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 Cosh

 http://functions.wolfram.com/01.20.21.4021.01

 Input Form

 Integrate[z^n E^(b Sqrt[z] + e) Cos[a Sqrt[z] + q]^m Cosh[c Sqrt[z] + g]^v, z] == ((-2^(1 - m - v)) E^e Binomial[m, m/2] Binomial[v, v/2] Gamma[2 (1 + n), (-b) Sqrt[z]] (1 - Mod[m, 2]) (1 - Mod[v, 2]))/ b^(2 (1 + n)) - 2^(1 - m - v) Binomial[v, v/2] (1 - Mod[v, 2]) Sum[Binomial[m, k] ((E^(e + I (-2 k + m) q) Gamma[2 (1 + n), (-b - I a (-2 k + m)) Sqrt[z]])/(-b - I a (-2 k + m))^(2 (1 + n)) + (E^(e - I (-2 k + m) q) Gamma[2 (1 + n), (-b + I a (-2 k + m)) Sqrt[z]])/(-b + I a (-2 k + m))^(2 (1 + n))), {k, 0, Floor[(1/2) (-1 + m)]}] - 2^(1 - m - v) Binomial[m, m/2] (1 - Mod[m, 2]) Sum[Binomial[v, s] ((E^(e + g (-2 s + v)) Gamma[2 (1 + n), (-b - c (-2 s + v)) Sqrt[z]])/ (-b - c (-2 s + v))^(2 (1 + n)) + (E^(e - g (-2 s + v)) Gamma[2 (1 + n), (-b + c (-2 s + v)) Sqrt[z]])/ (-b + c (-2 s + v))^(2 (1 + n))), {s, 0, Floor[(1/2) (-1 + v)]}] - 2^(1 - m - v) Sum[Binomial[m, k] Sum[Binomial[v, s] ((E^(e + I (-2 k + m) q + g (-2 s + v)) Gamma[2 (1 + n), (-b - I a (-2 k + m) - c (-2 s + v)) Sqrt[z]])/ (-b - I a (-2 k + m) - c (-2 s + v))^(2 (1 + n)) + (E^(e - I (-2 k + m) q + g (-2 s + v)) Gamma[2 (1 + n), (-b + I a (-2 k + m) - c (-2 s + v)) Sqrt[z]])/ (-b + I a (-2 k + m) - c (-2 s + v))^(2 (1 + n)) + (E^(e + I (-2 k + m) q - g (-2 s + v)) Gamma[2 (1 + n), (-b - I a (-2 k + m) + c (-2 s + v)) Sqrt[z]])/ (-b - I a (-2 k + m) + c (-2 s + v))^(2 (1 + n)) + (E^(e - I (-2 k + m) q - g (-2 s + v)) Gamma[2 (1 + n), (-b + I a (-2 k + m) + c (-2 s + v)) Sqrt[z]])/ (-b + I a (-2 k + m) + c (-2 s + v))^(2 (1 + n))), {s, 0, Floor[(1/2) (-1 + v)]}], {k, 0, Floor[(1/2) (-1 + 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 z b + e cos m ( z a + q ) cosh v ( z c + g ) z - 2 - m - v + 1 e ( m m 2 ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["m", Identity, Rule[Editable, True]]], List[TagBox[FractionBox["m", "2"], Identity, Rule[Editable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False]] ( v v 2 ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["v", Identity, Rule[Editable, True]]], List[TagBox[FractionBox["v", "2"], Identity, Rule[Editable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False]] Γ ( 2 ( n + 1 ) , - b z ) ( 1 - m mod 2 \$CellContext`m 2 ) ( 1 - v mod 2 \$CellContext`v 2 ) b - 2 ( n + 1 ) - 2 - m - v + 1 ( v v 2 ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["v", Identity, Rule[Editable, True]]], List[TagBox[FractionBox["v", "2"], Identity, Rule[Editable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False]] ( 1 - v mod 2 \$CellContext`v 2 ) k = 0 m - 1 2 ( m k ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["m", Identity, Rule[Editable, True]]], List[TagBox["k", Identity, Rule[Editable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False]] ( e - ( m - 2 k ) q Γ ( 2 ( n + 1 ) , ( a ( m - 2 k ) - b ) z ) ( a ( m - 2 k ) - b ) - 2 ( n + 1 ) + e + ( m - 2 k ) q ( - b - a ( m - 2 k ) ) - 2 ( n + 1 ) Γ ( 2 ( n + 