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 Cosh

 http://functions.wolfram.com/01.20.21.0375.01

 Input Form

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

 Standard Form

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 MathML Form

 z n e + d z cosh ( c z ) z 2 - 2 n - 2 d - 2 n - 2 e - c 2 4 d ( k = 0 n h = 0 k ( - 1 ) k - h 4 k ( - c ) - h - k + 2 n ( 2 d z - c ) h + k ( - ( 2 d z - c ) 2 d ) 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]]]]] ( 2 d - ( 2 d z - c ) 2 d Γ ( 1 2 ( h + k + 2 ) , - ( 2 d z - c ) 2 4 d ) - c ( 2 d z - c ) Γ ( 1 2 ( h + k + 1 ) , - ( 2 d z - c ) 2 4 d ) ) + k = 0 n h = 0 k ( - 1 ) k - h 4 k c - h - k + 2 n ( c + 2 d z ) h + k ( - ( c + 2 d z ) 2 d ) 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]]]]] ( c ( c + 2 d z ) Γ ( 1 2 ( h + k + 1 ) , - ( c + 2 d z ) 2 4 d ) + 2 - ( c + 2 d z ) 2 d d Γ ( 1 2 ( h + k + 2 ) , - ( c + 2 d z ) 2 4 d ) ) ) /; n Condition z z n e d z c z 1 2 2 -2 n -2 d -2 n -2 e -1 c 2 4 d -1 h 0 k k 0 n -1 k -1 h 4 k -1 c -1 h -1 k 2 n 2 d z 1 2 -1 c h k -1 2 d z 1 2 -1 c 2 d -1 1 2 -1 h -1 k -1 Binomial k h Binomial n k 2 d -1 2 d z 1 2 -1 c 2 d -1 1 2 Gamma 1 2 h k 2 -1 2 d z 1 2 -1 c 2 4 d -1 -1 c 2 d z 1 2 -1 c Gamma 1 2 h k 1 -1 2 d z 1 2 -1 c 2 4 d -1 h 0 k k 0 n -1 k -1 h 4 k c -1 h -1 k 2 n c 2 d z 1 2 h k -1 c 2 d z 1 2 2 d -1 1 2 -1 h -1 k -1 Binomial k h Binomial n k c c 2 d z 1 2 Gamma 1 2 h k 1 -1 c 2 d z 1 2 2 4 d -1 2 -1 c 2 d z 1 2 2 d -1 1 2 d Gamma 1 2 h k 2 -1 c 2 d z 1 2 2 4 d -1 n [/itex]

 Rule Form

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

 2002-12-18