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 Cosh

 http://functions.wolfram.com/01.20.21.0367.01

 Input Form

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

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["\[Integral]", RowBox[List[SuperscriptBox["z", "n"], SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["b", " ", SqrtBox["z"]]], "+", "e"]]], RowBox[List["Cosh", "[", RowBox[List[RowBox[List["f", " ", "z"]], "+", "g"]], "]"]], RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", RowBox[List[SuperscriptBox["2", RowBox[List[RowBox[List["-", "2"]], "-", RowBox[List["2", " ", "n"]]]]], " ", SuperscriptBox["f", RowBox[List[RowBox[List["-", "2"]], "-", RowBox[List["2", " ", "n"]]]]], " ", RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["e", "+", FractionBox[SuperscriptBox["b", "2"], RowBox[List["4", " ", "f"]]], "-", "g"]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "n"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["h", "=", "0"]], "k"], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List[RowBox[List["-", "h"]], "+", 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 MathML Form

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

 Rule Form

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

 2002-12-18