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 Cosh

 http://functions.wolfram.com/01.20.21.1515.01

 Input Form

 Integrate[Sinh[b z^2 + d z + e] Cosh[c z^2 + f z], z] == (1/8) Sqrt[Pi] ((Sqrt[-b - c] E^(-e - (d + f)^2/(-4 b - 4 c)) Erfi[(-d - f - 2 b z - 2 c z)/(2 Sqrt[-b - c])])/(b + c) + (1/(b - c)) (E^(-e - (-d + f)^2/(-4 b + 4 c)) (Sqrt[b - c] E^(2 (e + (-d + f)^2/(-4 b + 4 c))) Erfi[(d - f + 2 b z - 2 c z)/(2 Sqrt[b - c])] + Sqrt[-b + c] Erfi[(-d + f - 2 b z + 2 c z)/(2 Sqrt[-b + c])])) + (E^(e + (d + f)^2/(-4 b - 4 c)) Erfi[(d + f + 2 b z + 2 c z)/ (2 Sqrt[b + c])])/Sqrt[b + c]) /; Element[v, Integers] && v > 0

 Standard Form

 Cell[BoxData[RowBox[List[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["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[RowBox[List["(", RowBox[List["d", "+", "f"]], ")"]], "2"], RowBox[List[RowBox[List[RowBox[List["-", "4"]], " ", "b"]], "-", RowBox[List["4", " ", "c"]]]]]]]], " ", RowBox[List["Erfi", "[", FractionBox[RowBox[List[RowBox[List["-", "d"]], "-", "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[RowBox[List["(", RowBox[List[RowBox[List["-", "d"]], "+", "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[RowBox[List["(", RowBox[List[RowBox[List["-", "d"]], "+", "f"]], ")"]], "2"], RowBox[List[RowBox[List[RowBox[List["-", "4"]], " ", "b"]], "+", RowBox[List["4", " ", "c"]]]]]]], ")"]]]]], " ", RowBox[List["Erfi", "[", FractionBox[RowBox[List["d", "-", "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[RowBox[List["-", "d"]], "+", "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[RowBox[List["(", RowBox[List["d", "+", "f"]], ")"]], "2"], RowBox[List[RowBox[List[RowBox[List["-", "4"]], " ", "b"]], "-", RowBox[List["4", " ", "c"]]]]]]]], " ", RowBox[List["Erfi", "[", FractionBox[RowBox[List["d", "+", "f", "+", RowBox[List["2", " ", "b", " ", "z"]], "+", RowBox[List["2", " ", "c", " ", "z"]]]], RowBox[List["2", " ", SqrtBox[RowBox[List["b", "+", "c"]]]]]], "]"]]]], SqrtBox[RowBox[List["b", "+", "c"]]]]]], ")"]]]]]], "/;", RowBox[List[RowBox[List["v", "\[Element]", "Integers"]], "&&", RowBox[List["v", ">", "0"]]]]]]]]

 MathML Form

 sinh ( b z 2 + d z + e ) cosh ( c z 2 + f z ) z 1 8 π ( - b - c - ( d + f ) 2 - 4 b - 4 c - e erfi ( - d - f - 2 b z - 2 c z 2 - b - c ) b + c + - ( f - d ) 2 4 c - 4 b - e ( b - c 2 ( ( f - d ) 2 4 c - 4 b + e ) erfi ( d - f + 2 b z - 2 c z 2 b - c ) + c - b erfi ( - d + f - 2 b z + 2 c z 2 c - b ) ) b - c + ( d + f ) 2 - 4 b - 4 c + e erfi ( d + f + 2 b z + 2 c z 2 b + c ) b + c ) z b z 2 d z e c z 2 f z 1 8 1 2 -1 b -1 c 1 2 -1 d f 2 -4 b -1 4 c -1 -1 e Erfi -1 d -1 f -1 2 b z -1 2 c z 2 -1 b -1 c 1 2 -1 b c -1 -1 f -1 d 2 4 c -1 4 b -1 -1 e b -1 c 1 2 2 f -1 d 2 4 c -1 4 b -1 e Erfi d -1 f 2 b z -1 2 c z 2 b -1 c 1 2 -1 c -1 b 1 2 Erfi -1 d f -1 2 b z 2 c z 2 c -1 b 1 2 -1 b -1 c -1 d f 2 -4 b -1 4 c -1 e Erfi d 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"]]], "+", RowBox[List["d_", " ", "z_"]], "+", "e_"]], "]"]], " ", RowBox[List["Cosh", "[", RowBox[List[RowBox[List["c_", " ", SuperscriptBox["z_", "2"]]], "+", RowBox[List["f_", " ", "z_"]]]], "]"]]]], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List[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[RowBox[List["(", RowBox[List["d", "+", "f"]], ")"]], "2"], RowBox[List[RowBox[List[RowBox[List["-", "4"]], " ", "b"]], "-", RowBox[List["4", " ", "c"]]]]]]]], " ", RowBox[List["Erfi", "[", FractionBox[RowBox[List[RowBox[List["-", "d"]], "-", "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[RowBox[List["(", RowBox[List[RowBox[List["-", "d"]], "+", "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[RowBox[List["(", RowBox[List[RowBox[List["-", "d"]], "+", "f"]], ")"]], "2"], RowBox[List[RowBox[List[RowBox[List["-", "4"]], " ", "b"]], "+", RowBox[List["4", " ", "c"]]]]]]], ")"]]]]], " ", RowBox[List["Erfi", "[", FractionBox[RowBox[List["d", "-", "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[RowBox[List["-", "d"]], "+", "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[RowBox[List["(", RowBox[List["d", "+", "f"]], ")"]], "2"], RowBox[List[RowBox[List[RowBox[List["-", "4"]], " ", "b"]], "-", RowBox[List["4", " ", "c"]]]]]]]], " ", RowBox[List["Erfi", "[", FractionBox[RowBox[List["d", "+", "f", "+", RowBox[List["2", " ", "b", " ", "z"]], "+", RowBox[List["2", " ", "c", " ", "z"]]]], RowBox[List["2", " ", SqrtBox[RowBox[List["b", "+", "c"]]]]]], "]"]]]], SqrtBox[RowBox[List["b", "+", "c"]]]]]], ")"]]]], "/;", RowBox[List[RowBox[List["v", "\[Element]", "Integers"]], "&&", RowBox[List["v", ">", "0"]]]]]]]]]]

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

 2002-12-18