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 Cosh

 http://functions.wolfram.com/01.20.21.1511.01

 Input Form

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

 Standard Form

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

 MathML Form

 sinh ( b z 2 + e ) cosh ( c z 2 + f z ) z 1 8 π ( - b - c - f 2 - 4 b - 4 c - e erfi ( - f - 2 b z - 2 c z 2 - b - c ) b + c + - f 2 4 c - 4 b - e ( b - c 2 ( f 2 4 c - 4 b + e ) erfi ( - f + 2 b z - 2 c z 2 b - c ) + c - b erfi ( f - 2 b z + 2 c z 2 c - b ) ) b - c + f 2 - 4 b - 4 c + e erfi ( f + 2 b z + 2 c z 2 b + c ) b + c ) z b z 2 e c z 2 f z 1 8 1 2 -1 b -1 c 1 2 -1 f 2 -4 b -1 4 c -1 -1 e Erfi -1 f -1 2 b z -1 2 c z 2 -1 b -1 c 1 2 -1 b c -1 -1 f 2 4 c -1 4 b -1 -1 e b -1 c 1 2 2 f 2 4 c -1 4 b -1 e Erfi -1 f 2 b z -1 2 c z 2 b -1 c 1 2 -1 c -1 b 1 2 Erfi f -1 2 b z 2 c z 2 c -1 b 1 2 -1 b -1 c -1 f 2 -4 b -1 4 c -1 e Erfi f 2 b z 2 c z 2 b c 1 2 -1 b c 1 2 -1 [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["\[Integral]", RowBox[List[RowBox[List[RowBox[List["Sinh", "[", RowBox[List[RowBox[List["b_", " ", SuperscriptBox["z_", "2"]]], "+", "e_"]], "]"]], " ", RowBox[List["Cosh", "[", RowBox[List[RowBox[List["c_", " ", SuperscriptBox["z_", "2"]]], "+", RowBox[List["f_", " ", "z_"]]]], "]"]]]], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox["1", "8"], " ", SqrtBox["\[Pi]"], " ", RowBox[List["(", RowBox[List[FractionBox[RowBox[List[SqrtBox[RowBox[List[RowBox[List["-", "b"]], "-", "c"]]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "e"]], "-", FractionBox[SuperscriptBox["f", "2"], RowBox[List[RowBox[List[RowBox[List["-", "4"]], " ", "b"]], "-", RowBox[List["4", " ", "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[RowBox[List["-", "b"]], "-", "c"]]]]]], "]"]]]], RowBox[List["b", "+", "c"]]], "+", FractionBox[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "e"]], "-", FractionBox[SuperscriptBox["f", "2"], RowBox[List[RowBox[List[RowBox[List["-", "4"]], " ", "b"]], "+", RowBox[List["4", " ", "c"]]]]]]]], " ", RowBox[List["(", RowBox[List[RowBox[List[SqrtBox[RowBox[List["b", "-", "c"]]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", RowBox[List["(", RowBox[List["e", "+", FractionBox[SuperscriptBox["f", "2"], RowBox[List[RowBox[List[RowBox[List["-", "4"]], " ", "b"]], "+", RowBox[List["4", " ", "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[SqrtBox[RowBox[List[RowBox[List["-", "b"]], "+", "c"]]], " ", RowBox[List["Erfi", "[", FractionBox[RowBox[List["f", "-", RowBox[List["2", " ", "b", " ", "z"]], "+", RowBox[List["2", " ", "c", " ", "z"]]]], RowBox[List["2", " ", SqrtBox[RowBox[List[RowBox[List["-", "b"]], "+", "c"]]]]]], "]"]]]]]], ")"]]]], RowBox[List["b", "-", "c"]]], "+", FractionBox[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["e", "+", FractionBox[SuperscriptBox["f", "2"], RowBox[List[RowBox[List[RowBox[List["-", "4"]], " ", "b"]], "-", RowBox[List["4", " ", "c"]]]]]]]], " ", RowBox[List["Erfi", "[", FractionBox[RowBox[List["f", "+", RowBox[List["2", " ", "b", " ", "z"]], "+", RowBox[List["2", " ", "c", " ", "z"]]]], RowBox[List["2", " ", SqrtBox[RowBox[List["b", "+", "c"]]]]]], "]"]]]], SqrtBox[RowBox[List["b", "+", "c"]]]]]], ")"]]]]]]]]

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

 2002-12-18