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; }

 ArcCsch

 http://functions.wolfram.com/01.29.21.0020.01

 Input Form

 Integrate[ArcCsch[Sec[z]], z] == (1/4) (4 z ArcCsch[Sec[z]] - 2 Pi ArcTanh[(Sqrt[2] Cos[z])/Sqrt[3 + Cos[2 z]]] + (1/2) (Pi - 2 z) (-Log[(1/2) (1 + E^(2 I z) - 2 E^(I z) Sqrt[1 + Cos[z]^2])] + Log[(1/2) (1 + E^(2 I z) + 2 E^(I z) Sqrt[1 + Cos[z]^2])] - Log[(E^(I z) - Sqrt[1 + Cos[z]^2] - I Sin[z])/E^(I z)] + Log[(E^(I z) + Sqrt[1 + Cos[z]^2] - I Sin[z])/E^(I z)]) - ArcTan[Sin[z]/Sqrt[1 + Cos[z]^2]] (Log[(1/2) (1 + E^(2 I z) - 2 E^(I z) Sqrt[1 + Cos[z]^2])] + Log[(1/2) (1 + E^(2 I z) + 2 E^(I z) Sqrt[1 + Cos[z]^2])] + Log[(E^(I z) - Sqrt[1 + Cos[z]^2] - I Sin[z])/E^(I z)] + Log[(E^(I z) + Sqrt[1 + Cos[z]^2] - I Sin[z])/E^(I z)]) + I (-PolyLog[2, (-E^((-I) z)) (Sqrt[1 + Cos[z]^2] - I Sin[z])] - PolyLog[2, E^(I z) (Sqrt[1 + Cos[z]^2] - I Sin[z])] + PolyLog[2, (Sqrt[1 + Cos[z]^2] + I Sin[z])/E^(I z)] + PolyLog[2, (-E^(I z)) (Sqrt[1 + Cos[z]^2] + I Sin[z])]))