1 ) , ( - b - a ( m - 2 k ) ) z ) ) - 2 - m - v + 1 ( m m 2 ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["m", Identity, Rule[Editable, True]]], List[TagBox[FractionBox["m", "2"], Identity, Rule[Editable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False]] ( 1 - m mod 2 \$CellContext`m 2 ) s = 0 v - 1 2 ( v s ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["v", Identity, Rule[Editable, True]]], List[TagBox["s", Identity, Rule[Editable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False]] ( e - g ( v - 2 s ) Γ ( 2 ( n + 1 ) , ( c ( v - 2 s ) - b ) z ) ( c ( v - 2 s ) - b ) - 2 ( n + 1 ) + e + g ( v - 2 s ) ( - b - c ( v - 2 s ) ) - 2 ( n + 1 ) Γ ( 2 ( n + 1 ) , ( - b - c ( v - 2 s ) ) z ) ) - 2 - m - v + 1 k = 0 m - 1 2 ( m k ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["m", Identity, Rule[Editable, True]]], List[TagBox["k", Identity, Rule[Editable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False]] s = 0 v - 1 2 ( v s ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["v", Identity, Rule[Editable, True]]], List[TagBox["s", Identity, Rule[Editable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False]] ( e - ( m - 2 k ) q - g ( v - 2 s ) Γ ( 2 ( n + 1 ) , ( - b + a ( m - 2 k ) + c ( v - 2 s ) ) z ) ( - b + a ( m - 2 k ) + c ( v - 2 s ) ) - 2 ( n + 1 ) + e + ( m - 2 k ) q - g ( v - 2 s ) ( - b - a ( m - 2 k ) + c ( v - 2 s ) ) - 2 ( n + 1 ) Γ ( 2 ( n + 1 ) , ( - b - a ( m - 2 k ) + c ( v - 2 s ) ) z ) + e - ( m - 2 k ) q + g ( v - 2 s ) ( - b + a ( m - 2 k ) - c ( v - 2 s ) ) - 2 ( n + 1 ) Γ ( 2 ( n + 1 ) , ( - b + a ( m - 2 k ) - c ( v - 2 s ) ) z ) + e + ( m - 2 k ) q + g ( v - 2 s ) ( - b - a ( m - 2 k ) - c ( v - 2 s ) ) - 2 ( n + 1 ) Γ ( 2 ( n + 1 ) , ( - b - a ( m - 2 k ) - c ( v - 2 s ) ) z ) ) /; m + v + n Condition z z n z 1 2 b e z 1 2 a q m z 1 2 c g v -1 2 -1 m -1 v 1 e Binomial m m 2 -1 Binomial v v 2 -1 Gamma 2 n 1 -1 b z 1 2 1 -1 \$CellContext`m 2 1 -1 \$CellContext`v 2 b -2 n 1 -1 2 -1 m -1 v 1 Binomial v v 2 -1 1 -1 \$CellContext`v 2 k 0 m -1 2 -1 Binomial m k e -1 m -1 2 k q Gamma 2 n 1 a m -1 2 k -1 b z 1 2 a m -1 2 k -1 b -2 n 1 e m -1 2 k q -1 b -1 a m -1 2 k -2 n 1 Gamma 2 n 1 -1 b -1 a m -1 2 k z 1 2 -1 2 -1 m -1 v 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 2 n 1 c v -1 2 s -1 b z 1 2 c v -1 2 s -1 b -2 n 1 e g v -1 2 s -1 b -1 c v -1 2 s -2 n 1 Gamma 2 n 1 -1 b -1 c v -1 2 s z 1 2 -1 2 -1 m -1 v 1 k 0 m -1 2 -1 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 Gamma 2 n 1 -1 b a m -1 2 k c v -1 2 s z 1 2 -1 b a m -1 2 k c v -1 2 s -2 n 1 e m -1 2 k q -1 g v -1 2 s -1 b -1 a m -1 2 k c v -1 2 s -2 n 1 Gamma 2 n 1 -1 b -1 a m -1 2 k c v -1 2 s z 1 2 e -1 m -1 2 k q g v -1 2 s -1 b a m -1 2 k -1 c v -1 2 s -2 n 1 Gamma 2 n 1 -1 b a m -1 2 k -1 c v -1 2 s z 1 2 e m -1 2 k q g v -1 2 s -1 b -1 a m -1 2 k -1 c v -1 2 s -2 n 1 Gamma 2 n 1 -1 b -1 a m -1 2 k -1 c v -1 2 s z 1 2 m SuperPlus v SuperPlus n [/itex]

 Rule Form

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

 2002-12-18