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 Cosh

 http://functions.wolfram.com/01.20.21.0512.01

 Input Form

 Integrate[z^n E^(b Sqrt[z] + d z + e) Cosh[c Sqrt[z] + f z], z] == (E^e (Sum[(-1)^(-h + k) 4^k (b - c)^(-h - k + 2 n) (b - c + 2 (d - f) Sqrt[z])^(h + k) (-((b - c + 2 (d - f) Sqrt[z])^2/(d - f)))^((1/2) (-1 - h - k)) Binomial[k, h] Binomial[n, k] ((b - c) (b - c + 2 (d - f) Sqrt[z]) Gamma[(1/2) (1 + h + k), -((b - c + 2 (d - f) Sqrt[z])^2/ (4 (d - f)))] + 2 (d - f) Sqrt[-((b - c + 2 (d - f) Sqrt[z])^2/ (d - f))] Gamma[(1/2) (2 + h + k), -((b - c + 2 (d - f) Sqrt[z])^2/(4 (d - f)))]), {k, 0, n}, {h, 0, k}]/(E^((b - c)^2/(4 (d - f))) (d - f)^(2 (1 + n))) + Sum[(-1)^(-h + k) 4^k (b + c)^(-h - k + 2 n) (b + c + 2 (d + f) Sqrt[z])^ (h + k) (-((b + c + 2 (d + f) Sqrt[z])^2/(d + f)))^ ((1/2) (-1 - h - k)) Binomial[k, h] Binomial[n, k] ((b + c) (b + c + 2 (d + f) Sqrt[z]) Gamma[(1/2) (1 + h + k), -((b + c + 2 (d + f) Sqrt[z])^2/(4 (d + f)))] + 2 (d + f) Sqrt[-((b + c + 2 (d + f) Sqrt[z])^2/(d + f))] Gamma[(1/2) (2 + h + k), -((b + c + 2 (d + f) Sqrt[z])^2/ (4 (d + f)))]), {k, 0, n}, {h, 0, k}]/(E^((b + c)^2/(4 (d + f))) (d + f)^(2 (1 + n)))))/2^(2 (1 + n)) /; Element[n, Integers] && n >= 0

 Standard Form

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 MathML Form

 z n z b + e + d z cosh ( z c + f z ) z 2 - 2 ( n + 1 ) e ( - ( b - c ) 2 4 ( d - f ) ( k = 0 n h = 0 k ( - 1 ) k - h 4 k ( b - c ) - h - k + 2 n ( b - c + 2 ( d - f ) z ) h + k ( - ( b - c + 2 ( d - f ) z ) 2 d - f ) 1 2 ( - h - k - 1 ) ( k h ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["k", Identity]], List[TagBox["h", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( n k ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["n", Identity]], List[TagBox["k", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( ( b - c ) ( b - c + 2 ( d - f ) z ) Γ ( 1 2 ( h + k + 1 ) , - ( b - c + 2 ( d - f ) z ) 2 4 ( d - f ) ) + 2 - ( b - c + 2 ( d - f ) z ) 2 d - f ( d - f ) Γ ( 1 2 ( h + k + 2 ) , - ( b - c + 2 ( d - f ) z ) 2 4 ( d - f ) ) ) ) ( d - f ) - 2 ( n + 1 ) + - ( b + c ) 2 4 ( d + f ) ( d + f ) - 2 ( n + 1 ) k = 0 n h = 0 k ( - 1 ) k - h 4 k ( b + c ) - h - k + 2 n ( b + c + 2 ( d + f ) z ) h + k ( - ( b + c + 2 ( d + f ) z ) 2 d + f ) 1 2 ( - h - k - 1 ) ( k h ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["k", Identity]], List[TagBox["h", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( n k ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["n", Identity]], List[TagBox["k", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( ( b + c ) ( b + c + 2 ( d + f ) z ) Γ ( 1 2 ( h + k + 1 ) , - ( b + c + 2 ( d + f ) z ) 2 4 ( d + f ) ) + 2 - ( b + c + 2 ( d + f ) z ) 2 d + f ( d + f ) Γ ( 1 2 ( h + k + 2 ) , - ( b + c + 2 ( d + f ) z ) 2 4 ( d + f ) ) ) ) /; n Condition z z n z 1 2 b e d z z 1 2 c f z 2 -2 n 1 e -1 b -1 c 2 4 d -1 f -1 h 0 k k 0 n -1 k -1 h 4 k b -1 c -1 h -1 k 2 n b -1 c 2 d -1 f z 1 2 h k -1 b -1 c 2 d -1 f z 1 2 2 d -1 f -1 1 2 -1 h -1 k -1 Binomial k h Binomial n k b -1 c b -1 c 2 d -1 f z 1 2 Gamma 1 2 h k 1 -1 b -1 c 2 d -1 f z 1 2 2 4 d -1 f -1 2 -1 b -1 c 2 d -1 f z 1 2 2 d -1 f -1 1 2 d -1 f Gamma 1 2 h k 2 -1 b -1 c 2 d -1 f z 1 2 2 4 d -1 f -1 d -1 f -2 n 1 -1 b c 2 4 d f -1 d f -2 n 1 h 0 k k 0 n -1 k -1 h 4 k b c -1 h -1 k 2 n b c 2 d f z 1 2 h k -1 b c 2 d f z 1 2 2 d f -1 1 2 -1 h -1 k -1 Binomial k h Binomial n k b c b c 2 d f z 1 2 Gamma 1 2 h k 1 -1 b c 2 d f z 1 2 2 4 d f -1 2 -1 b c 2 d f z 1 2 2 d f -1 1 2 d f Gamma 1 2 h k 2 -1 b c 2 d f z 1 2 2 4 d f -1 n [/itex]

 Rule Form

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

 2002-12-18