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/01.24.21.0402.01

 Input Form

 Integrate[Sech[c z]^4/Sqrt[a + b Cosh[2 c z]], z] == (2 I (a^2 - 2 a b - 3 b^2) Sqrt[(a + b Cosh[2 c z])/(a + b)] EllipticE[I c z, (2 b)/(a + b)] - 2 I (a^2 - 3 a b + 2 b^2) Sqrt[(a + b Cosh[2 c z])/(a + b)] EllipticF[I c z, (2 b)/(a + b)] + (1/2) (4 a^2 - 7 a b - 3 b^2 + 2 (a^2 - a b - 4 b^2) Cosh[2 c z] + (a - 3 b) b Cosh[4 c z]) Sech[c z]^2 Tanh[c z])/ (3 (a - b)^2 c Sqrt[a + b Cosh[2 c z]])

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List["\[Integral]", RowBox[List[FractionBox[SuperscriptBox[RowBox[List["Sech", "[", RowBox[List["c", " ", "z"]], "]"]], "4"], SqrtBox[RowBox[List["a", "+", RowBox[List["b", " ", RowBox[List["Cosh", "[", RowBox[List["2", " ", "c", " ", "z"]], "]"]]]]]]]], RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["2", " ", "\[ImaginaryI]", " ", RowBox[List["(", RowBox[List[SuperscriptBox["a", "2"], "-", RowBox[List["2", " ", "a", " ", "b"]], "-", RowBox[List["3", " ", SuperscriptBox["b", "2"]]]]], ")"]], " ", SqrtBox[FractionBox[RowBox[List["a", "+", RowBox[List["b", " ", RowBox[List["Cosh", "[", RowBox[List["2", " ", "c", " ", "z"]], "]"]]]]]], RowBox[List["a", "+", "b"]]]], " ", RowBox[List["EllipticE", "[", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "c", " ", "z"]], ",", FractionBox[RowBox[List["2", " ", "b"]], RowBox[List["a", "+", "b"]]]]], "]"]]]], "-", RowBox[List["2", " ", "\[ImaginaryI]", " ", RowBox[List["(", RowBox[List[SuperscriptBox["a", "2"], "-", RowBox[List["3", " ", "a", " ", "b"]], "+", RowBox[List["2", " ", SuperscriptBox["b", "2"]]]]], ")"]], " ", SqrtBox[FractionBox[RowBox[List["a", "+", RowBox[List["b", " ", RowBox[List["Cosh", "[", RowBox[List["2", " ", "c", " ", "z"]], "]"]]]]]], RowBox[List["a", "+", "b"]]]], " ", RowBox[List["EllipticF", "[", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "c", " ", "z"]], ",", FractionBox[RowBox[List["2", " ", "b"]], RowBox[List["a", "+", "b"]]]]], "]"]]]], "+", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["4", " ", SuperscriptBox["a", "2"]]], "-", RowBox[List["7", " ", "a", " ", "b"]], "-", RowBox[List["3", " ", SuperscriptBox["b", "2"]]], "+", RowBox[List["2", " ", RowBox[List["(", RowBox[List[SuperscriptBox["a", "2"], "-", RowBox[List["a", " ", "b"]], "-", RowBox[List["4", " ", SuperscriptBox["b", "2"]]]]], ")"]], " ", RowBox[List["Cosh", "[", RowBox[List["2", " ", "c", " ", "z"]], "]"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List["a", "-", RowBox[List["3", " ", "b"]]]], ")"]], " ", "b", " ", RowBox[List["Cosh", "[", RowBox[List["4", " ", "c", " ", "z"]], "]"]]]]]], ")"]], " ", SuperscriptBox[RowBox[List["Sech", "[", RowBox[List["c", " ", "z"]], "]"]], "2"], " ", RowBox[List["Tanh", "[", RowBox[List["c", " ", "z"]], "]"]]]]]], ")"]], "/", RowBox[List["(", RowBox[List["3", " ", SuperscriptBox[RowBox[List["(", RowBox[List["a", "-", "b"]], ")"]], "2"], " ", "c", " ", SqrtBox[RowBox[List["a", "+", RowBox[List["b", " ", RowBox[List["Cosh", "[", RowBox[List["2", " ", "c", " ", "z"]], "]"]]]]]]]]], ")"]]]]]]]]

