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http://functions.wolfram.com/01.11.21.0082.01
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Integrate[1/Sqrt[Sec[c z]], z] == (2 EllipticE[(c z)/2, 2])/
(c Sqrt[Cos[c z]] Sqrt[Sec[c z]])
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Cell[BoxData[RowBox[List[RowBox[List["\[Integral]", RowBox[List[FractionBox["1", SqrtBox[RowBox[List["Sec", "[", RowBox[List["c", " ", "z"]], "]"]]]], RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", FractionBox[RowBox[List["2", " ", RowBox[List["EllipticE", "[", RowBox[List[FractionBox[RowBox[List["c", " ", "z"]], "2"], ",", "2"]], "]"]]]], RowBox[List["c", " ", SqrtBox[RowBox[List["Cos", "[", RowBox[List["c", " ", "z"]], "]"]]], " ", SqrtBox[RowBox[List["Sec", "[", RowBox[List["c", " ", "z"]], "]"]]]]]]]]]]
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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> <mn> 1 </mn> <mrow> <msup> <mi> sec </mi> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </msup> <mo> ( </mo> <mrow> <mi> c </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mfrac> <mo> ⁢ </mo> <mrow> <mo> ⅆ </mo> <mi> z </mi> </mrow> </mrow> </mrow> <mo> ⩵ </mo> <mfrac> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mrow> <mi> E </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mfrac> <mrow> <mi> c </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mn> 2 </mn> </mfrac> <mo> ❘ </mo> <mn> 2 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mrow> <mi> c </mi> <mo> ⁢ </mo> <mrow> <msup> <mi> cos </mi> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </msup> <mo> ( </mo> <mrow> <mi> c </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <msup> <mi> sec </mi> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </msup> <mo> ( </mo> <mrow> <mi> c </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mfrac> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <int /> <bvar> <ci> z </ci> </bvar> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <power /> <apply> <sec /> <apply> <times /> <ci> c </ci> <ci> z </ci> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <ci> EllipticE </ci> <apply> <times /> <ci> c </ci> <ci> z </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <times /> <ci> c </ci> <apply> <power /> <apply> <cos /> <apply> <times /> <ci> c </ci> <ci> z </ci> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <sec /> <apply> <times /> <ci> c </ci> <ci> z </ci> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </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[FractionBox["1", SqrtBox[RowBox[List["Sec", "[", RowBox[List["c_", " ", "z_"]], "]"]]]], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", FractionBox[RowBox[List["2", " ", RowBox[List["EllipticE", "[", RowBox[List[FractionBox[RowBox[List["c", " ", "z"]], "2"], ",", "2"]], "]"]]]], RowBox[List["c", " ", SqrtBox[RowBox[List["Cos", "[", RowBox[List["c", " ", "z"]], "]"]]], " ", SqrtBox[RowBox[List["Sec", "[", RowBox[List["c", " ", "z"]], "]"]]]]]]]]]] |
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Date Added to functions.wolfram.com (modification date)
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