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List["\[Integral]", RowBox[List[RowBox[List["ArcCsch", "[", RowBox[List["Sec", "[", "z", "]"]], "]"]], RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", RowBox[List[FractionBox["1", "4"], " ", RowBox[List["(", RowBox[List[RowBox[List["4", " ", "z", " ", RowBox[List["ArcCsch", "[", RowBox[List["Sec", "[", "z", "]"]], "]"]]]], "-", RowBox[List["2", " ", "\[Pi]", " ", RowBox[List["ArcTanh", "[", FractionBox[RowBox[List[SqrtBox["2"], " ", RowBox[List["Cos", "[", "z", "]"]]]], SqrtBox[RowBox[List["3", "+", RowBox[List["Cos", "[", RowBox[List["2", " ", "z"]], "]"]]]]]], "]"]]]], "+", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["\[Pi]", "-", RowBox[List["2", " ", "z"]]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", RowBox[List["Log", "[", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["1", "+", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "\[ImaginaryI]", " ", "z"]]], "-", RowBox[List["2", " ", SuperscriptBox["\[ExponentialE]", RowBox[List["\[ImaginaryI]", " ", "z"]]], " ", SqrtBox[RowBox[List["1", "+", SuperscriptBox[RowBox[List["Cos", "[", "z", "]"]], "2"]]]]]]]], ")"]]]], "]"]]]], "+", RowBox[List["Log", "[", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["1", "+", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "\[ImaginaryI]", " ", "z"]]], "+", RowBox[List["2", " ", SuperscriptBox["\[ExponentialE]", RowBox[List["\[ImaginaryI]", " ", "z"]]], " ", SqrtBox[RowBox[List["1", "+", SuperscriptBox[RowBox[List["Cos", "[", "z", "]"]], "2"]]]]]]]], ")"]]]], "]"]], "-", RowBox[List["Log", "[", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "z"]]], " ", RowBox[List["(", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["\[ImaginaryI]", " ", "z"]]], "-", SqrtBox[RowBox[List["1", "+", SuperscriptBox[RowBox[List["Cos", "[", "z", "]"]], "2"]]]], "-", RowBox[List["\[ImaginaryI]", " ", RowBox[List["Sin", "[", "z", "]"]]]]]], ")"]]]], "]"]], "+", RowBox[List["Log", "[", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "z"]]], " ", RowBox[List["(", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["\[ImaginaryI]", " ", "z"]]], "+", SqrtBox[RowBox[List["1", "+", SuperscriptBox[RowBox[List["Cos", "[", "z", "]"]], "2"]]]], "-", RowBox[List["\[ImaginaryI]", " ", RowBox[List["Sin", "[", "z", "]"]]]]]], ")"]]]], "]"]]]], ")"]]]], "-", RowBox[List[RowBox[List["ArcTan", "[", FractionBox[RowBox[List["Sin", "[", "z", "]"]], SqrtBox[RowBox[List["1", "+", SuperscriptBox[RowBox[List["Cos", "[", "z", "]"]], "2"]]]]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["Log", "[", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["1", "+", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "\[ImaginaryI]", " ", "z"]]], "-", RowBox[List["2", " ", SuperscriptBox["\[ExponentialE]", RowBox[List["\[ImaginaryI]", " ", "z"]]], " ", SqrtBox[RowBox[List["1", "+", SuperscriptBox[RowBox[List["Cos", "[", "z", "]"]], "2"]]]]]]]], ")"]]]], "]"]], "+", RowBox[List["Log", "[", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["1", "+", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "\[ImaginaryI]", " ", "z"]]], "+", RowBox[List["2", " ", SuperscriptBox["\[ExponentialE]", RowBox[List["\[ImaginaryI]", " ", "z"]]], " ", SqrtBox[RowBox[List["1", "+", SuperscriptBox[RowBox[List["Cos", "[", "z", "]"]], "2"]]]]]]]], ")"]]]], "]"]], "+", RowBox[List["Log", "[", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "z"]]], " ", RowBox[List["(", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["\[ImaginaryI]", " ", "z"]]], "-", SqrtBox[RowBox[List["1", "+", SuperscriptBox[RowBox[List["Cos", "[", "z", "]"]], "2"]]]], "-", RowBox[List["\[ImaginaryI]", " ", RowBox[List["Sin", "[", "z", "]"]]]]]], ")"]]]], "]"]], "+", RowBox[List["Log", "[", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "z"]]], " ", RowBox[List["(", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["\[ImaginaryI]", " ", "z"]]], "+", SqrtBox[RowBox[List["1", "+", SuperscriptBox[RowBox[List["Cos", "[", "z", "]"]], "2"]]]], "-", RowBox[List["\[ImaginaryI]", " ", RowBox[List["Sin", "[", "z", "]"]]]]]], ")"]]]], "]"]]]], ")"]]]], "+", RowBox[List["\[ImaginaryI]", " ", RowBox[List["(", RowBox[List[RowBox[List["-", RowBox[List["PolyLog", "[", RowBox[List["2", ",", RowBox[List[RowBox[List["-", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "z"]]]]], " ", RowBox[List["(", RowBox[List[SqrtBox[RowBox[List["1", "+", SuperscriptBox[RowBox[List["Cos", "[", "z", "]"]], "2"]]]], "-", RowBox[List["\[ImaginaryI]", " ", RowBox[List["Sin", "[", "z", "]"]]]]]], ")"]]]]]], "]"]]]], "-", RowBox[List["PolyLog", "[", RowBox[List["2", ",", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["\[ImaginaryI]", " ", "z"]]], " ", RowBox[List["(", RowBox[List[SqrtBox[RowBox[List["1", "+", SuperscriptBox[RowBox[List["Cos", "[", "z", "]"]], "2"]]]], "-", RowBox[List["\[ImaginaryI]", " ", RowBox[List["Sin", "[", "z", "]"]]]]]], ")"]]]]]], "]"]], "+", RowBox[List["PolyLog", "[", RowBox[List["2", ",", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "z"]]], " ", RowBox[List["(", RowBox[List[SqrtBox[RowBox[List["1", "+", SuperscriptBox[RowBox[List["Cos", "[", "z", "]"]], "2"]]]], "+", RowBox[List["\[ImaginaryI]", " ", RowBox[List["Sin", "[", "z", "]"]]]]]], ")"]]]]]], "]"]], "+", RowBox[List["PolyLog", "[", RowBox[List["2", ",", RowBox[List[RowBox[List["-", SuperscriptBox["\[ExponentialE]", RowBox[List["\[ImaginaryI]", " ", "z"]]]]], " ", RowBox[List["(", RowBox[List[SqrtBox[RowBox[List["1", "+", SuperscriptBox[RowBox[List["Cos", "[", "z", "]"]], "2"]]]], "+", RowBox[List["\[ImaginaryI]", " ", RowBox[List["Sin", "[", "z", "]"]]]]]], ")"]]]]]], "]"]]]], ")"]]]]]], ")"]]]]]]]]

