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/01.30.21.0027.01

 Input Form

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

 Standard Form

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 MathML Form

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

 Rule Form

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

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

 2001-10-29