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ArcSech






Mathematica Notation

Traditional Notation









Elementary Functions > ArcSech[z] > Integration > Indefinite integration > Involving one direct function and elementary functions > Involving logarithm > Involving log





http://functions.wolfram.com/01.30.21.0027.01









  


  










Input Form





Integrate[Log[b z] ArcSech[a z], z] == z ArcSech[a z] (-1 + Log[b z]) + (1/(2 a^(3/2) (-1 + a z))) (Sqrt[-a] Sqrt[(1 - a z)/(1 + a z)] Sqrt[1 - a^2 z^2] ArcSin[a z]^2 - 2 I Sqrt[a] (-1 + a z) (1 + Log[z] - Log[b z]) Log[-2 I a z + 2 Sqrt[(1 - a z)/(1 + a z)] (1 + a z)] + 2 Sqrt[-a] Sqrt[(1 - a z)/(1 + a z)] Sqrt[1 - a^2 z^2] Log[z] Log[Sqrt[-a^2] z + Sqrt[1 - a^2 z^2]] + 2 Sqrt[a] Sqrt[(1 - a z)/(1 + a z)] Sqrt[1 - a^2 z^2] ArcSin[a z] Log[2 Sqrt[a] z (a^(3/2) z + Sqrt[-a] Sqrt[1 - a^2 z^2])]) + (1/(a^(3/2) (-2 + 2 a z))) (Sqrt[-a] Sqrt[(1 - a z)/(1 + a z)] Sqrt[1 - a^2 z^2] PolyLog[2, 1 - 2 a^2 z^2 - 2 Sqrt[-a^2] z Sqrt[1 - a^2 z^2]])










Standard Form





Cell[BoxData[RowBox[List[RowBox[List["\[Integral]", RowBox[List[RowBox[List["Log", "[", RowBox[List["b", " ", "z"]], "]"]], RowBox[List["ArcSech", "[", RowBox[List["a", " ", "z"]], "]"]], RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", RowBox[List[RowBox[List["z", " ", RowBox[List["ArcSech", "[", RowBox[List["a", " ", "z"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", RowBox[List["Log", "[", RowBox[List["b", " ", "z"]], "]"]]]], ")"]]]], "+", RowBox[List[FractionBox["1", RowBox[List["2", " ", SuperscriptBox["a", RowBox[List["3", "/", "2"]]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", RowBox[List["a", " ", "z"]]]], ")"]]]]], RowBox[List["(", RowBox[List[RowBox[List[SqrtBox[RowBox[List["-", "a"]]], " ", SqrtBox[FractionBox[RowBox[List["1", "-", RowBox[List["a", " ", "z"]]]], RowBox[List["1", "+", RowBox[List["a", " ", "z"]]]]]], " ", SqrtBox[RowBox[List["1", "-", RowBox[List[SuperscriptBox["a", "2"], " ", SuperscriptBox["z", "2"]]]]]], " ", SuperscriptBox[RowBox[List["ArcSin", "[", RowBox[List["a", " ", "z"]], "]"]], "2"]]], "-", RowBox[List["2", " ", "\[ImaginaryI]", " ", SqrtBox["a"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", RowBox[List["a", " ", "z"]]]], ")"]], " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List["Log", "[", "z", "]"]], "-", RowBox[List["Log", "[", RowBox[List["b", " ", "z"]], "]"]]]], ")"]], " ", RowBox[List["Log", "[", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "\[ImaginaryI]", " ", "a", " ", "z"]], "+", RowBox[List["2", " ", SqrtBox[FractionBox[RowBox[List["1", "-", RowBox[List["a", " ", "z"]]]], RowBox[List["1", "+", RowBox[List["a", " ", "z"]]]]]], " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List["a", " ", "z"]]]], ")"]]]]]], "]"]]]], "+", RowBox[List["2", " ", SqrtBox[RowBox[List["-", "a"]]], " ", SqrtBox[FractionBox[RowBox[List["1", "-", RowBox[List["a", " ", "z"]]]], RowBox[List["1", "+", RowBox[List["a", " ", "z"]]]]]], " ", SqrtBox[RowBox[List["1", "-", RowBox[List[SuperscriptBox["a", "2"], " ", SuperscriptBox["z", "2"]]]]]], " ", RowBox[List["Log", "[", "z", "]"]], " ", RowBox[List["Log", "[", RowBox[List[RowBox[List[SqrtBox[RowBox[List["-", SuperscriptBox["a", "2"]]]], " ", "z"]], "+", SqrtBox[RowBox[List["1", "-", RowBox[List[SuperscriptBox["a", "2"], " ", SuperscriptBox["z", "2"]]]]]]]], "]"]]]], "+", RowBox[List["2", " ", SqrtBox["a"], " ", SqrtBox[FractionBox[RowBox[List["1", "-", RowBox[List["a", " ", "z"]]]], RowBox[List["1", "+", RowBox[List["a", " ", "z"]]]]]], " ", SqrtBox[RowBox[List["1", "-", RowBox[List[SuperscriptBox["a", "2"], " ", SuperscriptBox["z", "2"]]]]]], " ", RowBox[List["ArcSin", "[", RowBox[List["a", " ", "z"]], "]"]], " ", RowBox[List["Log", "[", RowBox[List["2", " ", SqrtBox["a"], " ", "z", " ", RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox["a", RowBox[List["3", "/", "2"]]], " ", "z"]], "+", RowBox[List[SqrtBox[RowBox[List["-", "a"]]], " ", SqrtBox[RowBox[List["1", "-", RowBox[List[SuperscriptBox["a", "2"], " ", SuperscriptBox["z", "2"]]]]]]]]]], ")"]]]], "]"]]]]]], ")"]]]], "+", RowBox[List[FractionBox["1", RowBox[List[SuperscriptBox["a", RowBox[List["3", "/", "2"]]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "2"]], "+", RowBox[List["2", " ", "a", " ", "z"]]]], ")"]]]]], RowBox[List["(", RowBox[List[SqrtBox[RowBox[List["-", "a"]]], " ", SqrtBox[FractionBox[RowBox[List["1", "-", RowBox[List["a", " ", "z"]]]], RowBox[List["1", "+", RowBox[List["a", " ", "z"]]]]]], " ", SqrtBox[RowBox[List["1", "-", RowBox[List[SuperscriptBox["a", "2"], " ", SuperscriptBox["z", "2"]]]]]], " ", RowBox[List["PolyLog", "[", RowBox[List["2", ",", RowBox[List["1", "-", RowBox[List["2", " ", SuperscriptBox["a", "2"], " ", SuperscriptBox["z", "2"]]], "-", RowBox[List["2", " ", SqrtBox[RowBox[List["-", SuperscriptBox["a", "2"]]]], " ", "z", " ", SqrtBox[RowBox[List["1", "-", RowBox[List[SuperscriptBox["a", "2"], " ", SuperscriptBox["z", "2"]]]]]]]]]]]], "]"]]]], ")"]]]]]]]]]]










