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http://functions.wolfram.com/01.10.21.0081.01
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Integrate[Csc[c z]^7, z] ==
(1/(48 c)) ((-Cot[c z]) Csc[c z] (15 + 10 Csc[c z]^2 + 8 Csc[c z]^4) +
15 (-Log[Cos[(c z)/2]] + Log[Sin[(c z)/2]]))
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Cell[BoxData[RowBox[List[RowBox[List["\[Integral]", RowBox[List[SuperscriptBox[RowBox[List["Csc", "[", RowBox[List["c", " ", "z"]], "]"]], "7"], RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", RowBox[List[FractionBox["1", RowBox[List["48", " ", "c"]]], RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", RowBox[List["Cot", "[", RowBox[List["c", " ", "z"]], "]"]]]], " ", RowBox[List["Csc", "[", RowBox[List["c", " ", "z"]], "]"]], " ", RowBox[List["(", RowBox[List["15", "+", RowBox[List["10", " ", SuperscriptBox[RowBox[List["Csc", "[", RowBox[List["c", " ", "z"]], "]"]], "2"]]], "+", RowBox[List["8", " ", SuperscriptBox[RowBox[List["Csc", "[", RowBox[List["c", " ", "z"]], "]"]], "4"]]]]], ")"]]]], "+", RowBox[List["15", " ", RowBox[List["(", RowBox[List[RowBox[List["-", RowBox[List["Log", "[", RowBox[List["Cos", "[", FractionBox[RowBox[List["c", " ", "z"]], "2"], "]"]], "]"]]]], "+", RowBox[List["Log", "[", RowBox[List["Sin", "[", FractionBox[RowBox[List["c", " ", "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> <mrow> <msup> <mi> csc </mi> <mn> 7 </mn> </msup> <mo> ( </mo> <mrow> <mi> c </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mo> ⅆ </mo> <mi> z </mi> </mrow> </mrow> </mrow> <mo> ⩵ </mo> <mfrac> <mrow> <mrow> <mn> 15 </mn> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> log </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> sin </mi> <mo> ⁡ </mo> <mo> ( </mo> <mfrac> <mrow> <mi> c </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mn> 2 </mn> </mfrac> <mo> ) </mo> </mrow> <mo> ) </mo> </mrow> <mo> - </mo> <mrow> <mi> log </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> cos </mi> <mo> ⁡ </mo> <mo> ( </mo> <mfrac> <mrow> <mi> c </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mn> 2 </mn> </mfrac> <mo> ) </mo> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <mrow> <mi> cot </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> c </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> csc </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> c </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 8 </mn> <mo> ⁢ </mo> <mrow> <msup> <mi> csc </mi> <mn> 4 </mn> </msup> <mo> ( </mo> <mrow> <mi> c </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mn> 10 </mn> <mo> ⁢ </mo> <mrow> <msup> <mi> csc </mi> <mn> 2 </mn> </msup> <mo> ( </mo> <mrow> <mi> c </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mn> 15 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mrow> <mn> 48 </mn> <mo> ⁢ </mo> <mi> c </mi> </mrow> </mfrac> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <int /> <bvar> <ci> z </ci> </bvar> <apply> <power /> <apply> <csc /> <apply> <times /> <ci> c </ci> <ci> z </ci> </apply> </apply> <cn type='integer'> 7 </cn> </apply> </apply> <apply> <times /> <apply> <plus /> <apply> <times /> <cn type='integer'> 15 </cn> <apply> <plus /> <apply> <ln /> <apply> <sin /> <apply> <times /> <ci> c </ci> <ci> z </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ln /> <apply> <cos /> <apply> <times /> <ci> c </ci> <ci> z </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <cot /> <apply> <times /> <ci> c </ci> <ci> z </ci> </apply> </apply> <apply> <csc /> <apply> <times /> <ci> c </ci> <ci> z </ci> </apply> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 8 </cn> <apply> <power /> <apply> <csc /> <apply> <times /> <ci> c </ci> <ci> z </ci> </apply> </apply> <cn type='integer'> 4 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 10 </cn> <apply> <power /> <apply> <csc /> <apply> <times /> <ci> c </ci> <ci> z </ci> </apply> </apply> <cn type='integer'> 2 </cn> </apply> </apply> <cn type='integer'> 15 </cn> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 48 </cn> <ci> c </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </annotation-xml> </semantics> </math>
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| Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["\[Integral]", RowBox[List[SuperscriptBox[RowBox[List["Csc", "[", RowBox[List["c_", " ", "z_"]], "]"]], "7"], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", FractionBox[RowBox[List[RowBox[List[RowBox[List["-", RowBox[List["Cot", "[", RowBox[List["c", " ", "z"]], "]"]]]], " ", RowBox[List["Csc", "[", RowBox[List["c", " ", "z"]], "]"]], " ", RowBox[List["(", RowBox[List["15", "+", RowBox[List["10", " ", SuperscriptBox[RowBox[List["Csc", "[", RowBox[List["c", " ", "z"]], "]"]], "2"]]], "+", RowBox[List["8", " ", SuperscriptBox[RowBox[List["Csc", "[", RowBox[List["c", " ", "z"]], "]"]], "4"]]]]], ")"]]]], "+", RowBox[List["15", " ", RowBox[List["(", RowBox[List[RowBox[List["-", RowBox[List["Log", "[", RowBox[List["Cos", "[", FractionBox[RowBox[List["c", " ", "z"]], "2"], "]"]], "]"]]]], "+", RowBox[List["Log", "[", RowBox[List["Sin", "[", FractionBox[RowBox[List["c", " ", "z"]], "2"], "]"]], "]"]]]], ")"]]]]]], RowBox[List["48", " ", "c"]]]]]]] |
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Date Added to functions.wolfram.com (modification date)
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