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 Cosh

 http://functions.wolfram.com/01.20.21.3787.01

 Input Form

 Integrate[E^(b Sqrt[z] + e) Sin[a Sqrt[z] + q]^m Cosh[c Sqrt[z] + g]^v, z] == (1/b^2) (2^(1 - m - v) E^(e + b Sqrt[z]) (-1 + b Sqrt[z]) Binomial[m, m/2] Binomial[v, v/2] (1 - Mod[m, 2]) (1 - Mod[v, 2])) + 2^(1 - m - v) Binomial[v, v/2] (1 - Mod[v, 2]) Sum[(-1)^k ((E^(e + (I m Pi)/2 - I (-2 k + m) q + (b - I a (-2 k + m)) Sqrt[z]) (-1 + b Sqrt[z] - I a (-2 k + m) Sqrt[z]))/ (b - I a (-2 k + m))^2 + (E^(e - (I m Pi)/2 + I (-2 k + m) q + (b + I a (-2 k + m)) Sqrt[z]) (-1 + b Sqrt[z] + I a (-2 k + m) Sqrt[z]))/(-b - I a (-2 k + m))^2) Binomial[m, k], {k, 0, Floor[(1/2) (-1 + m)]}] + 2^(1 - m - v) Binomial[m, m/2] (1 - Mod[m, 2]) Sum[((E^(e - g (-2 k + v) + (b - c (-2 k + v)) Sqrt[z]) (-1 + b Sqrt[z] - c (-2 k + v) Sqrt[z]))/(-b + c (-2 k + v))^2 + (E^(e + g (-2 k + v) + (b + c (-2 k + v)) Sqrt[z]) (-1 + b Sqrt[z] + c (-2 k + v) Sqrt[z]))/(b + c (-2 k + v))^2) Binomial[v, k], {k, 0, Floor[(1/2) (-1 + v)]}] + 2^(1 - m - v) Sum[(-1)^k Binomial[m, k] Sum[((E^(e + I (-2 k + m) q - (1/2) I Pi m - g (-2 s + v) + (b + I a (-2 k + m) - c (-2 s + v)) Sqrt[z]) (-1 + b Sqrt[z] - ((-I) a (-2 k + m) + c (-2 s + v)) Sqrt[z]))/ (-b - I a (-2 k + m) + c (-2 s + v))^2 + (E^(e - I (-2 k + m) q + (1/2) I Pi m + g (-2 s + v) + (b - I a (-2 k + m) + c (-2 s + v)) Sqrt[z]) (-1 + b Sqrt[z] + ((-I) a (-2 k + m) + c (-2 s + v)) Sqrt[z]))/ (b - I a (-2 k + m) + c (-2 s + v))^2 + (E^(e - I (-2 k + m) q + (1/2) I Pi m - g (-2 s + v) + (b - I a (-2 k + m) - c (-2 s + v)) Sqrt[z]) (-1 + b Sqrt[z] - (I a (-2 k + m) + c (-2 s + v)) Sqrt[z]))/ (-b + I a (-2 k + m) + c (-2 s + v))^2 + (E^(e + I (-2 k + m) q - (1/2) I Pi m + g (-2 s + v) + (b + I a (-2 k + m) + c (-2 s + v)) Sqrt[z]) (-1 + b Sqrt[z] + (I a (-2 k + m) + c (-2 s + v)) Sqrt[z]))/ (b + I a (-2 k + m) + c (-2 s + v))^2) Binomial[v, s], {s, 0, Floor[(1/2) (-1 + v)]}], {k, 0, Floor[(1/2) (-1 + m)]}] /; Element[m, Integers] && m > 0 && Element[v, Integers] && v > 0

