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http://functions.wolfram.com/01.06.21.1265.01
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Integrate[E^(b z) Sin[a z]^4, z] ==
(E^(b z) (192 a^4 + 60 a^2 b^2 + 3 b^4 - 4 b^2 (16 a^2 + b^2) Cos[2 a z] +
b^2 (4 a^2 + b^2) Cos[4 a z] - 128 a^3 b Sin[2 a z] -
8 a b^3 Sin[2 a z] + 16 a^3 b Sin[4 a z] + 4 a b^3 Sin[4 a z]))/
(8 (64 a^4 b + 20 a^2 b^3 + b^5))
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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> b </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> </msup> <mo> ⁢ </mo> <mrow> <msup> <mi> sin </mi> <mn> 4 </mn> </msup> <mo> ( </mo> <mrow> <mi> a </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ⁢ </mo> <mrow> <mo> ⅆ </mo> <mi> z </mi> </mrow> </mrow> </mrow> <mo> ⩵ </mo> <mrow> <mfrac> <mn> 1 </mn> <mrow> <mn> 8 </mn> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <msup> <mi> b </mi> <mn> 5 </mn> </msup> <mo> + </mo> <mrow> <mn> 20 </mn> <mo> ⁢ </mo> <msup> <mi> a </mi> <mn> 2 </mn> </msup> <mo> ⁢ </mo> <msup> <mi> b </mi> <mn> 3 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 64 </mn> <mo> ⁢ </mo> <msup> <mi> a </mi> <mn> 4 </mn> </msup> <mo> ⁢ </mo> <mi> b </mi> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mfrac> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <msup> <mi> ⅇ </mi> <mrow> <mi> b </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> </msup> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 192 </mn> <mo> ⁢ </mo> <msup> <mi> a </mi> <mn> 4 </mn> </msup> </mrow> <mo> - 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| Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["\[Integral]", RowBox[List[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["b_", " ", "z_"]]], " ", SuperscriptBox[RowBox[List["Sin", "[", RowBox[List["a_", " ", "z_"]], "]"]], "4"]]], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", FractionBox[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["b", " ", "z"]]], " ", RowBox[List["(", RowBox[List[RowBox[List["192", " ", SuperscriptBox["a", "4"]]], "+", RowBox[List["60", " ", SuperscriptBox["a", "2"], " ", SuperscriptBox["b", "2"]]], "+", RowBox[List["3", " ", SuperscriptBox["b", "4"]]], "-", RowBox[List["4", " ", SuperscriptBox["b", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["16", " ", SuperscriptBox["a", "2"]]], "+", SuperscriptBox["b", "2"]]], ")"]], " ", RowBox[List["Cos", "[", RowBox[List["2", " ", "a", " ", "z"]], "]"]]]], "+", RowBox[List[SuperscriptBox["b", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["4", " ", SuperscriptBox["a", "2"]]], "+", SuperscriptBox["b", "2"]]], ")"]], " ", RowBox[List["Cos", "[", RowBox[List["4", " ", "a", " ", "z"]], "]"]]]], "-", RowBox[List["128", " ", SuperscriptBox["a", "3"], " ", "b", " ", RowBox[List["Sin", "[", RowBox[List["2", " ", "a", " ", "z"]], "]"]]]], "-", RowBox[List["8", " ", "a", " ", SuperscriptBox["b", "3"], " ", RowBox[List["Sin", "[", RowBox[List["2", " ", "a", " ", "z"]], "]"]]]], "+", RowBox[List["16", " ", SuperscriptBox["a", "3"], " ", "b", " ", RowBox[List["Sin", "[", RowBox[List["4", " ", "a", " ", "z"]], "]"]]]], "+", RowBox[List["4", " ", "a", " ", SuperscriptBox["b", "3"], " ", RowBox[List["Sin", "[", RowBox[List["4", " ", "a", " ", "z"]], "]"]]]]]], ")"]]]], RowBox[List["8", " ", RowBox[List["(", RowBox[List[RowBox[List["64", " ", SuperscriptBox["a", "4"], " ", "b"]], "+", RowBox[List["20", " ", SuperscriptBox["a", "2"], " ", SuperscriptBox["b", "3"]]], "+", SuperscriptBox["b", "5"]]], ")"]]]]]]]]] |
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Date Added to functions.wolfram.com (modification date)
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){kind=small}
