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 Cosh

 http://functions.wolfram.com/01.20.21.0361.01

 Input Form

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

 Standard Form

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 MathML Form

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

 Rule Form

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

 2002-12-18