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http://functions.wolfram.com/06.25.21.0078.01
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Integrate[z^2 Sinh[b z] Erf[a z], z] == (1/(8 a^4 b^3 Sqrt[Pi]))
((8 a^3 b - 2 a b^3 - 8 a^3 b E^(2 b z) + 2 a b^3 E^(2 b z) + 4 a^3 b^2 z +
4 a^3 b^2 E^(2 b z) z + 4 a^4 E^(a^2 z^2) Sqrt[Pi]
(2 + 2 b z + b^2 z^2 + E^(2 b z) (2 - 2 b z + b^2 z^2)) Erf[a z] +
(8 a^4 - 2 a^2 b^2 + b^4) E^((b + 2 a^2 z)^2/(4 a^2)) Sqrt[Pi]
Erf[b/(2 a) - a z] - 8 a^4 E^((b + 2 a^2 z)^2/(4 a^2)) Sqrt[Pi]
Erf[b/(2 a) + a z] + 2 a^2 b^2 E^((b + 2 a^2 z)^2/(4 a^2)) Sqrt[Pi]
Erf[b/(2 a) + a z] - b^4 E^((b + 2 a^2 z)^2/(4 a^2)) Sqrt[Pi]
Erf[b/(2 a) + a z])/E^(z (b + a^2 z)))
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Cell[BoxData[RowBox[List[RowBox[List["\[Integral]", RowBox[List[SuperscriptBox["z", "2"], " ", RowBox[List["Sinh", "[", RowBox[List["b", " ", "z"]], "]"]], RowBox[List["Erf", "[", RowBox[List["a", " ", "z"]], "]"]], RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", RowBox[List[FractionBox["1", RowBox[List["8", " ", SuperscriptBox["a", "4"], " ", SuperscriptBox["b", "3"], " ", SqrtBox["\[Pi]"]]]], RowBox[List["(", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "z"]], " ", RowBox[List["(", RowBox[List["b", "+", RowBox[List[SuperscriptBox["a", "2"], " ", "z"]]]], ")"]]]]], " ", RowBox[List["(", RowBox[List[RowBox[List["8", " ", SuperscriptBox["a", "3"], " ", "b"]], "-", RowBox[List["2", " ", "a", " ", SuperscriptBox["b", "3"]]], "-", RowBox[List["8", " ", SuperscriptBox["a", "3"], " ", "b", " ", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "b", " ", "z"]]]]], "+", RowBox[List["2", " ", "a", " ", SuperscriptBox["b", "3"], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "b", " ", "z"]]]]], "+", RowBox[List["4", " ", SuperscriptBox["a", "3"], " ", SuperscriptBox["b", "2"], " ", "z"]], "+", RowBox[List["4", " ", SuperscriptBox["a", "3"], " ", SuperscriptBox["b", "2"], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "b", " ", "z"]]], " ", "z"]], "+", RowBox[List["4", " ", SuperscriptBox["a", "4"], " ", SuperscriptBox["\[ExponentialE]", RowBox[List[SuperscriptBox["a", "2"], " ", SuperscriptBox["z", "2"]]]], " ", SqrtBox["\[Pi]"], " ", RowBox[List["(", RowBox[List["2", "+", RowBox[List["2", " ", "b", " ", "z"]], "+", RowBox[List[SuperscriptBox["b", "2"], " ", SuperscriptBox["z", "2"]]], "+", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "b", " ", "z"]]], " ", RowBox[List["(", RowBox[List["2", "-", RowBox[List["2", " ", "b", " ", "z"]], "+", RowBox[List[SuperscriptBox["b", "2"], " ", SuperscriptBox["z", "2"]]]]], ")"]]]]]], ")"]], " ", RowBox[List["Erf", "[", RowBox[List["a", " ", "z"]], "]"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["8", " ", SuperscriptBox["a", "4"]]], "-", RowBox[List["2", " ", SuperscriptBox["a", "2"], " ", SuperscriptBox["b", "2"]]], "+", SuperscriptBox["b", "4"]]], ")"]], " ", SuperscriptBox["\[ExponentialE]", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List["b", "+", RowBox[List["2", " ", SuperscriptBox["a", "2"], " ", "z"]]]], ")"]], "2"], RowBox[List["4", " ", SuperscriptBox["a", "2"]]]]], " ", SqrtBox["\[Pi]"], " ", RowBox[List["Erf", "[", RowBox[List[FractionBox["b", RowBox[List["2", " ", "a"]]], "-", RowBox[List["a", " ", "z"]]]], "]"]]]], "-", RowBox[List["8", " ", SuperscriptBox["a", "4"], " ", SuperscriptBox["\[ExponentialE]", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List["b", "+", RowBox[List["2", " ", SuperscriptBox["a", "2"], " ", "z"]]]], ")"]], "2"], RowBox[List["4", " ", SuperscriptBox["a", "2"]]]]