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/01.24.21.0328.01

 Input Form

 Integrate[Sech[c z] Sqrt[(a + b Sech[c z]^2)^3], z] == (2 Cosh[c z]^3 Sqrt[(a + b Sech[c z]^2)^3] (2 I Sqrt[2] (2 a^2 + 3 a b + b^2) Sqrt[(a + 2 b + a Cosh[2 c z])/(a + b)] EllipticE[I c z, a/(a + b)] - I Sqrt[2] (3 a^2 + 5 a b + 2 b^2) Sqrt[(a + 2 b + a Cosh[2 c z])/(a + b)] EllipticF[I c z, a/(a + b)] + Tanh[c z] (4 a^2 + 11 a b + 4 b^2 + 2 a (2 a + b) Cosh[2 c z] + b (a + 2 b) Sech[c z]^2 + a b Tanh[c z]^2)))/ (3 c (a + 2 b + a Cosh[2 c z])^2)

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List["\[Integral]", RowBox[List[RowBox[List["Sech", "[", RowBox[List["c", " ", "z"]], "]"]], SqrtBox[SuperscriptBox[RowBox[List["(", RowBox[List["a", "+", RowBox[List["b", " ", SuperscriptBox[RowBox[List["Sech", "[", RowBox[List["c", " ", "z"]], "]"]], "2"]]]]], ")"]], "3"]], RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", RowBox[List[RowBox[List["(", RowBox[List["2", " ", SuperscriptBox[RowBox[List["Cosh", "[", RowBox[List["c", " ", "z"]], "]"]], "3"], " ", SqrtBox[SuperscriptBox[RowBox[List["(", RowBox[List["a", "+", RowBox[List["b", " ", SuperscriptBox[RowBox[List["Sech", "[", RowBox[List["c", " ", "z"]], "]"]], "2"]]]]], ")"]], "3"]], " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "\[ImaginaryI]", " ", SqrtBox["2"], " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", SuperscriptBox["a", "2"]]], "+", RowBox[List["3", " ", "a", " ", "b"]], "+", SuperscriptBox["b", "2"]]], ")"]], " ", SqrtBox[FractionBox[RowBox[List["a", "+", RowBox[List["2", " ", "b"]], "+", RowBox[List["a", " ", RowBox[List["Cosh", "[", RowBox[List["2", " ", "c", " ", "z"]], "]"]]]]]], RowBox[List["a", "+", "b"]]]], " ", RowBox[List["EllipticE", "[", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "c", " ", "z"]], ",", FractionBox["a", RowBox[List["a", "+", "b"]]]]], "]"]]]], "-", RowBox[List["\[ImaginaryI]", " ", SqrtBox["2"], " ", RowBox[List["(", RowBox[List[RowBox[List["3", " ", SuperscriptBox["a", "2"]]], "+", RowBox[List["5", " ", "a", " ", "b"]], "+", RowBox[List["2", " ", SuperscriptBox["b", "2"]]]]], ")"]], " ", SqrtBox[FractionBox[RowBox[List["a", "+", RowBox[List["2", " ", "b"]], "+", RowBox[List["a", " ", RowBox[List["Cosh", "[", RowBox[List["2", " ", "c", " ", "z"]], "]"]]]]]], RowBox[List["a", "+", "b"]]]], " ", RowBox[List["EllipticF", "[", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "c", " ", "z"]], ",", FractionBox["a", RowBox[List["a", "+", "b"]]]]], "]"]]]], "+", RowBox[List[RowBox[List["Tanh", "[", RowBox[List["c", " ", "z"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["4", " ", SuperscriptBox["a", "2"]]], "+", RowBox[List["11", " ", "a", " ", "b"]], "+", RowBox[List["4", " ", SuperscriptBox["b", "2"]]], "+", RowBox[List["2", " ", "a", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "a"]], "+", "b"]], ")"]], " ", RowBox[List["Cosh", "[", RowBox[List["2", " ", "c", " ", "z"]], "]"]]]], "+", RowBox[List["b", " ", RowBox[List["(", RowBox[List["a", "+", RowBox[List["2", " ", "b"]]]], ")"]], " ", SuperscriptBox[RowBox[List["Sech", "[", RowBox[List["c", " ", "z"]], "]"]], "2"]]], "+", RowBox[List["a", " ", "b", " ", SuperscriptBox[RowBox[List["Tanh", "[", RowBox[List["c", " ", "z"]], "]"]], "2"]]]]], ")"]]]]]], ")"]]]], ")"]], "/", RowBox[List["(", RowBox[List["3", " ", "c", " ", SuperscriptBox[RowBox[List["(", RowBox[List["a", "+", RowBox[List["2", " ", "b"]], "+", RowBox[List["a", " ", RowBox[List["Cosh", "[", RowBox[List["2", " ", "c", " ", "z"]], "]"]]]]]], ")"]], "2"]]], ")"]]]]]]]]

 MathML Form

 sech ( c z ) ( a + b sech 2 ( c z ) ) 3 z 1 3 c ( cosh ( 2 c z ) a + a + 2 b ) 2 ( 2 cosh 3 ( c z ) ( b sech 2 ( c z ) + a ) 3 ( 2 2 cosh ( 2 c z ) a + a + 2 b a + b ( 2 a 2 + 3 b a + b 2 ) E ( c z a a + b ) - 2 ( 3 a 2 + 5 b a + 2 b 2 ) cosh ( 2 c z ) a + a + 2 b a + b F ( c z a a + b ) + tanh ( c z ) ( 4 a 2 + b tanh 2 ( c z ) a + 11 b a + 2 ( 2 a + b ) cosh ( 2 c z ) a + 4 b 2 + b ( a + 2 b ) sech 2 ( c z ) ) ) ) z c z a b c z 2 3 1 2 1 3 c 2 c z a a 2 b 2 -1 2 c z 3 b c z 2 a 3 1 2 2 2 1 2 2 c z a a 2 b a b -1 1 2 2 a 2 3 b a b 2 EllipticE c z a a b -1 -1 2 1 2 3 a 2 5 b a 2 b 2 2 c z a a 2 b a b -1 1 2 EllipticF c z a a b -1 c z 4 a 2 b c z 2 a 11 b a 2 2 a b 2 c z a 4 b 2 b a 2 b c z 2 [/itex]

 Rule Form

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

 2002-12-18