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 Cosh

 http://functions.wolfram.com/01.20.21.0511.01

 Input Form

 Integrate[z^n E^(b z^2 + d z + e) Cosh[c z^2 + f z], z] == (-(1/(4 Sqrt[b - c]))) E^(e - (d - f)^2/(4 (b - c))) Sum[2^(-n + q) (b - c)^(-(1/2) - n) (-d + f)^(n - q) (d - f + 2 b z - 2 c z)^(1 + q) (-((d - f + 2 b z - 2 c z)^2/(b - c)))^ ((1/2) (-1 - q)) Binomial[n, q] Gamma[(1 + q)/2, -((d - f + 2 b z - 2 c z)^2/(4 (b - c)))], {q, 0, n}] - (1/(4 Sqrt[b + c])) E^(e - (d + f)^2/(4 (b + c))) Sum[2^(-n + q) (b + c)^(-(1/2) - n) (-d - f)^(n - q) (d + f + 2 (b + c) z)^(1 + q) (-((d + f + 2 (b + c) z)^2/(b + c)))^ ((1/2) (-1 - q)) Binomial[n, q] Gamma[(1 + q)/2, -((d + f + 2 (b + c) z)^2/(4 (b + c)))], {q, 0, n}] /; 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", " ", SuperscriptBox["z", "2"]]], "+", RowBox[List["d", " ", "z"]], "+", "e"]]], RowBox[List["Cosh", "[", RowBox[List[RowBox[List["c", " ", SuperscriptBox["z", "2"]]], "+", RowBox[List["f", " ", "z"]]]], "]"]], RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", RowBox[List[RowBox[List[RowBox[List["-", FractionBox["1", RowBox[List["4", " ", SqrtBox[RowBox[List["b", "-", "c"]]]]]]]], SuperscriptBox["\[ExponentialE]", RowBox[List["e", "-", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List["d", "-", "f"]], ")"]], "2"], RowBox[List["4", " ", RowBox[List["(", RowBox[List["b", "-", "c"]], ")"]]]]]]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["q", "=", "0"]], "n"], RowBox[List[SuperscriptBox["2", RowBox[List[RowBox[List["-", "n"]], "+", "q"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["b", "-", "c"]], ")"]], RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], "-", "n"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "d"]], "+", "f"]], ")"]], RowBox[List["n", "-", "q"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["d", "-", "f", "+", RowBox[List["2", " ", "b", " ", "z"]], "-", RowBox[List["2", " ", "c", " ", "z"]]]], ")"]], RowBox[List["1", "+", "q"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List["d", "-", "f", "+", RowBox[List["2", " ", "b", " ", "z"]], "-", RowBox[List["2", " ", "c", " ", "z"]]]], ")"]], "2"], RowBox[List["b", "-", "c"]]]]], ")"]], RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "-", "q"]], ")"]]]]], " ", RowBox[List["Binomial", "[", RowBox[List["n", ",", "q"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox[RowBox[List["1", "+", "q"]], "2"], ",", RowBox[List["-", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List["d", "-", "f", "+", RowBox[List["2", " ", "b", " ", "z"]], "-", RowBox[List["2", " ", "c", " ", "z"]]]], ")"]], "2"], RowBox[List["4", " ", RowBox[List["(", RowBox[List["b", "-", "c"]], ")"]]]]]]]]], "]"]]]]]]]], "-", RowBox[List[FractionBox["1", RowBox[List["4", " ", SqrtBox[RowBox[List["b", "+", "c"]]]]]], SuperscriptBox["\[ExponentialE]", RowBox[List["e", "-", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List["d", "+", "f"]], ")"]], "2"], RowBox[List["4", " ", RowBox[List["(", RowBox[List["b", "+", "c"]], ")"]]]]]]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["q", "=", "0"]], "n"], RowBox[List[SuperscriptBox["2", RowBox[List[RowBox[List["-", "n"]], "+", "q"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["b", "+", "c"]], ")"]], RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], "-", "n"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "d"]], "-", "f"]], ")"]], RowBox[List["n", "-", "q"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["d", "+", "f", "+", RowBox[List["2", " ", RowBox[List["(", RowBox[List["b", "+", "c"]], ")"]], " ", "z"]]]], ")"]], RowBox[List["1", "+", "q"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List["d", "+", "f", "+", RowBox[List["2", " ", RowBox[List["(", RowBox[List["b", "+", "c"]], ")"]], " ", "z"]]]], ")"]], "2"], RowBox[List["b", "+", "c"]]]]], ")"]], RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "-", "q"]], ")"]]]]], " ", RowBox[List["Binomial", "[", RowBox[List["n", ",", "q"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox[RowBox[List["1", "+", "q"]], "2"], ",", RowBox[List["-", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List["d", "+", "f", "+", RowBox[List["2", " ", RowBox[List["(", RowBox[List["b", "+", "c"]], ")"]], " ", "z"]]]], ")"]], "2"], RowBox[List["4", " ", RowBox[List["(", RowBox[List["b", "+", "c"]], ")"]]]]]]]]], "]"]]]]]]]]]]]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "\[And]", RowBox[List["n", "\[GreaterEqual]", "0"]]]]]]]]

 MathML Form

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

 Rule Form

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

 2002-12-18