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; }

 ArcSec

 http://functions.wolfram.com/01.18.21.0007.01

 Input Form

 Integrate[ArcSec[a z + b]/z, z] == -4 I ArcSin[Sqrt[1 - 1/b]/Sqrt[2]] ArcTan[((1 + b) Tan[(1/2) ArcSec[b + a z]])/Sqrt[1 - b^2]] + (ArcSec[b + a z] - 2 ArcSin[Sqrt[1 - 1/b]/Sqrt[2]]) Log[1 + ((-1 + Sqrt[1 - b^2]) E^(I ArcSec[b + a z]))/b] + (ArcSec[b + a z] + 2 ArcSin[Sqrt[1 - 1/b]/Sqrt[2]]) Log[(b - (1 + Sqrt[1 - b^2]) E^(I ArcSec[b + a z]))/b] - ArcSec[b + a z] Log[1 + E^(2 I ArcSec[b + a z])] - I (PolyLog[2, -(((-1 + Sqrt[1 - b^2]) E^(I ArcSec[b + a z]))/b)] + PolyLog[2, ((1 + Sqrt[1 - b^2]) E^(I ArcSec[b + a z]))/b]) + (1/2) I PolyLog[2, -E^(2 I ArcSec[b + a z])]

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List["\[Integral]", RowBox[List[FractionBox[RowBox[List["ArcSec", "[", RowBox[List[RowBox[List["a", " ", "z"]], "+", "b"]], "]"]], "z"], RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", RowBox[List[RowBox[List[RowBox[List["-", "4"]], " ", "\[ImaginaryI]", " ", RowBox[List["ArcSin", "[", FractionBox[SqrtBox[RowBox[List["1", "-", FractionBox["1", "b"]]]], SqrtBox["2"]], "]"]], " ", RowBox[List["ArcTan", "[", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["1", "+", "b"]], ")"]], " ", RowBox[List["Tan", "[", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["ArcSec", "[", RowBox[List["b", "+", RowBox[List["a", " ", "z"]]]], "]"]]]], "]"]]]], SqrtBox[RowBox[List["1", "-", SuperscriptBox["b", "2"]]]]], "]"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["ArcSec", "[", RowBox[List["b", "+", RowBox[List["a", " ", "z"]]]], "]"]], "-", RowBox[List["2", " ", RowBox[List["ArcSin", "[", FractionBox[SqrtBox[RowBox[List["1", "-", FractionBox["1", "b"]]]], SqrtBox["2"]], "]"]]]]]], ")"]], " ", RowBox[List["Log", "[", RowBox[List["1", "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", SqrtBox[RowBox[List["1", "-", SuperscriptBox["b", "2"]]]]]], ")"]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["\[ImaginaryI]", " ", RowBox[List["ArcSec", "[", RowBox[List["b", "+", RowBox[List["a", " ", "z"]]]], "]"]]]]]]], "b"]]], "]"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["ArcSec", "[", RowBox[List["b", "+", RowBox[List["a", " ", "z"]]]], "]"]], "+", RowBox[List["2", " ", RowBox[List["ArcSin", "[", FractionBox[SqrtBox[RowBox[List["1", "-", FractionBox["1", "b"]]]], SqrtBox["2"]], "]"]]]]]], ")"]], " ", RowBox[List["Log", "[", FractionBox[RowBox[List["b", "-", RowBox[List[RowBox[List["(", RowBox[List["1", "+", SqrtBox[RowBox[List["1", "-", SuperscriptBox["b", "2"]]]]]], ")"]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["\[ImaginaryI]", " ", RowBox[List["ArcSec", "[", RowBox[List["b", "+", RowBox[List["a", " ", "z"]]]], "]"]]]]]]]]], "b"], "]"]]]], "-", RowBox[List[RowBox[List["ArcSec", "[", RowBox[List["b", "+", RowBox[List["a", " ", "z"]]]], "]"]], " ", RowBox[List["Log", "[", RowBox[List["1", "+", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "\[ImaginaryI]", " ", RowBox[List["ArcSec", "[", RowBox[List["b", "+", RowBox[List["a", " ", "z"]]]], "]"]]]]]]], "]"]]]], "-", RowBox[List["\[ImaginaryI]", " ", RowBox[List["(", RowBox[List[RowBox[List["PolyLog", "[", RowBox[List["2", ",", RowBox[List["-", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", SqrtBox[RowBox[List["1", "-", SuperscriptBox["b", "2"]]]]]], ")"]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["\[ImaginaryI]", " ", RowBox[List["ArcSec", "[", RowBox[List["b", "+", RowBox[List["a", " ", "z"]]]], "]"]]]]]]], "b"]]]]], "]"]], "+", RowBox[List["PolyLog", "[", RowBox[List["2", ",", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["1", "+", SqrtBox[RowBox[List["1", "-", SuperscriptBox["b", "2"]]]]]], ")"]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["\[ImaginaryI]", " ", RowBox[List["ArcSec", "[", RowBox[List["b", "+", RowBox[List["a", " ", "z"]]]], "]"]]]]]]], "b"]]], "]"]]]], ")"]]]], "+", RowBox[List[FractionBox["1", "2"], " ", "\[ImaginaryI]", " ", RowBox[List["PolyLog", "[", RowBox[List["2", ",", RowBox[List["-", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "\[ImaginaryI]", " ", RowBox[List["ArcSec", "[", RowBox[List["b", "+", RowBox[List["a", " ", "z"]]]], "]"]]]]]]]]], "]"]]]]]]]]]]

