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 Cosh

 http://functions.wolfram.com/01.20.21.1470.01

 Input Form

 Integrate[Sinh[b Sqrt[z]] Cosh[f z + g], z] == (1/8) ((-((4 E^g Cosh[g - b Sqrt[z] + f z])/f) + (b E^(b^2/(4 f)) Sqrt[Pi] Erfi[(-b + 2 f Sqrt[z])/(2 Sqrt[-f])])/ (-f)^(3/2) - (b E^(-(b^2/(4 f)) + 2 g) Sqrt[Pi] Erfi[(-b + 2 f Sqrt[z])/(2 Sqrt[f])])/f^(3/2))/E^g - (-((4 E^g Cosh[g + b Sqrt[z] + f z])/f) - (b E^(b^2/(4 f)) Sqrt[Pi] Erfi[(b + 2 f Sqrt[z])/(2 Sqrt[-f])])/ (-f)^(3/2) + (b E^(-(b^2/(4 f)) + 2 g) Sqrt[Pi] Erfi[(b + 2 f Sqrt[z])/(2 Sqrt[f])])/f^(3/2))/E^g)

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List["\[Integral]", RowBox[List[RowBox[List["Sinh", "[", RowBox[List["b", " ", SqrtBox["z"]]], "]"]], RowBox[List["Cosh", "[", RowBox[List[RowBox[List["f", " ", "z"]], "+", "g"]], "]"]], RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", RowBox[List[FractionBox["1", "8"], " ", RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["-", "g"]]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", FractionBox[RowBox[List["4", " ", SuperscriptBox["\[ExponentialE]", "g"], " ", RowBox[List["Cosh", "[", RowBox[List["g", "-", RowBox[List["b", " ", SqrtBox["z"]]], "+", RowBox[List["f", " ", "z"]]]], "]"]]]], "f"]]], "+", FractionBox[RowBox[List["b", " ", SuperscriptBox["\[ExponentialE]", FractionBox[SuperscriptBox["b", "2"], RowBox[List["4", " ", "f"]]]], " ", SqrtBox["\[Pi]"], " ", RowBox[List["Erfi", "[", FractionBox[RowBox[List[RowBox[List["-", "b"]], "+", RowBox[List["2", " ", "f", " ", SqrtBox["z"]]]]], RowBox[List["2", " ", SqrtBox[RowBox[List["-", "f"]]]]]], "]"]]]], SuperscriptBox[RowBox[List["(", RowBox[List["-", "f"]], ")"]], RowBox[List["3", "/", "2"]]]], "-", FractionBox[RowBox[List["b", " ", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", FractionBox[SuperscriptBox["b", "2"], RowBox[List["4", " ", "f"]]]]], "+", RowBox[List["2", " ", "g"]]]]], " ", SqrtBox["\[Pi]"], " ", RowBox[List["Erfi", "[", FractionBox[RowBox[List[RowBox[List["-", "b"]], "+", RowBox[List["2", " ", "f", " ", SqrtBox["z"]]]]], RowBox[List["2", " ", SqrtBox["f"]]]], "]"]]]], SuperscriptBox["f", RowBox[List["3", "/", "2"]]]]]], ")"]]]], "-", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["-", "g"]]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", FractionBox[RowBox[List["4", " ", SuperscriptBox["\[ExponentialE]", "g"], " ", RowBox[List["Cosh", "[", RowBox[List["g", "+", RowBox[List["b", " ", SqrtBox["z"]]], "+", RowBox[List["f", " ", "z"]]]], "]"]]]], "f"]]], "-", FractionBox[RowBox[List["b", " ", SuperscriptBox["\[ExponentialE]", FractionBox[SuperscriptBox["b", "2"], RowBox[List["4", " ", "f"]]]], " ", SqrtBox["\[Pi]"], " ", RowBox[List["Erfi", "[", FractionBox[RowBox[List["b", "+", RowBox[List["2", " ", "f", " ", SqrtBox["z"]]]]], RowBox[List["2", " ", SqrtBox[RowBox[List["-", "f"]]]]]], "]"]]]], SuperscriptBox[RowBox[List["(", RowBox[List["-", "f"]], ")"]], RowBox[List["3", "/", "2"]]]], "+", FractionBox[RowBox[List["b", " ", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", FractionBox[SuperscriptBox["b", "2"], RowBox[List["4", " ", "f"]]]]], "+", RowBox[List["2", " ", "g"]]]]], " ", SqrtBox["\[Pi]"], " ", RowBox[List["Erfi", "[", FractionBox[RowBox[List["b", "+", RowBox[List["2", " ", "f", " ", SqrtBox["z"]]]]], RowBox[List["2", " ", SqrtBox["f"]]]], "]"]]]], SuperscriptBox["f", RowBox[List["3", "/", "2"]]]]]], ")"]]]]]], ")"]]]]]]]]

