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 Cosh

 http://functions.wolfram.com/01.20.21.1475.01

 Input Form

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

 Standard Form

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

 MathML Form

 sinh ( b z 2 + d z + e ) cosh ( g + f z ) z 1 8 π ( - b ( d + f ) 2 4 b - e - g erfi ( - d - f - 2 b z 2 - b ) b + ( f - d ) 2 4 b - e + g ( - b erfi ( - d + f - 2 b z 2 - b ) + b 2 ( - ( f - d ) 2 4 b + e - g ) erfi ( d - f + 2 b z 2 b ) ) b + - ( d + f ) 2 4 b + e + g erfi ( d + f + 2 b z 2 b ) b ) z b z 2 d z e g f z 1 8 1 2 -1 b 1 2 d f 2 4 b -1 -1 e -1 g Erfi -1 d -1 f -1 2 b z 2 -1 b 1 2 -1 b -1 f -1 d 2 4 b -1 -1 e g -1 b 1 2 Erfi -1 d f -1 2 b z 2 -1 b 1 2 -1 b 1 2 2 -1 f -1 d 2 4 b -1 e -1 g Erfi d -1 f 2 b z 2 b 1 2 -1 b -1 -1 d f 2 4 b -1 e g Erfi d f 2 b z 2 b 1 2 -1 b 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"]]], "+", RowBox[List["d_", " ", "z_"]], "+", "e_"]], "]"]], " ", RowBox[List["Cosh", "[", RowBox[List[RowBox[List["f_", " ", "z_"]], "+", "g_"]], "]"]]]], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox["1", "8"], " ", SqrtBox["\[Pi]"], " ", RowBox[List["(", RowBox[List[FractionBox[RowBox[List[SqrtBox[RowBox[List["-", "b"]]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "e"]], "+", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List["d", "+", "f"]], ")"]], "2"], RowBox[List["4", " ", "b"]]], "-", "g"]]], " ", RowBox[List["Erfi", "[", FractionBox[RowBox[List[RowBox[List["-", "d"]], "-", "f", "-", RowBox[List["2", " ", "b", " ", "z"]]]], RowBox[List["2", " ", SqrtBox[RowBox[List["-", "b"]]]]]], "]"]]]], "b"], "+", FractionBox[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "e"]], "+", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "d"]], "+", "f"]], ")"]], "2"], RowBox[List["4", " ", "b"]]], "+", "g"]]], " ", RowBox[List["(", RowBox[List[RowBox[List[SqrtBox[RowBox[List["-", "b"]]], " ", RowBox[List["Erfi", "[", FractionBox[RowBox[List[RowBox[List["-", "d"]], "+", "f", "-", RowBox[List["2", " ", "b", " ", "z"]]]], RowBox[List["2", " ", SqrtBox[RowBox[List["-", "b"]]]]]], "]"]]]], "+", RowBox[List[SqrtBox["b"], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", RowBox[List["(", RowBox[List["e", "-", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "d"]], "+", "f"]], ")"]], "2"], RowBox[List["4", " ", "b"]]], "-", "g"]], ")"]]]]], " ", RowBox[List["Erfi", "[", FractionBox[RowBox[List["d", "-", "f", "+", RowBox[List["2", " ", "b", " ", "z"]]]], RowBox[List["2", " ", SqrtBox["b"]]]], "]"]]]]]], ")"]]]], "b"], "+", FractionBox[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["e", "-", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List["d", "+", "f"]], ")"]], "2"], RowBox[List["4", " ", "b"]]], "+", "g"]]], " ", RowBox[List["Erfi", "[", FractionBox[RowBox[List["d", "+", "f", "+", RowBox[List["2", " ", "b", " ", "z"]]]], RowBox[List["2", " ", SqrtBox["b"]]]], "]"]]]], SqrtBox["b"]]]], ")"]]]]]]]]

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

 2002-12-18