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 Cosh

 http://functions.wolfram.com/01.20.21.0765.01

 Input Form

 Integrate[Cos[b z^2 + d z + e] Cosh[c z^2 + f z + g], z] == (1/8) E^((-I) e - g) Sqrt[Pi] ((I E^(2 I e - (I (d + I f)^2)/(4 (b + I c))) Erf[(d + I f + 2 b z + 2 I c z)/(2 Sqrt[I b - c])])/Sqrt[I b - c] + (E^((I (d + I f)^2)/(4 (b + I c)) + 2 g) Erfi[((-I) d + f - 2 I b z + 2 c z)/(2 Sqrt[(-I) b + c])])/ Sqrt[(-I) b + c] - Erfi[(I d + f + 2 I b z + 2 c z)/(2 Sqrt[(-I) b - c])]/ (E^((I d + f)^2/(-4 I b - 4 c)) Sqrt[(-I) b - c]) + (E^(2 I e + (I d + f)^2/(-4 I b - 4 c) + 2 g) Erfi[(I d + f + 2 I b z + 2 c z)/(2 Sqrt[I b + c])])/Sqrt[I b + c])

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List["\[Integral]", RowBox[List[RowBox[List["Cos", "[", 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"]], "+", "g"]], "]"]], RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", RowBox[List[FractionBox["1", "8"], " ", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "e"]], "-", "g"]]], " ", SqrtBox["\[Pi]"], " ", RowBox[List["(", RowBox[List[FractionBox[RowBox[List["\[ImaginaryI]", " ", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["2", " ", "\[ImaginaryI]", " ", "e"]], "-", FractionBox[RowBox[List["\[ImaginaryI]", " ", SuperscriptBox[RowBox[List["(", RowBox[List["d", "+", RowBox[List["\[ImaginaryI]", " ", "f"]]]], ")"]], "2"]]], RowBox[List["4", " ", RowBox[List["(", RowBox[List["b", "+", RowBox[List["\[ImaginaryI]", " ", "c"]]]], ")"]]]]]]]], " ", RowBox[List["Erf", "[", FractionBox[RowBox[List["d", "+", RowBox[List["\[ImaginaryI]", " ", "f"]], "+", RowBox[List["2", " ", "b", " ", "z"]], "+", RowBox[List["2", " ", "\[ImaginaryI]", " ", "c", " ", "z"]]]], RowBox[List["2", " ", SqrtBox[RowBox[List[RowBox[List["\[ImaginaryI]", " ", "b"]], "-", "c"]]]]]], "]"]]]], SqrtBox[RowBox[List[RowBox[List["\[ImaginaryI]", " ", "b"]], "-", "c"]]]], "+", FractionBox[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[FractionBox[RowBox[List["\[ImaginaryI]", " ", SuperscriptBox[RowBox[List["(", RowBox[List["d", "+", RowBox[List["\[ImaginaryI]", " ", "f"]]]], ")"]], "2"]]], RowBox[List["4", " ", RowBox[List["(", RowBox[List["b", "+", RowBox[List["\[ImaginaryI]", " ", "c"]]]], ")"]]]]], "+", RowBox[List["2", " ", "g"]]]]], " ", RowBox[List["Erfi", "[", FractionBox[RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "d"]], "+", "f", "-", RowBox[List["2", " ", "\[ImaginaryI]", " ", "b", " ", "z"]], "+", RowBox[List["2", " ", "c", " ", "z"]]]], RowBox[List["2", " ", SqrtBox[RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "b"]], "+", "c"]]]]]], "]"]]]], SqrtBox[RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "b"]], "+", "c"]]]], "-", FractionBox[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["-", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "d"]], "+", "f"]], ")"]], "2"], RowBox[List[RowBox[List[RowBox[List["-", "4"]], " ", "\[ImaginaryI]", " ", "b"]], "-", RowBox[List["4", " ", "c"]]]]]]]], " ", RowBox[List["Erfi", "[", FractionBox[RowBox[List[RowBox[List["\[ImaginaryI]", " ", "d"]], "+", "f", "+", RowBox[List["2", " ", "\[ImaginaryI]", " ", "b", " ", "z"]], "+", RowBox[List["2", " ", "c", " ", "z"]]]], RowBox[List["2", " ", SqrtBox[RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "b"]], "-", "c"]]]]]], "]"]]]], SqrtBox[RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "b"]], "-", "c"]]]], "+", FractionBox[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["2", " ", "\[ImaginaryI]", " ", "e"]], "+", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "d"]], "+", "f"]], ")"]], "2"], RowBox[List[RowBox[List[RowBox[List["-", "4"]], " ", "\[ImaginaryI]", " ", "b"]], "-", RowBox[List["4", " ", "c"]]]]], "+", RowBox[List["2", " ", "g"]]]]], " ", RowBox[List["Erfi", "[", FractionBox[RowBox[List[RowBox[List["\[ImaginaryI]", " ", "d"]], "+", "f", "+", RowBox[List["2", " ", "\[ImaginaryI]", " ", "b", " ", "z"]], "+", RowBox[List["2", " ", "c", " ", "z"]]]], RowBox[List["2", " ", SqrtBox[RowBox[List[RowBox[List["\[ImaginaryI]", " ", "b"]], "+", "c"]]]]]], "]"]]]], SqrtBox[RowBox[List[RowBox[List["\[ImaginaryI]", " ", "b"]], "+", "c"]]]]]], ")"]]]]]]]]

