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ArcSec






Mathematica Notation

Traditional Notation









Elementary Functions > ArcSec[z] > Integration > Indefinite integration > For the direct function itself





http://functions.wolfram.com/01.18.21.0005.01









  


  










Input Form





Integrate[ArcSec[a z + b], z] == z ArcSec[b + a z] + (b/a) ArcSin[1/(-b - a z)] - (1/a) Log[2 (b + a z) + 2 (b + a z) Sqrt[(-1 + b^2 + 2 a b z + a^2 z^2)/ (b + a z)^2]]










Standard Form





Cell[BoxData[RowBox[List[RowBox[List["\[Integral]", RowBox[List[RowBox[List["ArcSec", "[", RowBox[List[RowBox[List["a", " ", "z"]], "+", "b"]], "]"]], RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", RowBox[List[RowBox[List["z", " ", RowBox[List["ArcSec", "[", RowBox[List["b", "+", RowBox[List["a", " ", "z"]]]], "]"]]]], "+", RowBox[List[FractionBox["b", "a"], RowBox[List["ArcSin", "[", FractionBox["1", RowBox[List[RowBox[List["-", "b"]], "-", RowBox[List["a", " ", "z"]]]]], "]"]]]], "-", RowBox[List[FractionBox["1", "a"], RowBox[List["Log", "[", RowBox[List[RowBox[List["2", " ", RowBox[List["(", RowBox[List["b", "+", RowBox[List["a", " ", "z"]]]], ")"]]]], "+", RowBox[List["2", " ", RowBox[List["(", RowBox[List["b", "+", RowBox[List["a", " ", "z"]]]], ")"]], " ", SqrtBox[FractionBox[RowBox[List[RowBox[List["-", "1"]], "+", SuperscriptBox["b", "2"], "+", RowBox[List["2", " ", "a", " ", "b", " ", "z"]], "+", RowBox[List[SuperscriptBox["a", "2"], " ", SuperscriptBox["z", "2"]]]]], SuperscriptBox[RowBox[List["(", RowBox[List["b", "+", RowBox[List["a", " ", "z"]]]], ")"]], "2"]]]]]]], "]"]]]]]]]]]]










MathML Form







<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mo> &#8747; </mo> <mrow> <mrow> <msup> <mi> sec </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mrow> <mi> b </mi> <mo> + </mo> <mrow> <mi> a </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> &#8518; </mo> <mi> z </mi> </mrow> </mrow> </mrow> <mo> &#10869; </mo> <mrow> <mrow> <mi> z </mi> <mo> &#8290; </mo> <mrow> <msup> <mi> sec </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mrow> <mi> b </mi> <mo> + </mo> <mrow> <mi> a </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mfrac> <mi> b </mi> <mi> a </mi> </mfrac> <mo> &#8290; </mo> <mrow> <msup> <mi> sin </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mfrac> <mn> 1 </mn> <mrow> <mrow> <mo> - </mo> <mi> b </mi> </mrow> <mo> - </mo> <mrow> <mi> a </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> </mrow> </mfrac> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <mfrac> <mn> 1 </mn> <mi> a </mi> </mfrac> <mo> &#8290; </mo> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <msqrt> <mfrac> <mrow> <msup> <mi> b </mi> <mn> 2 </mn> </msup> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> a </mi> <mo> &#8290; </mo> <mi> z </mi> <mo> &#8290; </mo> <mi> b </mi> </mrow> <mo> + </mo> <mrow> <msup> <mi> a </mi> <mn> 2 </mn> </msup> <mo> &#8290; </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mi> b </mi> <mo> + </mo> <mrow> <mi> a </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> </mfrac> </msqrt> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> b </mi> <mo> + </mo> <mrow> <mi> a </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> b </mi> <mo> + </mo> <mrow> <mi> a </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <int /> <bvar> <ci> z </ci> </bvar> <apply> <arcsec /> <apply> <plus /> <ci> b </ci> <apply> <times /> <ci> a </ci> <ci> z </ci> </apply> </apply> </apply> </apply> <apply> <plus /> <apply> <times /> <ci> z </ci> <apply> <arcsec /> <apply> <plus /> <ci> b </ci> <apply> <times /> <ci> a </ci> <ci> z </ci> </apply> </apply> </apply> </apply> <apply> <times /> <apply> <times /> <ci> b </ci> <apply> <power /> <ci> a </ci> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <arcsin /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> b </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <ci> a </ci> <ci> z </ci> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <ci> a </ci> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <ln /> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <apply> <times /> <apply> <plus /> <apply> <power /> <ci> b </ci> <cn type='integer'> 2 </cn> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <ci> a </ci> <ci> z </ci> <ci> b </ci> </apply> <apply> <times /> <apply> <power /> <ci> a </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> <apply> <power /> <apply> <power /> <apply> <plus /> <ci> b </ci> <apply> <times /> <ci> a </ci> <ci> z </ci> </apply> </apply> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <plus /> <ci> b </ci> <apply> <times /> <ci> a </ci> <ci> z </ci> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <plus /> <ci> b </ci> <apply> <times /> <ci> a </ci> <ci> z </ci> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["\[Integral]", RowBox[List[RowBox[List["ArcSec", "[", RowBox[List[RowBox[List["a_", " ", "z_"]], "+", "b_"]], "]"]], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List["z", " ", RowBox[List["ArcSec", "[", RowBox[List["b", "+", RowBox[List["a", " ", "z"]]]], "]"]]]], "+", FractionBox[RowBox[List["b", " ", RowBox[List["ArcSin", "[", FractionBox["1", RowBox[List[RowBox[List["-", "b"]], "-", RowBox[List["a", " ", "z"]]]]], "]"]]]], "a"], "-", FractionBox[RowBox[List["Log", "[", RowBox[List[RowBox[List["2", " ", RowBox[List["(", RowBox[List["b", "+", RowBox[List["a", " ", "z"]]]], ")"]]]], "+", RowBox[List["2", " ", RowBox[List["(", RowBox[List["b", "+", RowBox[List["a", " ", "z"]]]], ")"]], " ", SqrtBox[FractionBox[RowBox[List[RowBox[List["-", "1"]], "+", SuperscriptBox["b", "2"], "+", RowBox[List["2", " ", "a", " ", "b", " ", "z"]], "+", RowBox[List[SuperscriptBox["a", "2"], " ", SuperscriptBox["z", "2"]]]]], SuperscriptBox[RowBox[List["(", RowBox[List["b", "+", RowBox[List["a", " ", "z"]]]], ")"]], "2"]]]]]]], "]"]], "a"]]]]]]]










Date Added to functions.wolfram.com (modification date)





2001-10-29





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