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 Cosh

 http://functions.wolfram.com/01.20.21.0486.01

 Input Form

 Integrate[E^(b z^2) Cosh[c z^2 + f z + g], z] == (1/(4 (b - c) (b + c))) ((Sqrt[Pi] (Sqrt[b - c] (b + c) E^((b f^2)/(4 (b - c) (b + c))) Erfi[(-f + 2 b z - 2 c z)/(2 Sqrt[b - c])] (Cosh[g] - Sinh[g]) + (b - c) Sqrt[b + c] E^(((b + 2 c) f^2)/(4 (b - c) (b + c))) Erfi[(f + 2 (b + c) z)/(2 Sqrt[b + c])] (Cosh[g] + Sinh[g])))/ E^((2 b f^2 + c f^2)/(4 (b^2 - c^2))))

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List["\[Integral]", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["b", " ", SuperscriptBox["z", "2"]]]], 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", RowBox[List["4", " ", RowBox[List["(", RowBox[List["b", "-", "c"]], ")"]], " ", RowBox[List["(", RowBox[List["b", "+", "c"]], ")"]]]]], RowBox[List["(", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["-", FractionBox[RowBox[List[RowBox[List["2", " ", "b", " ", SuperscriptBox["f", "2"]]], "+", RowBox[List["c", " ", SuperscriptBox["f", "2"]]]]], RowBox[List["4", " ", RowBox[List["(", RowBox[List[SuperscriptBox["b", "2"], "-", SuperscriptBox["c", "2"]]], ")"]]]]]]]], " ", SqrtBox["\[Pi]"], " ", RowBox[List["(", RowBox[List[RowBox[List[SqrtBox[RowBox[List["b", "-", "c"]]], " ", RowBox[List["(", RowBox[List["b", "+", "c"]], ")"]], " ", SuperscriptBox["\[ExponentialE]", FractionBox[RowBox[List["b", " ", SuperscriptBox["f", "2"]]], RowBox[List["4", " ", RowBox[List["(", RowBox[List["b", "-", "c"]], ")"]], " ", RowBox[List["(", RowBox[List["b", "+", "c"]], ")"]]]]]], " ", RowBox[List["Erfi", "[", FractionBox[RowBox[List[RowBox[List["-", "f"]], "+", RowBox[List["2", " ", "b", " ", "z"]], "-", RowBox[List["2", " ", "c", " ", "z"]]]], RowBox[List["2", " ", SqrtBox[RowBox[List["b", "-", "c"]]]]]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["Cosh", "[", "g", "]"]], "-", RowBox[List["Sinh", "[", "g", "]"]]]], ")"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List["b", "-", "c"]], ")"]], " ", SqrtBox[RowBox[List["b", "+", "c"]]], " ", SuperscriptBox["\[ExponentialE]", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["b", "+", RowBox[List["2", " ", "c"]]]], ")"]], " ", SuperscriptBox["f", "2"]]], RowBox[List["4", " ", RowBox[List["(", RowBox[List["b", "-", "c"]], ")"]], " ", RowBox[List["(", RowBox[List["b", "+", "c"]], ")"]]]]]], " ", RowBox[List["Erfi", "[", FractionBox[RowBox[List["f", "+", RowBox[List["2", " ", RowBox[List["(", RowBox[List["b", "+", "c"]], ")"]], " ", "z"]]]], RowBox[List["2", " ", SqrtBox[RowBox[List["b", "+", "c"]]]]]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["Cosh", "[", "g", "]"]], "+", RowBox[List["Sinh", "[", "g", "]"]]]], ")"]]]]]], ")"]]]], ")"]]]]]]]]

