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 ArcSech

 http://functions.wolfram.com/01.30.21.0036.01

 Input Form

 Integrate[z^3 ArcSech[a z]^3, z] == (1/4) (z^4 ArcSech[a z]^3 + (1/a^4) (Sqrt[(1 - a z)/(1 + a z)] (1 + a z) - (-2 + 2 Sqrt[(1 - a z)/(1 + a z)] + 2 a z Sqrt[(1 - a z)/(1 + a z)] + a^2 z^2 Sqrt[(1 - a z)/(1 + a z)] + a^3 z^3 Sqrt[(1 - a z)/(1 + a z)]) ArcSech[a z]^2 + ArcSech[a z] ((-a^2) z^2 + 4 Log[1 + E^(-2 ArcSech[a z])]) - 2 PolyLog[2, -E^(-2 ArcSech[a z])]))

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List["\[Integral]", RowBox[List[SuperscriptBox["z", "3"], " ", SuperscriptBox[RowBox[List["ArcSech", "[", RowBox[List["a", " ", "z"]], "]"]], "3"], RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", RowBox[List[FractionBox["1", "4"], " ", RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox["z", "4"], " ", SuperscriptBox[RowBox[List["ArcSech", "[", RowBox[List["a", " ", "z"]], "]"]], "3"]]], "+", RowBox[List[FractionBox["1", SuperscriptBox["a", "4"]], RowBox[List["(", RowBox[List[RowBox[List[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[RowBox[List["(", RowBox[List[RowBox[List["-", "2"]], "+", RowBox[List["2", " ", SqrtBox[FractionBox[RowBox[List["1", "-", RowBox[List["a", " ", "z"]]]], RowBox[List["1", "+", RowBox[List["a", " ", "z"]]]]]]]], "+", RowBox[List["2", " ", "a", " ", "z", " ", SqrtBox[FractionBox[RowBox[List["1", "-", RowBox[List["a", " ", "z"]]]], RowBox[List["1", "+", RowBox[List["a", " ", "z"]]]]]]]], "+", RowBox[List[SuperscriptBox["a", "2"], " ", SuperscriptBox["z", "2"], " ", SqrtBox[FractionBox[RowBox[List["1", "-", RowBox[List["a", " ", "z"]]]], RowBox[List["1", "+", RowBox[List["a", " ", "z"]]]]]]]], "+", RowBox[List[SuperscriptBox["a", "3"], " ", SuperscriptBox["z", "3"], " ", SqrtBox[FractionBox[RowBox[List["1", "-", RowBox[List["a", " ", "z"]]]], RowBox[List["1", "+", RowBox[List["a", " ", "z"]]]]]]]]]], ")"]], " ", SuperscriptBox[RowBox[List["ArcSech", "[", RowBox[List["a", " ", "z"]], "]"]], "2"]]], "+", RowBox[List[RowBox[List["ArcSech", "[", RowBox[List["a", " ", "z"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", SuperscriptBox["a", "2"]]], " ", SuperscriptBox["z", "2"]]], "+", RowBox[List["4", " ", RowBox[List["Log", "[", RowBox[List["1", "+", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "2"]], " ", RowBox[List["ArcSech", "[", RowBox[List["a", " ", "z"]], "]"]]]]]]], "]"]]]]]], ")"]]]], "-", RowBox[List["2", " ", RowBox[List["PolyLog", "[", RowBox[List["2", ",", RowBox[List["-", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "2"]], " ", RowBox[List["ArcSech", "[", RowBox[List["a", " ", "z"]], "]"]]]]]]]]], "]"]]]]]], ")"]]]]]], ")"]]]]]]]]

 MathML Form

 z 3 sech - 1 ( a z ) 3 z 1 4 ( sech - 1 ( a z ) 3 z 4 + 1 a 4 ( - ( a 3 z 3 1 - a z 1 + a z + a 2 z 2 1 - a z 1 + a z + 2 a z 1 - a z 1 + a z + 2 1 - a z 1 + a z - 2 ) sech - 1 ( a z ) 2 + ( 4 log ( 1 + - 2 sech - 1 ( a z ) ) - a 2 z 2 ) sech - 1 ( a z ) + 1 - a z 1 + a z ( a z + 1 ) - 2 Li PolyLog 2 ( - - 2 sech - 1 ( a z ) ) ) ) z z 3 a z 3 1 4 a z 3 z 4 1 a 4 -1 -1 a 3 z 3 1 -1 a z 1 a z -1 1 2 a 2 z 2 1 -1 a z 1 a z -1 1 2 2 a z 1 -1 a z 1 a z -1 1 2 2 1 -1 a z 1 a z -1 1 2 -2 a z 2 4 1 -2 a z -1 a 2 z 2 a z 1 -1 a z 1 a z -1 1 2 a z 1 -1 2 PolyLog 2 -1 -2 a z [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["\[Integral]", RowBox[List[RowBox[List[SuperscriptBox["z_", "3"], " ", SuperscriptBox[RowBox[List["ArcSech", "[", RowBox[List["a_", " ", "z_"]], "]"]], "3"]]], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox["1", "4"], " ", RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox["z", "4"], " ", SuperscriptBox[RowBox[List["ArcSech", "[", RowBox[List["a", " ", "z"]], "]"]], "3"]]], "+", FractionBox[RowBox[List[RowBox[List[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[RowBox[List["(", RowBox[List[RowBox[List["-", "2"]], "+", RowBox[List["2", " ", SqrtBox[FractionBox[RowBox[List["1", "-", RowBox[List["a", " ", "z"]]]], RowBox[List["1", "+", RowBox[List["a", " ", "z"]]]]]]]], "+", RowBox[List["2", " ", "a", " ", "z", " ", SqrtBox[FractionBox[RowBox[List["1", "-", RowBox[List["a", " ", "z"]]]], RowBox[List["1", "+", RowBox[List["a", " ", "z"]]]]]]]], "+", RowBox[List[SuperscriptBox["a", "2"], " ", SuperscriptBox["z", "2"], " ", SqrtBox[FractionBox[RowBox[List["1", "-", RowBox[List["a", " ", "z"]]]], RowBox[List["1", "+", RowBox[List["a", " ", "z"]]]]]]]], "+", RowBox[List[SuperscriptBox["a", "3"], " ", SuperscriptBox["z", "3"], " ", SqrtBox[FractionBox[RowBox[List["1", "-", RowBox[List["a", " ", "z"]]]], RowBox[List["1", "+", RowBox[List["a", " ", "z"]]]]]]]]]], ")"]], " ", SuperscriptBox[RowBox[List["ArcSech", "[", RowBox[List["a", " ", "z"]], "]"]], "2"]]], "+", RowBox[List[RowBox[List["ArcSech", "[", RowBox[List["a", " ", "z"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", SuperscriptBox["a", "2"]]], " ", SuperscriptBox["z", "2"]]], "+", RowBox[List["4", " ", RowBox[List["Log", "[", RowBox[List["1", "+", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "2"]], " ", RowBox[List["ArcSech", "[", RowBox[List["a", " ", "z"]], "]"]]]]]]], "]"]]]]]], ")"]]]], "-", RowBox[List["2", " ", RowBox[List["PolyLog", "[", RowBox[List["2", ",", RowBox[List["-", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "2"]], " ", RowBox[List["ArcSech", "[", RowBox[List["a", " ", "z"]], "]"]]]]]]]]], "]"]]]]]], SuperscriptBox["a", "4"]]]], ")"]]]]]]]]

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

 2001-10-29