], " ", SqrtBox["\[Pi]"], " ", RowBox[List["Erf", "[", RowBox[List[FractionBox["b", RowBox[List["2", " ", "a"]]], "+", RowBox[List["a", " ", "z"]]]], "]"]]]], "+", RowBox[List["2", " ", SuperscriptBox["a", "2"], " ", SuperscriptBox["b", "2"], " ", SuperscriptBox["\[ExponentialE]", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List["b", "+", RowBox[List["2", " ", SuperscriptBox["a", "2"], " ", "z"]]]], ")"]], "2"], RowBox[List["4", " ", SuperscriptBox["a", "2"]]]]], " ", SqrtBox["\[Pi]"], " ", RowBox[List["Erf", "[", RowBox[List[FractionBox["b", RowBox[List["2", " ", "a"]]], "+", RowBox[List["a", " ", "z"]]]], "]"]]]], "-", RowBox[List[SuperscriptBox["b", "4"], " ", SuperscriptBox["\[ExponentialE]", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List["b", "+", RowBox[List["2", " ", SuperscriptBox["a", "2"], " ", "z"]]]], ")"]], "2"], RowBox[List["4", " ", SuperscriptBox["a", "2"]]]]], " ", SqrtBox["\[Pi]"], " ", RowBox[List["Erf", "[", RowBox[List[FractionBox["b", RowBox[List["2", " ", "a"]]], "+", RowBox[List["a", " ", "z"]]]], "]"]]]]]], ")"]]]], ")"]]]]]]]]
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type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <ci> a </ci> <ci> z </ci> </apply> </apply> </apply> <apply> <power /> <ci> a </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <ci> b </ci> <cn type='integer'> 3 </cn> </apply> <apply> <power /> <exponentiale /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> b </ci> <ci> z </ci> </apply> </apply> <ci> a </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <ci> b </ci> <cn type='integer'> 3 </cn> </apply> <ci> a </ci> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <power /> <ci> b </ci> <cn type='integer'> 4 </cn> </apply> <apply> <power /> <exponentiale /> <apply> <times /> <apply> <power /> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> z </ci> <apply> <power /> <ci> a </ci> <cn type='integer'> 2 </cn> </apply> </apply> <ci> b </ci> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 4 </cn> <apply> <power /> <ci> a </ci> <cn type='integer'> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <power /> <pi /> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <ci> Erf </ci> <apply> <plus /> <apply> <times /> <ci> b </ci> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> a </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <ci> a </ci> <ci> z </ci> </apply> </apply> </apply> </apply> </apply> <apply> <times /> <apply> <plus /> <apply> <times /> <cn type='integer'> 8 </cn> <apply> <power /> <ci> a </ci> <cn type='integer'> 4 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <ci> b </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <ci> a </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <apply> <power /> <ci> b </ci> <cn type='integer'> 4 </cn> </apply> </apply> <apply> <power /> <exponentiale /> <apply> <times /> <apply> <power /> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> z </ci> <apply> <power /> <ci> a </ci> <cn type='integer'> 2 </cn> </apply> </apply> <ci> b </ci> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 4 </cn> <apply> <power /> <ci> a </ci> <cn type='integer'> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <power /> <pi /> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <ci> Erf </ci> <apply> <plus /> <apply> <times /> <ci> b </ci> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> a </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <ci> a </ci> <ci> z </ci> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>
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