 MathML Form

 sec - 1 ( a z + b ) z z - 4 sin - 1 ( 1 2 1 - 1 b ) tan - 1 ( b + 1 1 - b 2 tan ( 1 2 sec - 1 ( b + a z ) ) ) + ( sec - 1 ( b + a z ) - 2 sin - 1 ( 1 2 1 - 1 b ) ) log ( sec - 1 ( b + a z ) ( 1 - b 2 - 1 ) b + 1 ) + ( sec - 1 ( b + a z ) + 2 sin - 1 ( 1 2 1 - 1 b ) ) log ( b - ( 1 - b 2 + 1 ) sec - 1 ( b + a z ) b ) - sec - 1 ( b + a z ) log ( 1 + 2 sec - 1 ( b + a z ) ) - ( Li PolyLog 2 ( ( 1 - b 2 + 1 ) sec - 1 ( b + a z ) b ) + Li PolyLog 2 ( - ( 1 - b 2 - 1 ) sec - 1 ( b + a z ) b ) ) + 1 2 Li PolyLog 2 ( - 2 sec - 1 ( b + a z ) ) z a z b z -1 -4 1 2 1 2 -1 1 -1 1 b -1 1 2 b 1 1 -1 b 2 1 2 -1 1 2 b a z b a z -1 2 1 2 1 2 -1 1 -1 1 b -1 1 2 b a z 1 -1 b 2 1 2 -1 b -1 1 b a z 2 1 2 1 2 -1 1 -1 1 b -1 1 2 b -1 1 -1 b 2 1 2 1 b a z b -1 -1 b a z 1 2 b a z -1 PolyLog 2 1 -1 b 2 1 2 1 b a z b -1 PolyLog 2 -1 1 -1 b 2 1 2 -1 b a z b -1 1 2 PolyLog 2 -1 2 b a z [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["\[Integral]", RowBox[List[FractionBox[RowBox[List["ArcSec", "[", RowBox[List[RowBox[List["a_", " ", "z_"]], "+", "b_"]], "]"]], "z_"], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List["-", "4"]], " ", "\[ImaginaryI]", " ", RowBox[List["ArcSin", "[", FractionBox[SqrtBox[RowBox[List["1", "-", FractionBox["1", "b"]]]], SqrtBox["2"]], "]"]], " ", RowBox[List["ArcTan", "[", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["1", "+", "b"]], ")"]], " ", RowBox[List["Tan", "[", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["ArcSec", "[", RowBox[List["b", "+", RowBox[List["a", " ", "z"]]]], "]"]]]], "]"]]]], SqrtBox[RowBox[List["1", "-", SuperscriptBox["b", "2"]]]]], "]"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["ArcSec", "[", RowBox[List["b", "+", RowBox[List["a", " ", "z"]]]], "]"]], "-", RowBox[List["2", " ", RowBox[List["ArcSin", "[", FractionBox[SqrtBox[RowBox[List["1", "-", FractionBox["1", "b"]]]], SqrtBox["2"]], "]"]]]]]], ")"]], " ", RowBox[List["Log", "[", RowBox[List["1", "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", SqrtBox[RowBox[List["1", "-", SuperscriptBox["b", "2"]]]]]], ")"]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["\[ImaginaryI]", " ", RowBox[List["ArcSec", "[", RowBox[List["b", "+", RowBox[List["