MathML Form







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<power /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> a </ci> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <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> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <arcsin /> <apply> <times /> <ci> a </ci> <ci> z </ci> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <imaginaryi /> <apply> <power /> <ci> a </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <plus /> <apply> <times /> <ci> a </ci> <ci> z </ci> </apply> <cn type='integer'> -1 </cn> </apply> <apply> <plus /> <apply> <ln /> <ci> z </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ln /> <apply> <times /> <ci> b </ci> <ci> z </ci> </apply> </apply> </apply> <cn type='integer'> 1 </cn> </apply> <apply> <ln /> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <apply> <times /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <ci> a </ci> <ci> z </ci> </apply> </apply> </apply> <apply> <power /> <apply> <plus /> <apply> <times /> <ci> a </ci> <ci> z </ci> </apply> <cn type='integer'> 1 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <plus /> <apply> <times /> <ci> a </ci> <ci> z </ci> </apply> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <imaginaryi /> <ci> a </ci> <ci> z </ci> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> a </ci> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <times /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <ci> a </ci> <ci> z </ci> </apply> </apply> </apply> <apply> <power /> <apply> <plus /> <apply> <times /> <ci> a </ci> <ci> z </ci> </apply> <cn type='integer'> 1 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <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> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <ln /> <ci> z </ci> </apply> <apply> <ln /> <apply> <plus /> <apply> <times /> <apply> <power /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <ci> a </ci> <cn type='integer'> 2 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <ci> z </ci> </apply> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <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> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <times /> <apply> <times /> <apply> <power /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> a </ci> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <times /> <apply> <power /> <ci> a </ci> <cn type='rational'> 3 <sep /> 2 </cn> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> a </ci> <ci> z </ci> </apply> <cn type='integer'> -2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <apply> <times /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <ci> a </ci> <ci> z </ci> </apply> </apply> </apply> <apply> <power /> <apply> <plus /> <apply> <times /> <ci> a </ci> <ci> z </ci> </apply> <cn type='integer'> 1 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <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> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <ci> PolyLog </ci> <cn type='integer'> 2 </cn> <apply> <plus /> <apply> <times /> <cn type='integer'> -2 </cn> <apply> <power /> <ci> a </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <ci> a </ci> <cn type='integer'> 2 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <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> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <ci> z </ci> </apply> </apply> <cn type='integer'> 1 </cn> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





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





2001-10-29





© 1998- Wolfram Research, Inc.