ( m - 2 k ) q + g ( v - 2 s ) - m π 2 ( ( - b + a ( m - 2 k ) - c ( v - 2 s ) ) z 2 ) 1 2 ( - n - 1 ) Γ ( n + 1 2 , ( - b + a ( m - 2 k ) - c ( v - 2 s ) ) z 2 ) + e + ( m - 2 k ) q + g ( v - 2 s ) + m π 2 ( ( - b - a ( m - 2 k ) - c ( v - 2 s ) ) z 2 ) 1 2 ( - n - 1 ) Γ ( n + 1 2 , ( - b - a ( m - 2 k ) - c ( v - 2 s ) ) 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 -1 m e z n 1 -1 b z 2 1 2 -1 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 2 -1 m -1 v -1 z n 1 Binomial v v 2 -1 1 -1 \$CellContext`v 2 k 0 m -1 2 -1 -1 k Binomial m k -1 m e -1 m -1 2 k q Gamma n 1 2 -1 a m -1 2 k -1 b z 2 a m -1 2 k -1 b z 2 1 2 -1 n -1 e m -1 2 k q -1 b -1 a m -1 2 k z 2 1 2 -1 n -1 Gamma n 1 2 -1 -1 b -1 a m -1 2 k z 2 -1 2 -1 m -1 v -1 -1 m z n 1 Binomial m m 2 -1 1 -1 \$CellContext`m 2 s 0 v -1 2 -1 Binomial v s e -1 g v -1 2 s Gamma n 1 2 -1 c v -1 2 s -1 b z 2 c v -1 2 s -1 b z 2 1 2 -1 n -1 e g v -1 2 s -1 b -1 c v -1 2 s z 2 1 2 -1 n -1 Gamma n 1 2 -1 -1 b -1 c v -1 2 s z 2 -1 2 -1 m -1 v -1 -1 m z n 1 k 0 m -1 2 -1 -1 k Binomial m k s 0 v -1 2 -1 Binomial v s e -1 m -1 2 k q -1 g v -1 2 s -1 m 2 -1 Gamma n 1 2 -1 -1 b a m -1 2 k c v -1 2 s z 2 -1 b a m -1 2 k c v -1 2 s z 2 1 2 -1 n -1 e m -1 2 k q -1 g v -1 2 s m 2 -1 -1 b -1 a m -1 2 k c v -1 2 s z 2 1 2 -1 n -1 Gamma n 1 2 -1 -1 b -1 a m -1 2 k c v -1 2 s z 2 e -1 m -1 2 k q g v -1 2 s -1 m 2 -1 -1 b a m -1 2 k -1 c v -1 2 s z 2 1 2 -1 n -1 Gamma n 1 2 -1 -1 b a m -1 2 k -1 c v -1 2 s z 2 e m -1 2 k q g v -1 2 s m 2 -1 -1 b -1 a m -1 2 k -1 c v -1 2 s z 2 1 2 -1 n -1 Gamma n 1 2 -1 -1 b -1 a m -1 2 k -1 c v -1 2 s 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