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http://functions.wolfram.com/01.07.21.2258.01
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Integrate[1/(a Sin[e z]^2 + b Cos[e z]^2)^3, z] ==
(1/(8 a^(5/2) b^(5/2) e)) ((3 a^2 + 2 a b + 3 b^2)
ArcTan[(Sqrt[a] Tan[e z])/Sqrt[b]] -
(Sqrt[a] (a - b) Sqrt[b] (-3 a^2 - 10 a b - 3 b^2 +
3 (a^2 - b^2) Cos[2 e z]) Sin[2 e z])/(a + b + (-a + b) Cos[2 e z])^2)
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| Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["\[Integral]", RowBox[List[FractionBox["1", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["a_", " ", SuperscriptBox[RowBox[List["Sin", "[", RowBox[List["e_", " ", "z_"]], "]"]], "2"]]], "+", RowBox[List["b_", " ", SuperscriptBox[RowBox[List["Cos", "[", RowBox[List["e_", " ", "z_"]], "]"]], "2"]]]]], ")"]], "3"]], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", FractionBox[RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["3", " ", SuperscriptBox["a", "2"]]], "+", RowBox[List["2", " ", "a", " ", "b"]], "+", RowBox[List["3", " ", SuperscriptBox["b", "2"]]]]], ")"]], " ", RowBox[List["ArcTan", "[", FractionBox[RowBox[List[SqrtBox["a"], " ", RowBox[List["Tan", "[", RowBox[List["e", " ", "z"]], "]"]]]], SqrtBox["b"]], "]"]]]], "-", FractionBox[RowBox[List[SqrtBox["a"], " ", RowBox[List["(", RowBox[List["a", "-", "b"]], ")"]], " ", SqrtBox["b"], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "3"]], " ", SuperscriptBox["a", "2"]]], "-", RowBox[List["10", " ", "a", " ", "b"]], "-", RowBox[List["3", " ", SuperscriptBox["b", "2"]]], "+", RowBox[List["3", " ", RowBox[List["(", RowBox[List[SuperscriptBox["a", "2"], "-", SuperscriptBox["b", "2"]]], ")"]], " ", RowBox[List["Cos", "[", RowBox[List["2", " ", "e", " ", "z"]], "]"]]]]]], ")"]], " ", RowBox[List["Sin", "[", RowBox[List["2", " ", "e", " ", "z"]], "]"]]]], SuperscriptBox[RowBox[List["(", RowBox[List["a", "+", "b", "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "a"]], "+", "b"]], ")"]], " ", RowBox[List["Cos", "[", RowBox[List["2", " ", "e", " ", "z"]], "]"]]]]]], ")"]], "2"]]]], RowBox[List["8", " ", SuperscriptBox["a", RowBox[List["5", "/", "2"]]], " ", SuperscriptBox["b", RowBox[List["5", "/", "2"]]], " ", "e"]]]]]]] |
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Date Added to functions.wolfram.com (modification date)
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){kind=small}
