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ArcSech






Mathematica Notation

Traditional Notation









Elementary Functions > ArcSech[z] > Integration > Indefinite integration > Involving one direct function and elementary functions > Arguments involving exponential functions





http://functions.wolfram.com/01.30.21.0016.01









  


  










Input Form





Integrate[ArcSech[a^z], z] == z ArcSech[a^z] + (Sqrt[(1 - a^z)/(1 + a^z)] (1 + a^z) (ArcSech[-a^z] (ArcSech[-a^z] + 2 (z Log[a] + Log[1 + E^(-2 ArcSech[-a^z])])) - PolyLog[2, -E^(-2 ArcSech[-a^z])]))/(2 (-1 + a^z) Sqrt[(1 + a^z)/(1 - a^z)] Log[a])










Standard Form





Cell[BoxData[RowBox[List[RowBox[List["\[Integral]", RowBox[List[RowBox[List["ArcSech", "[", SuperscriptBox["a", "z"], "]"]], RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", RowBox[List[RowBox[List["z", " ", RowBox[List["ArcSech", "[", SuperscriptBox["a", "z"], "]"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List[SqrtBox[FractionBox[RowBox[List["1", "-", SuperscriptBox["a", "z"]]], RowBox[List["1", "+", SuperscriptBox["a", "z"]]]]], " ", RowBox[List["(", RowBox[List["1", "+", SuperscriptBox["a", "z"]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["ArcSech", "[", RowBox[List["-", SuperscriptBox["a", "z"]]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["ArcSech", "[", RowBox[List["-", SuperscriptBox["a", "z"]]], "]"]], "+", RowBox[List["2", " ", RowBox[List["(", RowBox[List[RowBox[List["z", " ", RowBox[List["Log", "[", "a", "]"]]]], "+", RowBox[List["Log", "[", RowBox[List["1", "+", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "2"]], " ", RowBox[List["ArcSech", "[", RowBox[List["-", SuperscriptBox["a", "z"]]], "]"]]]]]]], "]"]]]], ")"]]]]]], ")"]]]], "-", RowBox[List["PolyLog", "[", RowBox[List["2", ",", RowBox[List["-", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "2"]], " ", RowBox[List["ArcSech", "[", RowBox[List["-", SuperscriptBox["a", "z"]]], "]"]]]]]]]]], "]"]]]], ")"]]]], ")"]], "/", RowBox[List["(", RowBox[List["2", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", SuperscriptBox["a", "z"]]], ")"]], " ", SqrtBox[FractionBox[RowBox[List["1", "+", SuperscriptBox["a", "z"]]], RowBox[List["1", "-", SuperscriptBox["a", "z"]]]]], " ", RowBox[List["Log", "[", "a", "]"]]]], ")"]]]]]]]]]]










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> sech </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <msup> <mi> a </mi> <mi> z </mi> </msup> <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> sech </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <msup> <mi> a </mi> <mi> z </mi> </msup> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <msqrt> <mfrac> <mrow> <mn> 1 </mn> <mo> - </mo> <msup> <mi> a </mi> <mi> z </mi> </msup> </mrow> <mrow> <msup> <mi> a </mi> <mi> z </mi> </msup> <mo> + </mo> <mn> 1 </mn> </mrow> </mfrac> </msqrt> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <msup> <mi> a </mi> <mi> z </mi> </msup> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; 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Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["\[Integral]", RowBox[List[RowBox[List["ArcSech", "[", SuperscriptBox["a_", "z_"], "]"]], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List["z", " ", RowBox[List["ArcSech", "[", SuperscriptBox["a", "z"], "]"]]]], "+", FractionBox[RowBox[List[SqrtBox[FractionBox[RowBox[List["1", "-", SuperscriptBox["a", "z"]]], RowBox[List["1", "+", SuperscriptBox["a", "z"]]]]], " ", RowBox[List["(", RowBox[List["1", "+", SuperscriptBox["a", "z"]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["ArcSech", "[", RowBox[List["-", SuperscriptBox["a", "z"]]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["ArcSech", "[", RowBox[List["-", SuperscriptBox["a", "z"]]], "]"]], "+", RowBox[List["2", " ", RowBox[List["(", RowBox[List[RowBox[List["z", " ", RowBox[List["Log", "[", "a", "]"]]]], "+", RowBox[List["Log", "[", RowBox[List["1", "+", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "2"]], " ", RowBox[List["ArcSech", "[", RowBox[List["-", SuperscriptBox["a", "z"]]], "]"]]]]]]], "]"]]]], ")"]]]]]], ")"]]]], "-", RowBox[List["PolyLog", "[", RowBox[List["2", ",", RowBox[List["-", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "2"]], " ", RowBox[List["ArcSech", "[", RowBox[List["-", SuperscriptBox["a", "z"]]], "]"]]]]]]]]], "]"]]]], ")"]]]], RowBox[List["2", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", SuperscriptBox["a", "z"]]], ")"]], " ", SqrtBox[FractionBox[RowBox[List["1", "+", SuperscriptBox["a", "z"]]], RowBox[List["1", "-", SuperscriptBox["a", "z"]]]]], " ", RowBox[List["Log", "[", "a", "]"]]]]]]]]]]]










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





2001-10-29





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