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 Erf

 http://functions.wolfram.com/06.25.21.0105.01

 Input Form

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

 Standard Form

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 MathML Form

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

 Rule Form

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

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

 2001-10-29