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; }

 Sech

 http://functions.wolfram.com/01.24.21.0477.01

 Input Form

 Integrate[((3 + Sinh[z]^2) Tanh[z]^3)/((-2 + Cosh[z]^2) Sqrt[(5 - 4 Sech[z]^2)^3]), z] == (Sech[z]^2 (21 - 35 Cosh[2 z] + (1/(15 Sqrt[1 + Cosh[2 z]])) ((-3 + 5 Cosh[2 z])^(3/2) (36 Sqrt[5] ArcSinh[Sqrt[-3 + 5 Cosh[2 z]]/ (2 Sqrt[2])] Cosh[z] - 5 Sqrt[3] (14 ArcTanh[(Sqrt[6] Cosh[z])/Sqrt[-3 + 5 Cosh[2 z]]] Sqrt[Cosh[z]^2] + 3 Cosh[z] (-Log[4 Sqrt[3] + 5 Sqrt[6] Sqrt[Cosh[z]^2] - 3 Sqrt[-3 + 5 Cosh[2 z]]] - Log[2 Sqrt[3] - Sqrt[-3 + 5 Cosh[2 z]]] + Log[2 Sqrt[3] + Sqrt[-3 + 5 Cosh[2 z]]] + Log[4 Sqrt[3] + 5 Sqrt[6] Sqrt[Cosh[z]^2] + 3 Sqrt[-3 + 5 Cosh[2 z]]]))) Sech[z])))/ (120 Sqrt[(5 - 4 Sech[z]^2)^3])

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List["\[Integral]", RowBox[List[FractionBox[RowBox[List[RowBox[List["(", RowBox[List["3", "+", SuperscriptBox[RowBox[List["Sinh", "[", "z", "]"]], "2"]]], ")"]], " ", SuperscriptBox[RowBox[List["Tanh", "[", "z", "]"]], "3"]]], RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "2"]], "+", SuperscriptBox[RowBox[List["Cosh", "[", "z", "]"]], "2"]]], ")"]], " ", SqrtBox[SuperscriptBox[RowBox[List["(", RowBox[List["5", "-", RowBox[List["4", " ", SuperscriptBox[RowBox[List["Sech", "[", "z", "]"]], "2"]]]]], ")"]], "3"]]]]], RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["Sech", "[", "z", "]"]], "2"], " ", RowBox[List["(", RowBox[List["21", "-", RowBox[List["35", " ", RowBox[List["Cosh", "[", RowBox[List["2", " ", "z"]], "]"]]]], "+", RowBox[List[FractionBox["1", RowBox[List["15", " ", SqrtBox[RowBox[List["1", "+", RowBox[List["Cosh", "[", RowBox[List["2", " ", "z"]], "]"]]]]]]]], RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "3"]], "+", RowBox[List["5", " ", RowBox[List["Cosh", "[", RowBox[List["2", " ", "z"]], "]"]]]]]], ")"]], RowBox[List["3", "/", "2"]]], " ", RowBox[List["(", RowBox[List[RowBox[List["36", " ", SqrtBox["5"], " ", RowBox[List["ArcSinh", "[", FractionBox[SqrtBox[RowBox[List[RowBox[List["-", "3"]], "+", RowBox[List["5", " ", RowBox[List["Cosh", "[", RowBox[List["2", " ", "z"]], "]"]]]]]]], RowBox[List["2", " ", SqrtBox["2"]]]], "]"]], " ", RowBox[List["Cosh", "[", "z", "]"]]]], "-", RowBox[List["5", " ", SqrtBox["3"], " ", RowBox[List["(", RowBox[List[RowBox[List["14", " ", RowBox[List["ArcTanh", "[", FractionBox[RowBox[List[SqrtBox["6"], " ", RowBox[List["Cosh", "[", "z", "]"]]]], SqrtBox[RowBox[List[RowBox[List["-", "3"]], "+", RowBox[List["5", " ", RowBox[List["Cosh", "[", RowBox[List["2", " ", "z"]], "]"]]]]]]]], "]"]], " ", SqrtBox[SuperscriptBox[RowBox[List["Cosh", "[", "z", "]"]], "2"]]]], "+", RowBox[List["3", " ", RowBox[List["Cosh", "[", "z", "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", RowBox[List["Log", "[", RowBox[List[RowBox[List["4", " ", SqrtBox["3"]]], "+", RowBox[List["5", " ", SqrtBox["6"], " ", SqrtBox[SuperscriptBox[RowBox[List["Cosh", "[", "z", "]"]], "2"]]]], "-", RowBox[List["3", " ", SqrtBox[RowBox[List[RowBox[List["-", "3"]], "+", RowBox[List["5", " ", RowBox[List["Cosh", "[", RowBox[List["2", " ", "z"]], "]"]]]]]]]]]]], "]"]]]], "-", RowBox[List["Log", "[", RowBox[List[RowBox[List["2", " ", SqrtBox["3"]]], "-", SqrtBox[RowBox[List[RowBox[List["-", "3"]], "+", RowBox[List["5", " ", RowBox[List["Cosh", "[", RowBox[List["2", " ", "z"]], "]"]]]]]]]]], "]"]], "+", RowBox[List["Log", "[", RowBox[List[RowBox[List["2", " ", SqrtBox["3"]]], "+", SqrtBox[RowBox[List[RowBox[List["-", "3"]], "+", RowBox[List["5", " ", RowBox[List["Cosh", "[", RowBox[List["2", " ", "z"]], "]"]]]]]]]]], "]"]], "+", RowBox[List["Log", "[", RowBox[List[RowBox[List["4", " ", SqrtBox["3"]]], "+", RowBox[List["5", " ", SqrtBox["6"], " ", SqrtBox[SuperscriptBox[RowBox[List["Cosh", "[", "z", "]"]], "2"]]]], "+", RowBox[List["3", " ", SqrtBox[RowBox[List[RowBox[List["-", "3"]], "+", RowBox[List["5", " ", RowBox[List["Cosh", "[", RowBox[List["2", " ", "z"]], "]"]]]]]]]]]]], "]"]]]], ")"]]]]]], ")"]]]]]], ")"]], " ", RowBox[List["Sech", "[", "z", "]"]]]], ")"]]]]]], ")"]]]], ")"]], "/", RowBox[List["(", RowBox[List["120", " ", SqrtBox[SuperscriptBox[RowBox[List["(", RowBox[List["5", "-", RowBox[List["4", " ", SuperscriptBox[RowBox[List["Sech", "[", "z", "]"]], "2"]]]]], ")"]], "3"]]]], ")"]]]]]]]]