 Standard Form

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 MathML Form

 z b + e sin m ( z a + q ) cosh v ( z c + g ) z 2 - m - v + 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]]]]] ( k = 0 m - 1 2 ( - 1 ) k ( e + ( m - 2 k ) q + ( b + a ( m - 2 k ) ) z - m π 2 ( z b + a ( m - 2 k ) z - 1 ) ( - b - a ( m - 2 k ) ) 2 + e - ( m - 2 k ) q + ( b - a ( m - 2 k ) ) z + m π 2 ( z b - a ( m - 2 k ) z - 1 ) ( b - a ( m - 2 k ) ) 2 ) ( m k ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["m", Identity]], List[TagBox["k", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ) ( 1 - v mod 2 \$CellContext`v 2 ) + 2 - m - v + 1 z b + e ( b z - 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]]]]] ( 1 - m mod 2 \$CellContext`m 2 ) ( 1 - v mod 2 \$CellContext`v 2 ) b 2 + 2 - m - v + 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 ( e + g ( v - 2 k ) + ( b + c ( v - 2 k ) ) z ( z b + c ( v - 2 k ) z - 1 ) ( b + c ( v - 2 k ) ) 2 + e - g ( v - 2 k ) + ( b - c ( v - 2 k ) ) z ( z b - c ( v - 2 k ) z - 1 ) ( c ( v - 2 k ) - b ) 2 ) ( v k ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["v", Identity]], List[TagBox["k", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] + 2 - m - v + 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 ( ( e - ( m - 2 k ) q + g ( v - 2 s ) + ( b - a ( m - 2 k ) + c ( v - 2 s ) ) z + m π 2 ( z b + ( c ( v - 2 s ) - a ( m - 2 k ) ) z - 1 ) ) / ( b - a ( m - 2 k ) + c ( v - 2 s ) ) 2 + ( e - π m 2 + ( m - 2 k ) q + g ( v - 2 s ) + ( b + a ( m - 2 k ) + c ( v - 2 s ) ) z ( z b + ( a ( m - 2 k ) + c ( v - 2 s ) ) z - 1 ) ) / ( b + a ( m - 2 k ) + c ( v - 2 s ) ) 2 + ( e - π m 2 + ( m - 2 k ) q - g ( v - 2 s ) + ( b + a ( m - 2 k ) - c ( v - 2 s ) ) z ( z b - ( c ( v - 2 s ) - a ( m - 2 k ) ) z - 1 ) ) / ( - b - a ( m - 2 k ) + c ( v - 2 s ) ) 2 + ( e - ( m - 2 k ) q - g ( v - 2 s ) + ( b - a ( m - 2 k ) - c ( v - 2 s ) ) z + m π 2 ( z b - ( a ( m - 2 k ) + c ( v - 2 s ) ) z - 1 ) ) / ( - b + a ( m - 2 k ) + c ( v - 2 s ) ) 2 ) ( v s ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["v", Identity]], List[TagBox["s", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] /; m + v + Condition z z 1 2 b e z 1 2 a q m z 1 2 c g v 2 -1 m -1 v 1 Binomial v v 2 -1 k 0 m -1 2 -1 -1 k e m -1 2 k q b a m -1 2 k z 1 2 -1 m 2 -1 z 1 2 b a m -1 2 k z 1 2 -1 -1 b -1 a m -1 2 k 2 -1 e -1 m -1 2 k q b -1 a m -1 2 k z 1 2 m 2 -1 z 1 2 b -1 a m -1 2 k z 1 2 -1 b -1 a m -1 2 k 2 -1 Binomial m k 1 -1 \$CellContext`v 2 2 -1 m -1 v 1 z 1 2 b e b z 1 2 -1 Binomial m m 2 -1 Binomial v v 2 -1 1 -1 \$CellContext`m 2 1 -1 \$CellContext`v 2 b 2 -1 2 -1 m -1 v 1 Binomial m m 2 -1 1 -1 \$CellContext`m 2 k 0 v -1 2 -1 e g v -1 2 k b c v -1 2 k z 1 2 z 1 2 b c v -1 2 k z 1 2 -1 b c v -1 2 k 2 -1 e -1 g v -1 2 k b -1 c v -1 2 k z 1 2 z 1 2 b -1 c v -1 2 k z 1 2 -1 c v -1 2 k -1 b 2 -1 Binomial v k 2 -1 m -1 v 1 k 0 m -1 2 -1 -1 k Binomial m k s 0 v -1 2 -1 e -1 m -1 2 k q g v -1 2 s b -1 a m -1 2 k c v -1 2 s z 1 2 m 2 -1 z 1 2 b c v -1 2 s -1 a m -1 2 k z 1 2 -1 b -1 a m -1 2 k c v -1 2 s 2 -1 e -1 m 2 -1 m -1 2 k q g v -1 2 s b a m -1 2 k c v -1 2 s z 1 2 z 1 2 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 -1 e -1 m 2 -1 m -1 2 k q -1 g v -1 2 s b a m -1 2 k -1 c v -1 2 s z 1 2 z 1 2 b -1 c v -1 2 s -1 a m -1 2 k z 1 2 -1 -1 b -1 a m -1 2 k c v -1 2 s 2 -1 e -1 m -1 2 k q -1 g v -1 2 s b -1 a m -1 2 k -1 c v -1 2 s z 1 2 m 2 -1 z 1 2 b -1 a m -1 2 k c v -1 2 s z 1 2 -1 -1 b a m -1 2 k c v -1 2 s 2 -1 Binomial v s m SuperPlus v SuperPlus [/itex]

 Rule Form

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

 2002-12-18