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http://functions.wolfram.com/01.19.21.2182.01
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Integrate[E^(p z) Sinh[b z^2] Sinh[c z^2], z] ==
(1/8) Sqrt[Pi]
(-((E^(p^2/(4 b - 4 c)) Erf[(-p + 2 b z - 2 c z)/(2 Sqrt[b - c])])/
Sqrt[b - c]) + (E^(p^2/(4 (b + c)))
Erf[(-p + 2 (b + c) z)/(2 Sqrt[b + c])])/Sqrt[b + c] -
(E^(p^2/(-4 b + 4 c)) Erfi[(p + 2 b z - 2 c z)/(2 Sqrt[b - c])])/
Sqrt[b - c] + Erfi[(p + 2 (b + c) z)/(2 Sqrt[b + c])]/
(E^(p^2/(4 (b + c))) Sqrt[b + c]))
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Cell[BoxData[RowBox[List[RowBox[List["\[Integral]", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["p", " ", "z"]]], " ", RowBox[List["Sinh", "[", RowBox[List["b", " ", SuperscriptBox["z", "2"]]], "]"]], " ", RowBox[List["Sinh", "[", RowBox[List["c", " ", SuperscriptBox["z", "2"]]], "]"]], RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", RowBox[List[FractionBox["1", "8"], " ", SqrtBox["\[Pi]"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", FractionBox[RowBox[List[SuperscriptBox["\[ExponentialE]", FractionBox[SuperscriptBox["p", "2"], RowBox[List[RowBox[List["4", " ", "b"]], "-", RowBox[List["4", " ", "c"]]]]]], " ", RowBox[List["Erf", "[", FractionBox[RowBox[List[RowBox[List["-", "p"]], "+", RowBox[List["2", " ", "b", " ", "z"]], "-", RowBox[List["2", " ", "c", " ", "z"]]]], RowBox[List["2", " ", SqrtBox[RowBox[List["b", "-", "c"]]]]]], "]"]]]], SqrtBox[RowBox[List["b", "-", "c"]]]]]], "+", FractionBox[RowBox[List[SuperscriptBox["\[ExponentialE]", FractionBox[SuperscriptBox["p", "2"], RowBox[List["4", " ", RowBox[List["(", RowBox[List["b", "+", "c"]], ")"]]]]]], " ", RowBox[List["Erf", "[", FractionBox[RowBox[List[RowBox[List["-", "p"]], "+", RowBox[List["2", " ", RowBox[List["(", RowBox[List["b", "+", "c"]], ")"]], " ", "z"]]]], RowBox[List["2", " ", SqrtBox[RowBox[List["b", "+", "c"]]]]]], "]"]]]], SqrtBox[RowBox[List["b", "+", "c"]]]], "-", FractionBox[RowBox[List[SuperscriptBox["\[ExponentialE]", FractionBox[SuperscriptBox["p", "2"], RowBox[List[RowBox[List[RowBox[List["-", "4"]], " ", "b"]], "+", RowBox[List["4", " ", "c"]]]]]], " ", RowBox[List["Erfi", "[", FractionBox[RowBox[List["p", "+", RowBox[List["2", " ", "b", " ", "z"]], "-", RowBox[List["2", " ", "c", " ", "z"]]]], RowBox[List["2", " ", SqrtBox[RowBox[List["b", "-", "c"]]]]]], "]"]]]], SqrtBox[RowBox[List["b", "-", "c"]]]], "+", FractionBox[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["-", FractionBox[SuperscriptBox["p", "2"], RowBox[List["4", " ", RowBox[List["(", RowBox[List["b", "+", "c"]], ")"]]]]]]]], " ", RowBox[List["Erfi", "[", FractionBox[RowBox[List["p", "+", RowBox[List["2", " ", RowBox[List["(", RowBox[List["b", "+", "c"]], ")"]], " ", "z"]]]], RowBox[List["2", " ", SqrtBox[RowBox[List["b", "+", "c"]]]]]], "]"]]]], SqrtBox[RowBox[List["b", "+", "c"]]]]]], ")"]]]]]]]]
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<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mo> ∫ </mo> <mrow> <mrow> <msup> <mi> ⅇ </mi> <mrow> <mi> p </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> </msup> <mo> ⁢ </mo> <mrow> <mi> sinh </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> b </mi> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> sinh </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> c </mi> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ⁢ </mo> <mrow> <mo> ⅆ </mo> <mi> z </mi> </mrow> </mrow> </mrow> <mo> ⩵ </mo> <mrow> <mfrac> <msqrt> <mi> π </mi> </msqrt> <mn> 8 </mn> </mfrac> <mo> ⁢ </mo> <mtext> </mtext> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <mfrac> <mrow> <msup> <mi> ⅇ </mi> <mfrac> <msup> <mi> p </mi> <mn> 2 </mn> </msup> <mrow> <mrow> <mn> 4 </mn> <mo> ⁢ </mo> <mi> b </mi> </mrow> <mo> - </mo> <mrow> <mn> 4 </mn> <mo> ⁢ </mo> <mi> c </mi> </mrow> </mrow> </mfrac> </msup> <mo> ⁢ </mo> <mrow> <mi> erf </mi> <mo> ⁡ </mo> <mo> ( </mo> <mfrac> <mrow> <mrow> <mo> - </mo> <mi> p </mi> </mrow> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> b </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> - </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> c </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> </mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <msqrt> <mrow> <mi> b </mi> <mo> - </mo> <mi> c </mi> </mrow> </msqrt> </mrow> </mfrac> <mo> ) </mo> </mrow> </mrow> <msqrt> <mrow> <mi> b </mi> <mo> - </mo> <mi> c </mi> </mrow> </msqrt> </mfrac> </mrow> <mo> + </mo> <mfrac> <mrow> <msup> <mi> ⅇ </mi> <mfrac> <msup> <mi> p </mi> <mn> 2 </mn> </msup> <mrow> <mn> 4 </mn> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> b </mi> <mo> + </mo> <mi> c </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mfrac> </msup> <mo> ⁢ </mo> <mrow> <mi> erf </mi> <mo> ⁡ </mo> <mo> ( </mo> <mfrac> <mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> b </mi> <mo> + </mo> <mi> c </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> - </mo> <mi> p </mi> </mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <msqrt> <mrow> <mi> b </mi> <mo> + </mo> <mi> c </mi> </mrow> </msqrt> </mrow> </mfrac> <mo> ) </mo> </mrow> </mrow> <msqrt> <mrow> <mi> b </mi> <mo> + </mo> <mi> c </mi> </mrow> </msqrt> </mfrac> <mo> + </mo> <mfrac> <mrow> <msup> <mi> ⅇ </mi> <mrow> <mo> - </mo> <mfrac> <msup> <mi> p </mi> <mn> 2 </mn> </msup> <mrow> <mn> 4 </mn> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> b </mi> <mo> + </mo> <mi> c </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mfrac> </mrow> </msup> <mo> ⁢ </mo> <mrow> <mi> erfi </mi> <mo> ⁡ </mo> <mo> ( </mo> <mfrac> <mrow> <mi> p </mi> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> b </mi> <mo> + </mo> <mi> c </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mi> z </mi> </mrow> </mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <msqrt> <mrow> <mi> b </mi> <mo> + </mo> <mi> c </mi> </mrow> </msqrt> </mrow> </mfrac> <mo> ) </mo> </mrow> </mrow> <msqrt> <mrow> <mi> b </mi> <mo> + </mo> <mi> c </mi> </mrow> </msqrt> </mfrac> <mo> - </mo> <mfrac> <mrow> <msup> <mi> ⅇ </mi> <mfrac> <msup> <mi> p </mi> <mn> 2 </mn> </msup> <mrow> <mrow> <mn> 4 </mn> <mo> ⁢ </mo> <mi> c </mi> </mrow> <mo> - </mo> <mrow> <mn> 4 </mn> <mo> ⁢ </mo> <mi> b </mi> </mrow> </mrow> </mfrac> </msup> <mo> ⁢ </mo> <mrow> <mi> erfi </mi> <mo> ⁡ </mo> <mo> ( </mo> <mfrac> <mrow> <mi> p </mi> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> b </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> - </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> c </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> </mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <msqrt> <mrow> <mi> b </mi> <mo> - </mo> <mi> c </mi> </mrow> </msqrt> </mrow> </mfrac> <mo> ) </mo> </mrow> </mrow> <msqrt> <mrow> <mi> b </mi> <mo> - </mo> <mi> c </mi> </mrow> </msqrt> </mfrac> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <int /> <bvar> <ci> z </ci> </bvar> <apply> <times /> <apply> <power /> <exponentiale /> <apply> <times /> <ci> p </ci> <ci> z </ci> </apply> </apply> <apply> <sinh /> <apply> <times /> <ci> b </ci> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <apply> <sinh /> <apply> <times /> <ci> c </ci> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <times /> <apply> <times /> <apply> <power /> <pi /> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <cn type='integer'> 8 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <power /> <exponentiale /> <apply> <times /> <apply> <power /> <ci> p </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <plus /> <apply> <times /> <cn type='integer'> 4 </cn> <ci> b </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 4 </cn> <ci> c </ci> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <ci> Erf </ci> <apply> <times /> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> p </ci> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <ci> b </ci> <ci> z </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <ci> c </ci> <ci> z </ci> </apply> </apply> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <apply> <plus /> <ci> b </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> c </ci> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <power /> <apply> <power /> <apply> <plus /> <ci> b </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> c </ci> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <times /> <apply> <power /> <exponentiale /> <apply> <times /> <apply> <power /> <ci> p </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 4 </cn> <apply> <plus /> <ci> b </ci> <ci> c </ci> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <ci> Erf </ci> <apply> <times /> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <plus /> <ci> b </ci> <ci> c </ci> </apply> <ci> z </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> p </ci> </apply> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <apply> <plus /> <ci> b </ci> <ci> c </ci> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <power /> <apply> <power /> <apply> <plus /> <ci> b </ci> <ci> c </ci> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <apply> <power /> <exponentiale /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <power /> <ci> p </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 4 </cn> <apply> <plus /> <ci> b </ci> <ci> c </ci> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <ci> Erfi </ci> <apply> <times /> <apply> <plus /> <ci> p </ci> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <plus /> <ci> b </ci> <ci> c </ci> </apply> <ci> z </ci> </apply> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <apply> <plus /> <ci> b </ci> <ci> c </ci> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <power /> <apply> <power /> <apply> <plus /> <ci> b </ci> <ci> c </ci> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <power /> <exponentiale /> <apply> <times /> <apply> <power /> <ci> p </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <plus /> <apply> <times /> <cn type='integer'> 4 </cn> <ci> c </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 4 </cn> <ci> b </ci> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <ci> Erfi </ci> <apply> <times /> <apply> <plus /> <ci> p </ci> <apply> <times /> <cn type='integer'> 2 </cn> <ci> b </ci> <ci> z </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <ci> c </ci> <ci> z </ci> </apply> </apply> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <apply> <plus /> <ci> b </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> c </ci> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <power /> <apply> <power /> <apply> <plus /> <ci> b </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> c </ci> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>
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Date Added to functions.wolfram.com (modification date)
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