 MathML Form

 b z 2 cosh ( c z 2 + f z + g ) z 1 4 ( b - c ) ( b + c ) ( - 2 b f 2 + c f 2 4 ( b 2 - c 2 ) π ( b - c ( b + c ) b f 2 4 ( b - c ) ( b + c ) erfi ( - f + 2 b z - 2 c z 2 b - c ) ( cosh ( g ) - sinh ( g ) ) + ( b - c ) b + c ( b + 2 c ) f 2 4 ( b - c ) ( b + c ) erfi ( f + 2 ( b + c ) z 2 b + c ) ( cosh ( g ) + sinh ( g ) ) ) ) z b z 2 c z 2 f z g 1 4 b -1 c b c -1 -1 2 b f 2 c f 2 4 b 2 -1 c 2 -1 1 2 b -1 c 1 2 b c b f 2 4 b -1 c b c -1 Erfi -1 f 2 b z -1 2 c z 2 b -1 c 1 2 -1 g -1 g b -1 c b c 1 2 b 2 c f 2 4 b -1 c b c -1 Erfi f 2 b c z 2 b c 1 2 -1 g g [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["\[Integral]", RowBox[List[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["b_", " ", SuperscriptBox["z_", "2"]]]], " ", RowBox[List["Cosh", "[", RowBox[List[RowBox[List["c_", " ", SuperscriptBox["z_", "2"]]], "+", RowBox[List["f_", " ", "z_"]], "+", "g_"]], "]"]]]], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", FractionBox[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["-", FractionBox[RowBox[List[RowBox[List["2", " ", "b", " ", SuperscriptBox["f", "2"]]], "+", RowBox[List["c", " ", SuperscriptBox["f", "2"]]]]], RowBox[List["4", " ", RowBox[List["(", RowBox[List[SuperscriptBox["b", "2"], "-", SuperscriptBox["c", "2"]]], ")"]]]]]]]], " ", SqrtBox["\[Pi]"], " ", RowBox[List["(", RowBox[List[RowBox[List[SqrtBox[RowBox[List["b", "-", "c"]]], " ", RowBox[List["(", RowBox[List["b", "+", "c"]], ")"]], " ", SuperscriptBox["\[ExponentialE]", FractionBox[RowBox[List["b", " ", SuperscriptBox["f", "2"]]], RowBox[List["4", " ", RowBox[List["(", RowBox[List["b", "-", "c"]], ")"]], " ", RowBox[List["(", RowBox[List["b", "+", "c"]], ")"]]]]]], " ", RowBox[List["Erfi", "[", FractionBox[RowBox[List[RowBox[List["-", "f"]], "+", RowBox[List["2", " ", "b", " ", "z"]], "-", RowBox[List["2", " ", "c", " ", "z"]]]], RowBox[List["2", " ", SqrtBox[RowBox[List["b", "-", "c"]]]]]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["Cosh", "[", "g", "]"]], "-", RowBox[List["Sinh", "[", "g", "]"]]]], ")"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List["b", "-", "c"]], ")"]], " ", SqrtBox[RowBox[List["b", "+", "c"]]], " ", SuperscriptBox["\[ExponentialE]", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["b", "+", RowBox[List["2", " ", "c"]]]], ")"]], " ", SuperscriptBox["f", "2"]]], RowBox[List["4", " ", RowBox[List["(", RowBox[List["b", "-", "c"]], ")"]], " ", RowBox[List["(", RowBox[List["b", "+", "c"]], ")"]]]]]], " ", RowBox[List["Erfi", "[", FractionBox[RowBox[List["f", "+", RowBox[List["2", " ", RowBox[List["(", RowBox[List["b", "+", "c"]], ")"]], " ", "z"]]]], RowBox[List["2", " ", SqrtBox[RowBox[List["b", "+", "c"]]]]]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["Cosh", "[", "g", "]"]], "+", RowBox[List["Sinh", "[", "g", "]"]]]], ")"]]]]]], ")"]]]], RowBox[List["4", " ", RowBox[List["(", RowBox[List["b", "-", "c"]], ")"]], " ", RowBox[List["(", RowBox[List["b", "+", "c"]], ")"]]]]]]]]]

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

 2002-12-18