 MathML Form

 sech 4 ( c z ) a + b cosh ( 2 c z ) z ( 1 2 ( 4 a 2 - 7 b a - 3 b 2 + 2 ( a 2 - b a - 4 b 2 ) cosh ( 2 c z ) + ( a - 3 b ) b cosh ( 4 c z ) ) tanh ( c z ) sech 2 ( c z ) + 2 ( a 2 - 2 b a - 3 b 2 ) a + b cosh ( 2 c z ) a + b E ( c z 2 b a + b ) - 2 ( a 2 - 3 b a + 2 b 2 ) a + b cosh ( 2 c z ) a + b F ( c z 2 b a + b ) ) / ( 3 ( a - b ) 2 c a + b cosh ( 2 c z ) ) z c z 4 a b 2 c z 1 2 -1 1 2 4 a 2 -1 7 b a -1 3 b 2 2 a 2 -1 b a -1 4 b 2 2 c z a -1 3 b b 4 c z c z c z 2 2 a 2 -1 2 b a -1 3 b 2 a b 2 c z a b -1 1 2 EllipticE c z 2 b a b -1 -1 2 a 2 -1 3 b a 2 b 2 a b 2 c z a b -1 1 2 EllipticF c z 2 b a b -1 3 a -1 b 2 c a b 2 c z 1 2 -1 [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["\[Integral]", RowBox[List[FractionBox[SuperscriptBox[RowBox[List["Sech", "[", RowBox[List["c_", " ", "z_"]], "]"]], "4"], SqrtBox[RowBox[List["a_", "+", RowBox[List["b_", " ", RowBox[List["Cosh", "[", RowBox[List["2", " ", "c_", " ", "z_"]], "]"]]]]]]]], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", FractionBox[RowBox[List[RowBox[List["2", " ", "\[ImaginaryI]", " ", RowBox[List["(", RowBox[List[SuperscriptBox["a", "2"], "-", RowBox[List["2", " ", "a", " ", "b"]], "-", RowBox[List["3", " ", SuperscriptBox["b", "2"]]]]], ")"]], " ", SqrtBox[FractionBox[RowBox[List["a", "+", RowBox[List["b", " ", RowBox[List["Cosh", "[", RowBox[List["2", " ", "c", " ", "z"]], "]"]]]]]], RowBox[List["a", "+", "b"]]]], " ", RowBox[List["EllipticE", "[", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "c", " ", "z"]], ",", FractionBox[RowBox[List["2", " ", "b"]], RowBox[List["a", "+", "b"]]]]], "]"]]]], "-", RowBox[List["2", " ", "\[ImaginaryI]", " ", RowBox[List["(", RowBox[List[SuperscriptBox["a", "2"], "-", RowBox[List["3", " ", "a", " ", "b"]], "+", RowBox[List["2", " ", SuperscriptBox["b", "2"]]]]], ")"]], " ", SqrtBox[FractionBox[RowBox[List["a", "+", RowBox[List["b", " ", RowBox[List["Cosh", "[", RowBox[List["2", " ", "c", " ", "z"]], "]"]]]]]], RowBox[List["a", "+", "b"]]]], " ", RowBox[List["EllipticF", "[", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "c", " ", "z"]], ",", FractionBox[RowBox[List["2", " ", "b"]], RowBox[List["a", "+", "b"]]]]], "]"]]]], "+", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["4", " ", SuperscriptBox["a", "2"]]], "-", RowBox[List["7", " ", "a", " ", "b"]], "-", RowBox[List["3", " ", SuperscriptBox["b", "2"]]], "+", RowBox[List["2", " ", RowBox[List["(", RowBox[List[SuperscriptBox["a", "2"], "-", RowBox[List["a", " ", "b"]], "-", RowBox[List["4", " ", SuperscriptBox["b", "2"]]]]], ")"]], " ", RowBox[List["Cosh", "[", RowBox[List["2", " ", "c", " ", "z"]], "]"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List["a", "-", RowBox[List["3", " ", "b"]]]], ")"]], " ", "b", " ", RowBox[List["Cosh", "[", RowBox[List["4", " ", "c", " ", "z"]], "]"]]]]]], ")"]], " ", SuperscriptBox[RowBox[List["Sech", "[", RowBox[List["c", " ", "z"]], "]"]], "2"], " ", RowBox[List["Tanh", "[", RowBox[List["c", " ", "z"]], "]"]]]]]], RowBox[List["3", " ", SuperscriptBox[RowBox[List["(", RowBox[List["a", "-", "b"]], ")"]], "2"], " ", "c", " ", SqrtBox[RowBox[List["a", "+", RowBox[List["b", " ", RowBox[List["Cosh", "[", RowBox[List["2", " ", "c", " ", "z"]], "]"]]]]]]]]]]]]]]

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

 2002-12-18