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http://functions.wolfram.com/01.09.21.0181.01
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Integrate[Cot[c z] Sqrt[(a + b Cot[c z])^3], z] ==
(Sqrt[(a + b Cot[c z])^3]
(3 (a - I b)^(3/2) ArcTanh[Sqrt[a + b Cot[c z]]/Sqrt[a - I b]] +
3 (a + I b)^(3/2) ArcTanh[Sqrt[a + b Cot[c z]]/Sqrt[a + I b]] -
2 Sqrt[a + b Cot[c z]] (4 a + b Cot[c z])))/(3 c (a + b Cot[c z])^(3/2))
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Cell[BoxData[RowBox[List[RowBox[List["\[Integral]", RowBox[List[RowBox[List["Cot", "[", RowBox[List["c", " ", "z"]], "]"]], SqrtBox[SuperscriptBox[RowBox[List["(", RowBox[List["a", "+", RowBox[List["b", " ", RowBox[List["Cot", "[", RowBox[List["c", " ", "z"]], "]"]]]]]], ")"]], "3"]], RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", RowBox[List[RowBox[List["(", RowBox[List[SqrtBox[SuperscriptBox[RowBox[List["(", RowBox[List["a", "+", RowBox[List["b", " ", RowBox[List["Cot", "[", RowBox[List["c", " ", "z"]], "]"]]]]]], ")"]], "3"]], " ", RowBox[List["(", RowBox[List[RowBox[List["3", " ", SuperscriptBox[RowBox[List["(", RowBox[List["a", "-", RowBox[List["\[ImaginaryI]", " ", "b"]]]], ")"]], RowBox[List["3", "/", "2"]]], " ", RowBox[List["ArcTanh", "[", FractionBox[SqrtBox[RowBox[List["a", "+", RowBox[List["b", " ", RowBox[List["Cot", "[", RowBox[List["c", " ", "z"]], "]"]]]]]]], SqrtBox[RowBox[List["a", "-", RowBox[List["\[ImaginaryI]", " ", "b"]]]]]], "]"]]]], "+", RowBox[List["3", " ", SuperscriptBox[RowBox[List["(", RowBox[List["a", "+", RowBox[List["\[ImaginaryI]", " ", "b"]]]], ")"]], RowBox[List["3", "/", "2"]]], " ", RowBox[List["ArcTanh", "[", FractionBox[SqrtBox[RowBox[List["a", "+", RowBox[List["b", " ", RowBox[List["Cot", "[", RowBox[List["c", " ", "z"]], "]"]]]]]]], SqrtBox[RowBox[List["a", "+", RowBox[List["\[ImaginaryI]", " ", "b"]]]]]], "]"]]]], "-", RowBox[List["2", " ", SqrtBox[RowBox[List["a", "+", RowBox[List["b", " ", RowBox[List["Cot", "[", RowBox[List["c", " ", "z"]], "]"]]]]]]], " ", RowBox[List["(", RowBox[List[RowBox[List["4", " ", "a"]], "+", RowBox[List["b", " ", RowBox[List["Cot", "[", RowBox[List["c", " ", "z"]], "]"]]]]]], ")"]]]]]], ")"]]]], ")"]], "/", RowBox[List["(", RowBox[List["3", " ", "c", " ", SuperscriptBox[RowBox[List["(", RowBox[List["a", "+", RowBox[List["b", " ", RowBox[List["Cot", "[", RowBox[List["c", " ", "z"]], "]"]]]]]], ")"]], RowBox[List["3", "/", "2"]]]]], ")"]]]]]]]]
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| Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["\[Integral]", RowBox[List[RowBox[List[RowBox[List["Cot", "[", RowBox[List["c_", " ", "z_"]], "]"]], " ", SqrtBox[SuperscriptBox[RowBox[List["(", RowBox[List["a_", "+", RowBox[List["b_", " ", RowBox[List["Cot", "[", RowBox[List["c_", " ", "z_"]], "]"]]]]]], ")"]], "3"]]]], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", FractionBox[RowBox[List[SqrtBox[SuperscriptBox[RowBox[List["(", RowBox[List["a", "+", RowBox[List["b", " ", RowBox[List["Cot", "[", RowBox[List["c", " ", "z"]], "]"]]]]]], ")"]], "3"]], " ", RowBox[List["(", RowBox[List[RowBox[List["3", " ", SuperscriptBox[RowBox[List["(", RowBox[List["a", "-", RowBox[List["\[ImaginaryI]", " ", "b"]]]], ")"]], RowBox[List["3", "/", "2"]]], " ", RowBox[List["ArcTanh", "[", FractionBox[SqrtBox[RowBox[List["a", "+", RowBox[List["b", " ", RowBox[List["Cot", "[", RowBox[List["c", " ", "z"]], "]"]]]]]]], SqrtBox[RowBox[List["a", "-", RowBox[List["\[ImaginaryI]", " ", "b"]]]]]], "]"]]]], "+", RowBox[List["3", " ", SuperscriptBox[RowBox[List["(", RowBox[List["a", "+", RowBox[List["\[ImaginaryI]", " ", "b"]]]], ")"]], RowBox[List["3", "/", "2"]]], " ", RowBox[List["ArcTanh", "[", FractionBox[SqrtBox[RowBox[List["a", "+", RowBox[List["b", " ", RowBox[List["Cot", "[", RowBox[List["c", " ", "z"]], "]"]]]]]]], SqrtBox[RowBox[List["a", "+", RowBox[List["\[ImaginaryI]", " ", "b"]]]]]], "]"]]]], "-", RowBox[List["2", " ", SqrtBox[RowBox[List["a", "+", RowBox[List["b", " ", RowBox[List["Cot", "[", RowBox[List["c", " ", "z"]], "]"]]]]]]], " ", RowBox[List["(", RowBox[List[RowBox[List["4", " ", "a"]], "+", RowBox[List["b", " ", RowBox[List["Cot", "[", RowBox[List["c", " ", "z"]], "]"]]]]]], ")"]]]]]], ")"]]]], RowBox[List["3", " ", "c", " ", SuperscriptBox[RowBox[List["(", RowBox[List["a", "+", RowBox[List["b", " ", RowBox[List["Cot", "[", RowBox[List["c", " ", "z"]], "]"]]]]]], ")"]], RowBox[List["3", "/", "2"]]]]]]]]]] |
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Date Added to functions.wolfram.com (modification date)
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