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 Cosh

 http://functions.wolfram.com/01.20.21.1468.01

 Input Form

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

 Standard Form

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

 MathML Form

 sinh ( z b + e + d z ) cosh ( c z ) z 1 8 ( e ( 4 - e cosh ( z b + e + ( d - c ) z ) d - c + b - b 2 4 ( d - c ) π erfi ( 2 ( c - d ) z - b 2 d - c ) ( d - c ) 3 / 2 - b - b 2 4 ( c - d ) - 2 e π erfi ( 2 ( c - d ) z - b 2 c - d ) ( c - d ) 3 / 2 ) - - e ( - 4 e cosh ( z b + e + ( c + d ) z ) c + d + b 2 e - b 2 4 ( c + d ) π erfi ( b + 2 ( c + d ) z 2 c + d ) ( c + d ) 3 / 2 - b - b 2 4 ( - c - d ) π erfi ( b + 2 ( c + d ) z 2 - c - d ) ( - c - d ) 3 / 2 ) ) z z 1 2 b e d z c z 1 8 e 4 -1 e z 1 2 b e d -1 c z d -1 c -1 b -1 b 2 4 d -1 c -1 1 2 Erfi 2 c -1 d z 1 2 -1 b 2 d -1 c 1 2 -1 d -1 c 3 2 -1 -1 b -1 b 2 4 c -1 d -1 -1 2 e 1 2 Erfi 2 c -1 d z 1 2 -1 b 2 c -1 d 1 2 -1 c -1 d 3 2 -1 -1 -1 e -1 4 e z 1 2 b e c d z c d -1 b 2 e -1 b 2 4 c d -1 1 2 Erfi b 2 c d z 1 2 2 c d 1 2 -1 c d 3 2 -1 -1 b -1 b 2 4 -1 c -1 d -1 1 2 Erfi b 2 c d z 1 2 2 -1 c -1 d 1 2 -1 -1 c -1 d 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[RowBox[List["b_", " ", SqrtBox["z_"]]], "+", RowBox[List["d_", " ", "z_"]], "+", "e_"]], "]"]], " ", RowBox[List["Cosh", "[", RowBox[List["c_", " ", "z_"]], "]"]]]], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox["1", "8"], " ", RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox["\[ExponentialE]", "e"], " ", RowBox[List["(", RowBox[List[FractionBox[RowBox[List["4", " ", SuperscriptBox["\[ExponentialE]", RowBox[List["-", "e"]]], " ", RowBox[List["Cosh", "[", RowBox[List["e", "+", RowBox[List["b", " ", SqrtBox["z"]]], "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "c"]], "+", "d"]], ")"]], " ", "z"]]]], "]"]]]], RowBox[List[RowBox[List["-", "c"]], "+", "d"]]], "-", FractionBox[RowBox[List["b", " ", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", FractionBox[SuperscriptBox["b", "2"], RowBox[List["4", " ", RowBox[List["(", RowBox[List["c", "-", "d"]], ")"]]]]]]], "-", RowBox[List["2", " ", "e"]]]]], " ", SqrtBox["\[Pi]"], " ", RowBox[List["Erfi", "[", FractionBox[RowBox[List[RowBox[List["-", "b"]], "+", RowBox[List["2", " ", RowBox[List["(", RowBox[List["c", "-", "d"]], ")"]], " ", SqrtBox["z"]]]]], RowBox[List["2", " ", SqrtBox[RowBox[List["c", "-", "d"]]]]]], "]"]]]], SuperscriptBox[RowBox[List["(", RowBox[List["c", "-", "d"]], ")"]], RowBox[List["3", "/", "2"]]]], "+", FractionBox[RowBox[List["b", " ", SuperscriptBox["\[ExponentialE]", RowBox[List["-", FractionBox[SuperscriptBox["b", "2"], RowBox[List["4", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "c"]], "+", "d"]], ")"]]]]]]]], " ", SqrtBox["\[Pi]"], " ", RowBox[List["Erfi", "[", FractionBox[RowBox[List[RowBox[List["-", "b"]], "+", RowBox[List["2", " ", RowBox[List["(", RowBox[List["c", "-", "d"]], ")"]], " ", SqrtBox["z"]]]]], RowBox[List["2", " ", SqrtBox[RowBox[List[RowBox[List["-", "c"]], "+", "d"]]]]]], "]"]]]], SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "c"]], "+", "d"]], ")"]], RowBox[List["3", "/", "2"]]]]]], ")"]]]], "-", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["-", "e"]]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", FractionBox[RowBox[List["4", " ", SuperscriptBox["\[ExponentialE]", "e"], " ", RowBox[List["Cosh", "[", RowBox[List["e", "+", RowBox[List["b", " ", SqrtBox["z"]]], "+", RowBox[List[RowBox[List["(", RowBox[List["c", "+", "d"]], ")"]], " ", "z"]]]], "]"]]]], RowBox[List["c", "+", "d"]]]]], "-", FractionBox[RowBox[List["b", " ", SuperscriptBox["\[ExponentialE]", RowBox[List["-", FractionBox[SuperscriptBox["b", "2"], RowBox[List["4", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "c"]], "-", "d"]], ")"]]]]]]]], " ", SqrtBox["\[Pi]"], " ", RowBox[List["Erfi", "[", FractionBox[RowBox[List["b", "+", RowBox[List["2", " ", RowBox[List["(", RowBox[List["c", "+", "d"]], ")"]], " ", SqrtBox["z"]]]]], RowBox[List["2", " ", SqrtBox[RowBox[List[RowBox[List["-", "c"]], "-", "d"]]]]]], "]"]]]], SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "c"]], "-", "d"]], ")"]], RowBox[List["3", "/", "2"]]]], "+", FractionBox[RowBox[List["b", " ", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", FractionBox[SuperscriptBox["b", "2"], RowBox[List["4", " ", RowBox[List["(", RowBox[List["c", "+", "d"]], ")"]]]]]]], "+", RowBox[List["2", " ", "e"]]]]], " ", SqrtBox["\[Pi]"], " ", RowBox[List["Erfi", "[", FractionBox[RowBox[List["b", "+", RowBox[List["2", " ", RowBox[List["(", RowBox[List["c", "+", "d"]], ")"]], " ", SqrtBox["z"]]]]], RowBox[List["2", " ", SqrtBox[RowBox[List["c", "+", "d"]]]]]], "]"]]]], SuperscriptBox[RowBox[List["(", RowBox[List["c", "+", "d"]], ")"]], RowBox[List["3", "/", "2"]]]]]], ")"]]]]]], ")"]]]]]]]]

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

 2002-12-18