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http://functions.wolfram.com/08.05.21.0012.01
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Integrate[(Cot[z] Csc[z] EllipticF[z, m])/Sqrt[1 - m Sin[z]^2], z] ==
-((Sqrt[2 - m + m Cos[2 z]] Csc[z] EllipticF[z, m])/Sqrt[2]) -
Log[Cos[z/2]] + Log[Sin[z/2]]
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Cell[BoxData[RowBox[List[RowBox[List["\[Integral]", RowBox[List[FractionBox[RowBox[List[RowBox[List["Cot", "[", "z", "]"]], " ", RowBox[List["Csc", "[", "z", "]"]], " ", RowBox[List["EllipticF", "[", RowBox[List["z", ",", "m"]], "]"]]]], SqrtBox[RowBox[List["1", "-", RowBox[List["m", " ", SuperscriptBox[RowBox[List["Sin", "[", "z", "]"]], "2"]]]]]]], RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", RowBox[List[RowBox[List["-", FractionBox[RowBox[List[SqrtBox[RowBox[List["2", "-", "m", "+", RowBox[List["m", " ", RowBox[List["Cos", "[", RowBox[List["2", " ", "z"]], "]"]]]]]]], " ", RowBox[List["Csc", "[", "z", "]"]], " ", RowBox[List["EllipticF", "[", RowBox[List["z", ",", "m"]], "]"]]]], SqrtBox["2"]]]], "-", RowBox[List["Log", "[", RowBox[List["Cos", "[", FractionBox["z", "2"], "]"]], "]"]], "+", RowBox[List["Log", "[", RowBox[List["Sin", "[", FractionBox["z", "2"], "]"]], "]"]]]]]]]]
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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> <mfrac> <mrow> <mrow> <mi> cot </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> csc </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> F </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> z </mi> <mo> ❘ </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <msqrt> <mrow> <mn> 1 </mn> <mo> - </mo> <mrow> <mi> m </mi> <mo> ⁢ </mo> <mrow> <msup> <mi> sin </mi> <mn> 2 </mn> </msup> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> </mrow> </msqrt> </mfrac> <mo> ⁢ </mo> <mrow> <mo> ⅆ </mo> <mi> z </mi> </mrow> </mrow> </mrow> <mo> ⩵ </mo> <mrow> <mrow> <mo> - </mo> <mfrac> <mrow> <msqrt> <mrow> <mrow> <mrow> <mi> cos </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mi> m </mi> </mrow> <mo> - </mo> <mi> m </mi> <mo> + </mo> <mn> 2 </mn> </mrow> </msqrt> <mo> ⁢ </mo> <mrow> <mi> csc </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> F </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> z </mi> <mo> ❘ </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <msqrt> <mn> 2 </mn> </msqrt> </mfrac> </mrow> <mo> - </mo> <mrow> <mi> log </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> cos </mi> <mo> ⁡ </mo> <mo> ( </mo> <mfrac> <mi> z </mi> <mn> 2 </mn> </mfrac> <mo> ) </mo> </mrow> <mo> ) </mo> </mrow> <mo> + </mo> <mrow> <mi> log </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> sin </mi> <mo> ⁡ </mo> <mo> ( </mo> <mfrac> <mi> z </mi> <mn> 2 </mn> </mfrac> <mo> ) </mo> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <int /> <bvar> <ci> z </ci> </bvar> <apply> <times /> <apply> <cot /> <ci> z </ci> </apply> <apply> <csc /> <ci> z </ci> </apply> <apply> <ci> EllipticF </ci> <ci> z </ci> <ci> m </ci> </apply> <apply> <power /> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <ci> m </ci> <apply> <power /> <apply> <sin /> <ci> z </ci> </apply> <cn type='integer'> 2 </cn> </apply> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <power /> <apply> <plus /> <apply> <times /> <apply> <cos /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> z </ci> </apply> </apply> <ci> m </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> m </ci> </apply> <cn type='integer'> 2 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <csc /> <ci> z </ci> </apply> <apply> <ci> EllipticF </ci> <ci> z </ci> <ci> m </ci> </apply> <apply> <power /> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ln /> <apply> <cos /> <apply> <times /> <ci> z </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <ln /> <apply> <sin /> <apply> <times /> <ci> z </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>
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| Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["\[Integral]", RowBox[List[FractionBox[RowBox[List[RowBox[List["Cot", "[", "z_", "]"]], " ", RowBox[List["Csc", "[", "z_", "]"]], " ", RowBox[List["EllipticF", "[", RowBox[List["z_", ",", "m_"]], "]"]]]], SqrtBox[RowBox[List["1", "-", RowBox[List["m_", " ", SuperscriptBox[RowBox[List["Sin", "[", "z_", "]"]], "2"]]]]]]], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List["-", FractionBox[RowBox[List[SqrtBox[RowBox[List["2", "-", "m", "+", RowBox[List["m", " ", RowBox[List["Cos", "[", RowBox[List["2", " ", "z"]], "]"]]]]]]], " ", RowBox[List["Csc", "[", "z", "]"]], " ", RowBox[List["EllipticF", "[", RowBox[List["z", ",", "m"]], "]"]]]], SqrtBox["2"]]]], "-", RowBox[List["Log", "[", RowBox[List["Cos", "[", FractionBox["z", "2"], "]"]], "]"]], "+", RowBox[List["Log", "[", RowBox[List["Sin", "[", FractionBox["z", "2"], "]"]], "]"]]]]]]]] |
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Date Added to functions.wolfram.com (modification date)
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