 MathML Form

 cos ( b z 2 + d z + e ) cosh ( c z 2 + f z + g ) z 1 8 - g - e π ( 2 e - ( d + f ) 2 4 ( b + c ) erf ( d + f + 2 b z + 2 c z 2 b - c ) b - c + ( d + f ) 2 4 ( b + c ) + 2 g erfi ( f - d - 2 b z + 2 c z 2 c - b ) c - b + ( f + d ) 2 - 4 c - 4 b + 2 g + 2 e erfi ( f + d + 2 c z + 2 b z 2 c + b ) c + b - - ( f + d ) 2 - 4 c - 4 b erfi ( f + d + 2 c z + 2 b z 2 - c - b ) - c - b ) z b z 2 d z e c z 2 f z g 1 8 -1 g -1 e 1 2 2 e -1 d f 2 4 b c -1 Erf d f 2 b z 2 c z 2 b -1 c 1 2 -1 b -1 c 1 2 -1 d f 2 4 b c -1 2 g Erfi f -1 d -1 2 b z 2 c z 2 c -1 b 1 2 -1 c -1 b 1 2 -1 f d 2 -4 c -1 4 b -1 2 g 2 e Erfi f d 2 c z 2 b z 2 c b 1 2 -1 c b 1 2 -1 -1 -1 f d 2 -4 c -1 4 b -1 Erfi f d 2 c z 2 b z 2 -1 c -1 b 1 2 -1 -1 c -1 b 1 2 -1 [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["\[Integral]", RowBox[List[RowBox[List[RowBox[List["Cos", "[", 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_"]], "+", "g_"]], "]"]]]], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox["1", "8"], " ", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "e"]], "-", "g"]]], " ", SqrtBox["\[Pi]"], " ", RowBox[List["(", RowBox[List[FractionBox[RowBox[List["\[ImaginaryI]", " ", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["2", " ", "\[ImaginaryI]", " ", "e"]], "-", FractionBox[RowBox[List["\[ImaginaryI]", " ", SuperscriptBox[RowBox[List["(", RowBox[List["d", "+", RowBox[List["\[ImaginaryI]", " ", "f"]]]], ")"]], "2"]]], RowBox[List["4", " ", RowBox[List["(", RowBox[List["b", "+", RowBox[List["\[ImaginaryI]", " ", "c"]]]], ")"]]]]]]]], " ", RowBox[List["Erf", "[", FractionBox[RowBox[List["d", "+", RowBox[List["\[ImaginaryI]", " ", "f"]], "+", RowBox[List["2", " ", "b", " ", "z"]], "+", RowBox[List["2", " ", "\[ImaginaryI]", " ", "c", " ", "z"]]]], RowBox[List["2", " ", SqrtBox[RowBox[List[RowBox[List["\[ImaginaryI]", " ", "b"]], "-", "c"]]]]]], "]"]]]], SqrtBox[RowBox[List[RowBox[List["\[ImaginaryI]", " ", "b"]], "-", "c"]]]], "+", FractionBox[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[FractionBox[RowBox[List["\[ImaginaryI]", " ", SuperscriptBox[RowBox[List["(", RowBox[List["d", "+", RowBox[List["\[ImaginaryI]", " ", "f"]]]], ")"]], "2"]]], RowBox[List["4", " ", RowBox[List["(", RowBox[List["b", "+", RowBox[List["\[ImaginaryI]", " ", "c"]]]], ")"]]]]], "+", RowBox[List["2", " ", "g"]]]]], " ", RowBox[List["Erfi", "[", FractionBox[RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "d"]], "+", "f", "-", RowBox[List["2", " ", "\[ImaginaryI]", " ", "b", " ", "z"]], "+", RowBox[List["2", " ", "c", " ", "z"]]]], RowBox[List["2", " ", SqrtBox[RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "b"]], "+", "c"]]]]]], "]"]]]], SqrtBox[RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "b"]], "+", "c"]]]], "-", FractionBox[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["-", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "d"]], "+", "f"]], ")"]], "2"], RowBox[List[RowBox[List[RowBox[List["-", "4"]], " ", "\[ImaginaryI]", " ", "b"]], "-", RowBox[List["4", " ", "c"]]]]]]]], " ", RowBox[List["Erfi", "[", FractionBox[RowBox[List[RowBox[List["\[ImaginaryI]", " ", "d"]], "+", "f", "+", RowBox[List["2", " ", "\[ImaginaryI]", " ", "b", " ", "z"]], "+", RowBox[List["2", " ", "c", " ", "z"]]]], RowBox[List["2", " ", SqrtBox[RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "b"]], "-", "c"]]]]]], "]"]]]], SqrtBox[RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "b"]], "-", "c"]]]], "+", FractionBox[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["2", " ", "\[ImaginaryI]", " ", "e"]], "+", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "d"]], "+", "f"]], ")"]], "2"], RowBox[List[RowBox[List[RowBox[List["-", "4"]], " ", "\[ImaginaryI]", " ", "b"]], "-", RowBox[List["4", " ", "c"]]]]], "+", RowBox[List["2", " ", "g"]]]]], " ", RowBox[List["Erfi", "[", FractionBox[RowBox[List[RowBox[List["\[ImaginaryI]", " ", "d"]], "+", "f", "+", RowBox[List["2", " ", "\[ImaginaryI]", " ", "b", " ", "z"]], "+", RowBox[List["2", " ", "c", " ", "z"]]]], RowBox[List["2", " ", SqrtBox[RowBox[List[RowBox[List["\[ImaginaryI]", " ", "b"]], "+", "c"]]]]]], "]"]]]], SqrtBox[RowBox[List[RowBox[List["\[ImaginaryI]", " ", "b"]], "+", "c"]]]]]], ")"]]]]]]]]

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

 2002-12-18