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ArcCsch






Mathematica Notation

Traditional Notation









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





http://functions.wolfram.com/01.29.21.0016.01









  


  










Input Form





Integrate[ArcCsch[a^z], z] == z ArcCsch[a^z] + (1/(8 Sqrt[1 + a^(2 z)] Log[a])) (a^z Sqrt[1 + a^(-2 z)] (Log[-a^(2 z)]^2 + ArcTanh[Sqrt[1 + a^(2 z)]] (-8 z Log[a] + 4 Log[-a^(2 z)]) - 4 Log[-a^(2 z)] Log[(1/2) (1 + Sqrt[1 + a^(2 z)])] + 2 Log[(1/2) (1 + Sqrt[1 + a^(2 z)])]^2 - 4 PolyLog[2, 1/2 - (1/2) Sqrt[1 + a^(2 z)]]))










Standard Form





Cell[BoxData[RowBox[List[RowBox[List["\[Integral]", RowBox[List[RowBox[List["ArcCsch", "[", SuperscriptBox["a", "z"], "]"]], RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", RowBox[List[RowBox[List["z", " ", RowBox[List["ArcCsch", "[", SuperscriptBox["a", "z"], "]"]]]], "+", RowBox[List[FractionBox["1", RowBox[List["8", " ", SqrtBox[RowBox[List["1", "+", SuperscriptBox["a", RowBox[List["2", " ", "z"]]]]]], " ", RowBox[List["Log", "[", "a", "]"]]]]], RowBox[List["(", RowBox[List[SuperscriptBox["a", "z"], " ", SqrtBox[RowBox[List["1", "+", SuperscriptBox["a", RowBox[List[RowBox[List["-", "2"]], " ", "z"]]]]]], " ", RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["Log", "[", RowBox[List["-", SuperscriptBox["a", RowBox[List["2", " ", "z"]]]]], "]"]], "2"], "+", RowBox[List[RowBox[List["ArcTanh", "[", SqrtBox[RowBox[List["1", "+", SuperscriptBox["a", RowBox[List["2", " ", "z"]]]]]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "8"]], " ", "z", " ", RowBox[List["Log", "[", "a", "]"]]]], "+", RowBox[List["4", " ", RowBox[List["Log", "[", RowBox[List["-", SuperscriptBox["a", RowBox[List["2", " ", "z"]]]]], "]"]]]]]], ")"]]]], "-", RowBox[List["4", " ", RowBox[List["Log", "[", RowBox[List["-", SuperscriptBox["a", RowBox[List["2", " ", "z"]]]]], "]"]], " ", RowBox[List["Log", "[", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["1", "+", SqrtBox[RowBox[List["1", "+", SuperscriptBox["a", RowBox[List["2", " ", "z"]]]]]]]], ")"]]]], "]"]]]], "+", RowBox[List["2", " ", SuperscriptBox[RowBox[List["Log", "[", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["1", "+", SqrtBox[RowBox[List["1", "+", SuperscriptBox["a", RowBox[List["2", " ", "z"]]]]]]]], ")"]]]], "]"]], "2"]]], "-", RowBox[List["4", " ", RowBox[List["PolyLog", "[", RowBox[List["2", ",", RowBox[List[FractionBox["1", "2"], "-", RowBox[List[FractionBox["1", "2"], " ", SqrtBox[RowBox[List["1", "+", SuperscriptBox["a", RowBox[List["2", " ", "z"]]]]]]]]]]]], "]"]]]]]], ")"]]]], ")"]]]]]]]]]]










MathML Form







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Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["\[Integral]", RowBox[List[RowBox[List["ArcCsch", "[", SuperscriptBox["a_", "z_"], "]"]], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List["z", " ", RowBox[List["ArcCsch", "[", SuperscriptBox["a", "z"], "]"]]]], "+", FractionBox[RowBox[List[SuperscriptBox["a", "z"], " ", SqrtBox[RowBox[List["1", "+", SuperscriptBox["a", RowBox[List[RowBox[List["-", "2"]], " ", "z"]]]]]], " ", RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["Log", "[", RowBox[List["-", SuperscriptBox["a", RowBox[List["2", " ", "z"]]]]], "]"]], "2"], "+", RowBox[List[RowBox[List["ArcTanh", "[", SqrtBox[RowBox[List["1", "+", SuperscriptBox["a", RowBox[List["2", " ", "z"]]]]]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "8"]], " ", "z", " ", RowBox[List["Log", "[", "a", "]"]]]], "+", RowBox[List["4", " ", RowBox[List["Log", "[", RowBox[List["-", SuperscriptBox["a", RowBox[List["2", " ", "z"]]]]], "]"]]]]]], ")"]]]], "-", RowBox[List["4", " ", RowBox[List["Log", "[", RowBox[List["-", SuperscriptBox["a", RowBox[List["2", " ", "z"]]]]], "]"]], " ", RowBox[List["Log", "[", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["1", "+", SqrtBox[RowBox[List["1", "+", SuperscriptBox["a", RowBox[List["2", " ", "z"]]]]]]]], ")"]]]], "]"]]]], "+", RowBox[List["2", " ", SuperscriptBox[RowBox[List["Log", "[", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["1", "+", SqrtBox[RowBox[List["1", "+", SuperscriptBox["a", RowBox[List["2", " ", "z"]]]]]]]], ")"]]]], "]"]], "2"]]], "-", RowBox[List["4", " ", RowBox[List["PolyLog", "[", RowBox[List["2", ",", RowBox[List[FractionBox["1", "2"], "-", RowBox[List[FractionBox["1", "2"], " ", SqrtBox[RowBox[List["1", "+", SuperscriptBox["a", RowBox[List["2", " ", "z"]]]]]]]]]]]], "]"]]]]]], ")"]]]], RowBox[List["8", " ", SqrtBox[RowBox[List["1", "+", SuperscriptBox["a", RowBox[List["2", " ", "z"]]]]]], " ", RowBox[List["Log", "[", "a", "]"]]]]]]]]]]]










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





2001-10-29





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