a", " ", "z"]]]], "]"]]]]]]], "b"]]], "]"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["ArcSec", "[", RowBox[List["b", "+", RowBox[List["a", " ", "z"]]]], "]"]], "+", RowBox[List["2", " ", RowBox[List["ArcSin", "[", FractionBox[SqrtBox[RowBox[List["1", "-", FractionBox["1", "b"]]]], SqrtBox["2"]], "]"]]]]]], ")"]], " ", RowBox[List["Log", "[", FractionBox[RowBox[List["b", "-", RowBox[List[RowBox[List["(", RowBox[List["1", "+", SqrtBox[RowBox[List["1", "-", SuperscriptBox["b", "2"]]]]]], ")"]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["\[ImaginaryI]", " ", RowBox[List["ArcSec", "[", RowBox[List["b", "+", RowBox[List["a", " ", "z"]]]], "]"]]]]]]]]], "b"], "]"]]]], "-", RowBox[List[RowBox[List["ArcSec", "[", RowBox[List["b", "+", RowBox[List["a", " ", "z"]]]], "]"]], " ", RowBox[List["Log", "[", RowBox[List["1", "+", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "\[ImaginaryI]", " ", RowBox[List["ArcSec", "[", RowBox[List["b", "+", RowBox[List["a", " ", "z"]]]], "]"]]]]]]], "]"]]]], "-", RowBox[List["\[ImaginaryI]", " ", RowBox[List["(", RowBox[List[RowBox[List["PolyLog", "[", RowBox[List["2", ",", RowBox[List["-", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", SqrtBox[RowBox[List["1", "-", SuperscriptBox["b", "2"]]]]]], ")"]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["\[ImaginaryI]", " ", RowBox[List["ArcSec", "[", RowBox[List["b", "+", RowBox[List["a", " ", "z"]]]], "]"]]]]]]], "b"]]]]], "]"]], "+", RowBox[List["PolyLog", "[", RowBox[List["2", ",", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["1", "+", SqrtBox[RowBox[List["1", "-", SuperscriptBox["b", "2"]]]]]], ")"]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["\[ImaginaryI]", " ", RowBox[List["ArcSec", "[", RowBox[List["b", "+", RowBox[List["a", " ", "z"]]]], "]"]]]]]]], "b"]]], "]"]]]], ")"]]]], "+", RowBox[List[FractionBox["1", "2"], " ", "\[ImaginaryI]", " ", RowBox[List["PolyLog", "[", RowBox[List["2", ",", RowBox[List["-", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "\[ImaginaryI]", " ", RowBox[List["ArcSec", "[", RowBox[List["b", "+", RowBox[List["a", " ", "z"]]]], "]"]]]]]]]]], "]"]]]]]]]]]]

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

 2001-10-29