 MathML Form

 csch - 1 ( sec ( z ) ) z 1 4 ( 4 z csch - 1 ( sec ( z ) ) - 2 π tanh - 1 ( 2 cos ( z ) cos ( 2 z ) + 3 ) + 1 2 ( π - 2 z ) ( - log ( 1 2 ( - 2 z cos 2 ( z ) + 1 + 2 z + 1 ) ) + log ( 1 2 ( 2 z cos 2 ( z ) + 1 + 2 z + 1 ) ) - log ( - z ( - sin ( z ) + z - cos 2 ( z ) + 1 ) ) + log ( - z ( - sin ( z ) + z + cos 2 ( z ) + 1 ) ) ) - tan - 1 ( sin ( z ) cos 2 ( z ) + 1 ) ( log ( 1 2 ( - 2 z cos 2 ( z ) + 1 + 2 z + 1 ) ) + log ( 1 2 ( 2 z cos 2 ( z ) + 1 + 2 z + 1 ) ) + log ( - z ( - sin ( z ) + z - cos 2 ( z ) + 1 ) ) + log ( - z ( - sin ( z ) + z + cos 2 ( z ) + 1 ) ) ) + ( - Li PolyLog 2 ( - - z ( cos 2 ( z ) + 1 - sin ( z ) ) ) - Li PolyLog 2 ( z ( cos 2 ( z ) + 1 - sin ( z ) ) ) + Li PolyLog 2 ( - z ( sin ( z ) + cos 2 ( z ) + 1 ) ) + Li PolyLog 2 ( - z ( sin ( z ) + cos 2 ( z ) + 1 ) ) ) ) z z 1 4 4 z z -1 2 2 1 2 z 2 z 3 1 2 -1 1 2 -1 2 z -1 1 2 -2 z z 2 1 1 2 2 z 1 1 2 2 z z 2 1 1 2 2 z 1 -1 -1 z -1 z z -1 z 2 1 1 2 -1 z -1 z z z 2 1 1 2 -1 z z 2 1 1 2 -1 1 2 -2 z z 2 1 1 2 2 z 1 1 2 2 z z 2 1 1 2 2 z 1 -1 z -1 z z -1 z 2 1 1 2 -1 z -1 z z z 2 1 1 2 -1 PolyLog 2 -1 -1 z z 2 1 1 2 -1 z -1 PolyLog 2 z z 2 1 1 2 -1 z PolyLog 2 -1 z z z 2 1 1 2 PolyLog 2 -1 z z z 2 1 1 2 [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["\[Integral]", RowBox[List[RowBox[List["ArcCsch", "[", RowBox[List["Sec", "[", "z_", "]"]], "]"]], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox["1", "4"], " ", RowBox[List["(", RowBox[List[RowBox[List["4", " ", "z", " ", RowBox[List["ArcCsch", "[", RowBox[List["Sec", "[", "z", "]"]], "]"]]]], "-", RowBox[List["2", " ", "\[Pi]", " ", RowBox[List["ArcTanh", "[", FractionBox[RowBox[List[SqrtBox["2"], " ", RowBox[List["Cos", "[", "z", "]"]]]], SqrtBox[RowBox[List["3", "+", RowBox[List["Cos", "[", RowBox[List["2", " ", "z"]], "]"]]]]]], "]"]]]], "+", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["\[Pi]", "-", RowBox[List["2", " ", "z"]]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", RowBox[List["Log", "[", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["1", "+", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "\[ImaginaryI]", " ", "z"]]], "-", RowBox[List["2", " ", SuperscriptBox["\[ExponentialE]", RowBox[List["\[ImaginaryI]", " ", "z"]]], " ", SqrtBox[RowBox[List["1", "+", SuperscriptBox[RowBox[List["Cos", "[", "z", "]"]], "2"]]]]]]]], ")"]]]], "]"]]]], "+", RowBox[List["Log", "[", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["1", "+", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "\[ImaginaryI]", " ", "z"]]], "+", RowBox[List["2", " ", SuperscriptBox["\[ExponentialE]", RowBox[List["\[ImaginaryI]", " ", "z"]]], " ", SqrtBox[RowBox[List["1", "+", SuperscriptBox[RowBox[List["Cos", "[", "z", "]"]], "2"]]]]]]]], ")"]]]], "]"]], "-", RowBox[List["Log", "[", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "z"]]], " ", RowBox[List["(", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["\[ImaginaryI]", " ", "z"]]], "-", SqrtBox[RowBox[List["1", "+", SuperscriptBox[RowBox[List["Cos", "[", "z", "]"]], "2"]]]], "-", RowBox[List["\[ImaginaryI]", " ", RowBox[List["Sin", "[", "z", "]"]]]]]], ")"]]]], "]"]], "+", RowBox[List["Log", "[", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "z"]]], " ", RowBox[List["(", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["\[ImaginaryI]", " ", "z"]]], "+", SqrtBox[RowBox[List["1", "+", SuperscriptBox[RowBox[List["Cos", "[", "z", "]"]], "2"]]]], "-", RowBox[List["\[ImaginaryI]", " ", RowBox[List["Sin", "[", "z", "]"]]]]]], ")"]]]], "]"]]]], ")"]]]], "-", RowBox[List[RowBox[List["ArcTan", "[", FractionBox[RowBox[List["Sin", "[", "z", "]"]], SqrtBox[RowBox[List["1", "+", SuperscriptBox[RowBox[List["Cos", "[", "z", "]"]], "2"]]]]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["Log", "[", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["1", "+", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "\[ImaginaryI]", " ", "z"]]], "-", RowBox[List["2", " ", SuperscriptBox["\[ExponentialE]", RowBox[List["\[ImaginaryI]", " ", "z"]]], " ", SqrtBox[RowBox[List["1", "+", SuperscriptBox[RowBox[List["Cos", "[", "z", "]"]], "2"]]]]]]]], ")"]]]], "]"]], "+", RowBox[List["Log", "[", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["1", "+", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "\[ImaginaryI]", " ", "z"]]], "+", RowBox[List["2", " ", SuperscriptBox["\[ExponentialE]", RowBox[List["\[ImaginaryI]", " ", "z"]]], " ", SqrtBox[RowBox[List["1", "+", SuperscriptBox[RowBox[List["Cos", "[", "z", "]"]], "2"]]]]]]]], ")"]]]], "]"]], "+", RowBox[List["Log", "[", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "z"]]], " ", RowBox[List["(", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["\[ImaginaryI]", " ", "z"]]], "-", SqrtBox[RowBox[List["1", "+", SuperscriptBox[RowBox[List["Cos", "[", "z", "]"]], "2"]]]], "-", RowBox[List["\[ImaginaryI]", " ", RowBox[List["Sin", "[", "z", "]"]]]]]], ")"]]]], "]"]], "+", RowBox[List["Log", "[", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "z"]]], " ", RowBox[List["(", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["\[ImaginaryI]", " ", "z"]]], "+", SqrtBox[RowBox[List["1", "+", SuperscriptBox[RowBox[List["Cos", "[", "z", "]"]], "2"]]]], "-", RowBox[List["\[ImaginaryI]", " ", RowBox[List["Sin", "[", "z", "]"]]]]]], ")"]]]], "]"]]]], ")"]]]], "+", RowBox[List["\[ImaginaryI]", " ", RowBox[List["(", RowBox[List[RowBox[List["-", RowBox[List["PolyLog", "[", RowBox[List["2", ",", RowBox[List[RowBox[List["-", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "z"]]]]], " ", RowBox[List["(", RowBox[List[SqrtBox[RowBox[List["1", "+", SuperscriptBox[RowBox[List["Cos", "[", "z", "]"]], "2"]]]], "-", RowBox[List["\[ImaginaryI]", " ", RowBox[List["Sin", "[", "z", "]"]]]]]], ")"]]]]]], "]"]]]], "-", RowBox[List["PolyLog", "[", RowBox[List["2", ",", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["\[ImaginaryI]", " ", "z"]]], " ", RowBox[List["(", RowBox[List[SqrtBox[RowBox[List["1", "+", SuperscriptBox[RowBox[List["Cos", "[", "z", "]"]], "2"]]]], "-", RowBox[List["\[ImaginaryI]", " ", RowBox[List["Sin", "[", "z", "]"]]]]]], ")"]]]]]], "]"]], "+", RowBox[List["PolyLog", "[", RowBox[List["2", ",", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "z"]]], " ", RowBox[List["(", RowBox[List[SqrtBox[RowBox[List["1", "+", SuperscriptBox[RowBox[List["Cos", "[", "z", "]"]], "2"]]]], "+", RowBox[List["\[ImaginaryI]", " ", RowBox[List["Sin", "[", "z", "]"]]]]]], ")"]]]]]], "]"]], "+", RowBox[List["PolyLog", "[", RowBox[List["2", ",", RowBox[List[RowBox[List["-", SuperscriptBox["\[ExponentialE]", RowBox[List["\[ImaginaryI]", " ", "z"]]]]], " ", RowBox[List["(", RowBox[List[SqrtBox[RowBox[List["1", "+", SuperscriptBox[RowBox[List["Cos", "[", "z", "]"]], "2"]]]], "+", RowBox[List["\[ImaginaryI]", " ", RowBox[List["Sin", "[", "z", "]"]]]]]], ")"]]]]]], "]"]]]], ")"]]]]]], ")"]]]]]]]]

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

 2001-10-29