 MathML Form

 sinh ( b z ) cosh ( g + f z ) z 1 8 ( - g ( - 4 g cosh ( - z b + g + f z ) f + b b 2 4 f π erfi ( 2 f z - b 2 - f ) ( - f ) 3 / 2 - b 2 g - b 2 4 f π erfi ( 2 f z - b 2 f ) f 3 / 2 ) - - g ( - 4 g cosh ( z b + g + f z ) f + b 2 g - b 2 4 f π erfi ( b + 2 f z 2 f ) f 3 / 2 - b b 2 4 f π erfi ( b + 2 f z 2 - f ) ( - f ) 3 / 2 ) ) z b z 1 2 g f z 1 8 -1 g -1 4 g -1 z 1 2 b g f z f -1 b b 2 4 f -1 1 2 Erfi 2 f z 1 2 -1 b 2 -1 f 1 2 -1 -1 f 3 2 -1 -1 b 2 g -1 b 2 4 f -1 1 2 Erfi 2 f z 1 2 -1 b 2 f 1 2 -1 f 3 2 -1 -1 -1 g -1 4 g z 1 2 b g f z f -1 b 2 g -1 b 2 4 f -1 1 2 Erfi b 2 f z 1 2 2 f 1 2 -1 f 3 2 -1 -1 b b 2 4 f -1 1 2 Erfi b 2 f z 1 2 2 -1 f 1 2 -1 -1 f 3 2 -1 [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["\[Integral]", RowBox[List[RowBox[List[RowBox[List["Sinh", "[", RowBox[List["b_", " ", SqrtBox["z_"]]], "]"]], " ", RowBox[List["Cosh", "[", RowBox[List[RowBox[List["f_", " ", "z_"]], "+", "g_"]], "]"]]]], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox["1", "8"], " ", RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["-", "g"]]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", FractionBox[RowBox[List["4", " ", SuperscriptBox["\[ExponentialE]", "g"], " ", RowBox[List["Cosh", "[", RowBox[List["g", "-", RowBox[List["b", " ", SqrtBox["z"]]], "+", RowBox[List["f", " ", "z"]]]], "]"]]]], "f"]]], "+", FractionBox[RowBox[List["b", " ", SuperscriptBox["\[ExponentialE]", FractionBox[SuperscriptBox["b", "2"], RowBox[List["4", " ", "f"]]]], " ", SqrtBox["\[Pi]"], " ", RowBox[List["Erfi", "[", FractionBox[RowBox[List[RowBox[List["-", "b"]], "+", RowBox[List["2", " ", "f", " ", SqrtBox["z"]]]]], RowBox[List["2", " ", SqrtBox[RowBox[List["-", "f"]]]]]], "]"]]]], SuperscriptBox[RowBox[List["(", RowBox[List["-", "f"]], ")"]], RowBox[List["3", "/", "2"]]]], "-", FractionBox[RowBox[List["b", " ", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", FractionBox[SuperscriptBox["b", "2"], RowBox[List["4", " ", "f"]]]]], "+", RowBox[List["2", " ", "g"]]]]], " ", SqrtBox["\[Pi]"], " ", RowBox[List["Erfi", "[", FractionBox[RowBox[List[RowBox[List["-", "b"]], "+", RowBox[List["2", " ", "f", " ", SqrtBox["z"]]]]], RowBox[List["2", " ", SqrtBox["f"]]]], "]"]]]], SuperscriptBox["f", RowBox[List["3", "/", "2"]]]]]], ")"]]]], "-", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["-", "g"]]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", FractionBox[RowBox[List["4", " ", SuperscriptBox["\[ExponentialE]", "g"], " ", RowBox[List["Cosh", "[", RowBox[List["g", "+", RowBox[List["b", " ", SqrtBox["z"]]], "+", RowBox[List["f", " ", "z"]]]], "]"]]]], "f"]]], "-", FractionBox[RowBox[List["b", " ", SuperscriptBox["\[ExponentialE]", FractionBox[SuperscriptBox["b", "2"], RowBox[List["4", " ", "f"]]]], " ", SqrtBox["\[Pi]"], " ", RowBox[List["Erfi", "[", FractionBox[RowBox[List["b", "+", RowBox[List["2", " ", "f", " ", SqrtBox["z"]]]]], RowBox[List["2", " ", SqrtBox[RowBox[List["-", "f"]]]]]], "]"]]]], SuperscriptBox[RowBox[List["(", RowBox[List["-", "f"]], ")"]], RowBox[List["3", "/", "2"]]]], "+", FractionBox[RowBox[List["b", " ", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", FractionBox[SuperscriptBox["b", "2"], RowBox[List["4", " ", "f"]]]]], "+", RowBox[List["2", " ", "g"]]]]], " ", SqrtBox["\[Pi]"], " ", RowBox[List["Erfi", "[", FractionBox[RowBox[List["b", "+", RowBox[List["2", " ", "f", " ", SqrtBox["z"]]]]], RowBox[List["2", " ", SqrtBox["f"]]]], "]"]]]], SuperscriptBox["f", RowBox[List["3", "/", "2"]]]]]], ")"]]]]]], ")"]]]]]]]]

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

 2002-12-18