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 Erf

 http://functions.wolfram.com/06.25.21.0083.01

 Input Form

 Integrate[z^3 Sinh[b z^2] Erf[a z], z] == (1/(4 b^2)) ((a/(a^4 - b^2)) (Sqrt[a^2 - b] (a^2 + b) Erf[Sqrt[a^2 - b] z] + (-a^2 + b) Sqrt[a^2 + b] Erf[Sqrt[a^2 + b] z]) - ((a b z^3)/(2 Sqrt[Pi])) ((-Sqrt[Pi] - 2 E^((-a^2 + b) z^2) Sqrt[(a^2 - b) z^2] + Sqrt[Pi] Erf[Sqrt[(a^2 - b) z^2]])/((a^2 - b) z^2)^(3/2) + (-Sqrt[Pi] - (2 Sqrt[(a^2 + b) z^2])/E^((a^2 + b) z^2) + Sqrt[Pi] Erf[Sqrt[(a^2 + b) z^2]])/((a^2 + b) z^2)^(3/2)) + 2 Erf[a z] (b z^2 Cosh[b z^2] - Sinh[b z^2]))

 Standard Form

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

 MathML Form

 z 3 sinh ( b z 2 ) erf ( a z ) z 1 4 b 2 ( - a b z 3 2 π ( ( π erf ( ( a 2 - b ) z 2 ) - π - 2 ( b - a 2 ) z 2 ( a 2 - b ) z 2 ) / ( ( a 2 - b ) z 2 ) 3 / 2 + ( π erf ( ( a 2 + b ) z 2 ) - π - 2 - ( a 2 + b ) z 2 ( a 2 + b ) z 2 ) / ( ( a 2 + b ) z 2 ) 3 / 2 ) + a a 4 - b 2 ( a 2 - b ( a 2 + b ) erf ( a 2 - b z ) + ( b - a 2 ) a 2 + b erf ( a 2 + b z ) ) + 2 erf ( a z ) ( b z 2 cosh ( b z 2 ) - sinh ( b z 2 ) ) ) z z 3 b z 2 Erf a z 1 4 b 2 -1 -1 a b z 3 2 1 2 -1 1 2 Erf a 2 -1 b z 2 1 2 -1 1 2 -1 2 b -1 a 2 z 2 a 2 -1 b z 2 1 2 a 2 -1 b z 2 3 2 -1 1 2 Erf a 2 b z 2 1 2 -1 1 2 -1 2 -1 a 2 b z 2 a 2 b z 2 1 2 a 2 b z 2 3 2 -1 a a 4 -1 b 2 -1 a 2 -1 b 1 2 a 2 b Erf a 2 -1 b 1 2 z b -1 a 2 a 2 b 1 2 Erf a 2 b 1 2 z 2 Erf a z b z 2 b z 2 -1 b z 2 [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["\[Integral]", RowBox[List[RowBox[List[SuperscriptBox["z_", "3"], " ", RowBox[List["Sinh", "[", RowBox[List["b_", " ", SuperscriptBox["z_", "2"]]], "]"]], " ", RowBox[List["Erf", "[", RowBox[List["a_", " ", "z_"]], "]"]]]], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", FractionBox[RowBox[List[FractionBox[RowBox[List["a", " ", RowBox[List["(", RowBox[List[RowBox[List[SqrtBox[RowBox[List[SuperscriptBox["a", "2"], "-", "b"]]], " ", RowBox[List["(", RowBox[List[SuperscriptBox["a", "2"], "+", "b"]], ")"]], " ", RowBox[List["Erf", "[", RowBox[List[SqrtBox[RowBox[List[SuperscriptBox["a", "2"], "-", "b"]]], " ", "z"]], "]"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", SuperscriptBox["a", "2"]]], "+", "b"]], ")"]], " ", SqrtBox[RowBox[List[SuperscriptBox["a", "2"], "+", "b"]]], " ", RowBox[List["Erf", "[", RowBox[List[SqrtBox[RowBox[List[SuperscriptBox["a", "2"], "+", "b"]]], " ", "z"]], "]"]]]]]], ")"]]]], RowBox[List[SuperscriptBox["a", "4"], "-", SuperscriptBox["b", "2"]]]], "-", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["a", " ", "b", " ", SuperscriptBox["z", "3"]]], ")"]], " ", RowBox[List["(", RowBox[List[FractionBox[RowBox[List[RowBox[List["-", SqrtBox["\[Pi]"]]], "-", RowBox[List["2", " ", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", SuperscriptBox["a", "2"]]], "+", "b"]], ")"]], " ", SuperscriptBox["z", "2"]]]], " ", SqrtBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["a", "2"], "-", "b"]], ")"]], " ", SuperscriptBox["z", "2"]]]]]], "+", RowBox[List[SqrtBox["\[Pi]"], " ", RowBox[List["Erf", "[", SqrtBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["a", "2"], "-", "b"]], ")"]], " ", SuperscriptBox["z", "2"]]]], "]"]]]]]], SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["a", "2"], "-", "b"]], ")"]], " ", SuperscriptBox["z", "2"]]], ")"]], RowBox[List["3", "/", "2"]]]], "+", FractionBox[RowBox[List[RowBox[List["-", SqrtBox["\[Pi]"]]], "-", RowBox[List["2", " ", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", RowBox[List["(", RowBox[List[SuperscriptBox["a", "2"], "+", "b"]], ")"]]]], " ", SuperscriptBox["z", "2"]]]], " ", SqrtBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["a", "2"], "+", "b"]], ")"]], " ", SuperscriptBox["z", "2"]]]]]], "+", RowBox[List[SqrtBox["\[Pi]"], " ", RowBox[List["Erf", "[", SqrtBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["a", "2"], "+", "b"]], ")"]], " ", SuperscriptBox["z", "2"]]]], "]"]]]]]], SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["a", "2"], "+", "b"]], ")"]], " ", SuperscriptBox["z", "2"]]], ")"]], RowBox[List["3", "/", "2"]]]]]], ")"]]]], RowBox[List["2", " ", SqrtBox["\[Pi]"]]]], "+", RowBox[List["2", " ", RowBox[List["Erf", "[", RowBox[List["a", " ", "z"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["b", " ", SuperscriptBox["z", "2"], " ", RowBox[List["Cosh", "[", RowBox[List["b", " ", SuperscriptBox["z", "2"]]], "]"]]]], "-", RowBox[List["Sinh", "[", RowBox[List["b", " ", SuperscriptBox["z", "2"]]], "]"]]]], ")"]]]]]], RowBox[List["4", " ", SuperscriptBox["b", "2"]]]]]]]]

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

 2001-10-29