 MathML Form

 ( sinh 2 ( z ) + 3 ) tanh 3 ( z ) ( cosh 2 ( z ) - 2 ) ( 5 - 4 sech 2 ( z ) ) 3 z ( sech 2 ( z ) ( 1 15 cosh ( 2 z ) + 1 ( ( 36 5 sinh - 1 ( 5 cosh ( 2 z ) - 3 2 2 ) cosh ( z ) - 5 3 ( 14 cosh 2 ( z ) tanh - 1 ( 6 cosh ( z ) 5 cosh ( 2 z ) - 3 ) + 3 cosh ( z ) ( - log ( 5 6 cosh 2 ( z ) - 3 5 cosh ( 2 z ) - 3 + 4 3 ) - log ( 2 3 - 5 cosh ( 2 z ) - 3 ) + log ( 5 cosh ( 2 z ) - 3 + 2 3 ) + log ( 5 6 cosh 2 ( z ) + 3 5 cosh ( 2 z ) - 3 + 4 3 ) ) ) ) sech ( z ) ( 5 cosh ( 2 z ) - 3 ) 3 / 2 ) - 35 cosh ( 2 z ) + 21 ) ) / ( 120 ( 5 - 4 sech 2 ( z ) ) 3 ) z z 2 3 z 3 z 2 -2 5 -1 4 z 2 3 1 2 -1 z 2 1 15 2 z 1 1 2 -1 36 5 1 2 5 2 z -3 1 2 2 2 1 2 -1 z -1 5 3 1 2 14 z 2 1 2 6 1 2 z 5 2 z -3 1 2 -1 3 z -1 5 6 1 2 z 2 1 2 -1 3 5 2 z -3 1 2 4 3 1 2 -1 2 3 1 2 -1 5 2 z -3 1 2 5 2 z -3 1 2 2 3 1 2 5 6 1 2 z 2 1 2 3 5 2 z -3 1 2 4 3 1 2 z 5 2 z -3 3 2 -1 35 2 z 21 120 5 -1 4 z 2 3 1 2 -1 [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["\[Integral]", RowBox[List[FractionBox[RowBox[List[RowBox[List["(", RowBox[List["3", "+", SuperscriptBox[RowBox[List["Sinh", "[", "z_", "]"]], "2"]]], ")"]], " ", SuperscriptBox[RowBox[List["Tanh", "[", "z_", "]"]], "3"]]], RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "2"]], "+", SuperscriptBox[RowBox[List["Cosh", "[", "z_", "]"]], "2"]]], ")"]], " ", SqrtBox[SuperscriptBox[RowBox[List["(", RowBox[List["5", "-", RowBox[List["4", " ", SuperscriptBox[RowBox[List["Sech", "[", "z_", "]"]], "2"]]]]], ")"]], "3"]]]]], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", FractionBox[RowBox[List[SuperscriptBox[RowBox[List["Sech", "[", "z", "]"]], "2"], " ", RowBox[List["(", RowBox[List["21", "-", RowBox[List["35", " ", RowBox[List["Cosh", "[", RowBox[List["2", " ", "z"]], "]"]]]], "+", FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "3"]], "+", RowBox[List["5", " ", RowBox[List["Cosh", "[", RowBox[List["2", " ", "z"]], "]"]]]]]], ")"]], RowBox[List["3", "/", "2"]]], " ", RowBox[List["(", RowBox[List[RowBox[List["36", " ", SqrtBox["5"], " ", RowBox[List["ArcSinh", "[", FractionBox[SqrtBox[RowBox[List[RowBox[List["-", "3"]], "+", RowBox[List["5", " ", RowBox[List["Cosh", "[", RowBox[List["2", " ", "z"]], "]"]]]]]]], RowBox[List["2", " ", SqrtBox["2"]]]], "]"]], " ", RowBox[List["Cosh", "[", "z", "]"]]]], "-", RowBox[List["5", " ", SqrtBox["3"], " ", RowBox[List["(", RowBox[List[RowBox[List["14", " ", RowBox[List["ArcTanh", "[", FractionBox[RowBox[List[SqrtBox["6"], " ", RowBox[List["Cosh", "[", "z", "]"]]]], SqrtBox[RowBox[List[RowBox[List["-", "3"]], "+", RowBox[List["5", " ", RowBox[List["Cosh", "[", RowBox[List["2", " ", "z"]], "]"]]]]]]]], "]"]], " ", SqrtBox[SuperscriptBox[RowBox[List["Cosh", "[", "z", "]"]], "2"]]]], "+", RowBox[List["3", " ", RowBox[List["Cosh", "[", "z", "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", RowBox[List["Log", "[", RowBox[List[RowBox[List["4", " ", SqrtBox["3"]]], "+", RowBox[List["5", " ", SqrtBox["6"], " ", SqrtBox[SuperscriptBox[RowBox[List["Cosh", "[", "z", "]"]], "2"]]]], "-", RowBox[List["3", " ", SqrtBox[RowBox[List[RowBox[List["-", "3"]], "+", RowBox[List["5", " ", RowBox[List["Cosh", "[", RowBox[List["2", " ", "z"]], "]"]]]]]]]]]]], "]"]]]], "-", RowBox[List["Log", "[", RowBox[List[RowBox[List["2", " ", SqrtBox["3"]]], "-", SqrtBox[RowBox[List[RowBox[List["-", "3"]], "+", RowBox[List["5", " ", RowBox[List["Cosh", "[", RowBox[List["2", " ", "z"]], "]"]]]]]]]]], "]"]], "+", RowBox[List["Log", "[", RowBox[List[RowBox[List["2", " ", SqrtBox["3"]]], "+", SqrtBox[RowBox[List[RowBox[List["-", "3"]], "+", RowBox[List["5", " ", RowBox[List["Cosh", "[", RowBox[List["2", " ", "z"]], "]"]]]]]]]]], "]"]], "+", RowBox[List["Log", "[", RowBox[List[RowBox[List["4", " ", SqrtBox["3"]]], "+", RowBox[List["5", " ", SqrtBox["6"], " ", SqrtBox[SuperscriptBox[RowBox[List["Cosh", "[", "z", "]"]], "2"]]]], "+", RowBox[List["3", " ", SqrtBox[RowBox[List[RowBox[List["-", "3"]], "+", RowBox[List["5", " ", RowBox[List["Cosh", "[", RowBox[List["2", " ", "z"]], "]"]]]]]]]]]]], "]"]]]], ")"]]]]]], ")"]]]]]], ")"]], " ", RowBox[List["Sech", "[", "z", "]"]]]], RowBox[List["15", " ", SqrtBox[RowBox[List["1", "+", RowBox[List["Cosh", "[", RowBox[List["2", " ", "z"]], "]"]]]]]]]]]], ")"]]]], RowBox[List["120", " ", SqrtBox[SuperscriptBox[RowBox[List["(", RowBox[List["5", "-", RowBox[List["4", " ", SuperscriptBox[RowBox[List["Sech", "[", "z", "]"]], "2"]]]]], ")"]], "3"]]]]]]]]]

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

 2002-12-18