ArcSec[1/Sqrt[a z^c + 1]] == (-((Sqrt[a] z^(c/2))/Sqrt[(-a) z^c])) ArcSinh[Sqrt[a] z^(c/2)]

Cell[BoxData[RowBox[List[RowBox[List["ArcSec", "[", FractionBox["1", SqrtBox[RowBox[List[RowBox[List["a", " ", SuperscriptBox["z", "c"]]], "+", "1"]]]], "]"]], "\[Equal]", RowBox[List[RowBox[List["-", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SqrtBox["a"], " ", SuperscriptBox["z", RowBox[List["c", "/", "2"]]]]], ")"]], " "]], SqrtBox[RowBox[List[RowBox[List["-", "a"]], " ", SuperscriptBox["z", "c"]]]]]]], RowBox[List["ArcSinh", "[", RowBox[List[SqrtBox["a"], " ", SuperscriptBox["z", RowBox[List["c", "/", "2"]]]]], "]"]]]]]]]]

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

Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["ArcSec", "[", FractionBox["1", SqrtBox[RowBox[List[RowBox[List["a_", " ", SuperscriptBox["z_", "c_"]]], "+", "1"]]]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List["-", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SqrtBox["a"], " ", SuperscriptBox["z", RowBox[List["c", "/", "2"]]]]], ")"]], " ", RowBox[List["ArcSinh", "[", RowBox[List[SqrtBox["a"], " ", SuperscriptBox["z", RowBox[List["c", "/", "2"]]]]], "]"]]]], SqrtBox[RowBox[List[RowBox[List["-", "a"]], " ", SuperscriptBox["z", "c"]]]]]]]]]]]

2001-10-29