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variants of this functions
Erf






Mathematica Notation

Traditional Notation









Gamma, Beta, Erf > Erf[z] > Integration > Indefinite integration > Involving one direct function and elementary functions > Involving hyperbolic functions and a power function > Involving sinh and power





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





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MathML Form







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<times /> <cn type='integer'> 2 </cn> <apply> <ci> Erf </ci> <apply> <times /> <ci> a </ci> <ci> z </ci> </apply> </apply> <apply> <plus /> <apply> <times /> <ci> b </ci> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <apply> <cosh /> <apply> <times /> <ci> b </ci> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <sinh /> <apply> <times /> <ci> b </ci> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










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