html, body, form { margin: 0; padding: 0; width: 100%; } #calculate { position: relative; width: 177px; height: 110px; background: transparent url(/images/alphabox/embed_functions_inside.gif) no-repeat scroll 0 0; } #i { position: relative; left: 18px; top: 44px; width: 133px; border: 0 none; outline: 0; font-size: 11px; } #eq { width: 9px; height: 10px; background: transparent; position: absolute; top: 47px; right: 18px; cursor: pointer; }

 ArcSech

 http://functions.wolfram.com/01.30.21.0014.01

 Input Form

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

 Standard Form

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

 MathML Form

 sech - 1 ( a z r ) z z 1 8 ( 8 sech - 1 ( a z r ) log ( z ) - 1 r ( a z r - 1 ) ( 1 - a z r a z r + 1 ( 4 a 2 z 2 r - 1 tan - 1 ( a 2 z 2 r - 1 ) ( 2 r log ( z ) - log ( a 2 z 2 r ) ) + 1 - a 2 z 2 r ( log 2 ( a 2 z 2 r ) - 4 log ( 1 2 ( 1 - a 2 z 2 r + 1 ) ) log ( a 2 z 2 r ) + 2 log 2 ( 1 2 ( 1 - a 2 z 2 r + 1 ) ) ) - 4 1 - a 2 z 2 r Li PolyLog 2 ( 1 2 - 1 2 1 - a 2 z 2 r ) ) ) ) z a z r z -1 1 8 8 a z r z -1 1 r a z r -1 -1 1 -1 a z r a z r 1 -1 1 2 4 a 2 z 2 r -1 1 2 a 2 z 2 r -1 1 2 2 r z -1 a 2 z 2 r 1 -1 a 2 z 2 r 1 2 a 2 z 2 r 2 -1 4 1 2 1 -1 a 2 z 2 r 1 2 1 a 2 z 2 r 2 1 2 1 -1 a 2 z 2 r 1 2 1 2 -1 4 1 -1 a 2 z 2 r 1 2 PolyLog 2 1 2 -1 1 2 1 -1 a 2 z 2 r 1 2 [/itex]

 Rule Form

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

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

 2001-10-29