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Sech






Mathematica Notation

Traditional Notation









Elementary Functions > Sech[z] > Integration > Indefinite integration > Involving functions of the direct function > Involving algebraic functions of the direct function > Involving (a+b sech2(c z))beta





http://functions.wolfram.com/01.24.21.0322.01









  


  










Input Form





Integrate[Sech[c z] Sqrt[a + b Sech[c z]^2], z] == (Sqrt[a + b Sech[c z]^2] (2 I Sqrt[2] (a + b) Cosh[c z] Sqrt[(a + 2 b + a Cosh[2 c z])/(a + b)] EllipticE[I c z, a/(a + b)] - 2 I Sqrt[2] (a + b) Cosh[c z] Sqrt[(a + 2 b + a Cosh[2 c z])/(a + b)] EllipticF[I c z, a/(a + b)] + 2 (a + 2 b + a Cosh[2 c z]) Sinh[c z]))/ (2 c (a + 2 b + a Cosh[2 c z]))










Standard Form





Cell[BoxData[RowBox[List[RowBox[List["\[Integral]", RowBox[List[RowBox[List["Sech", "[", RowBox[List["c", " ", "z"]], "]"]], SqrtBox[RowBox[List["a", "+", RowBox[List["b", " ", SuperscriptBox[RowBox[List["Sech", "[", RowBox[List["c", " ", "z"]], "]"]], "2"]]]]]], RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", RowBox[List[RowBox[List["(", RowBox[List[SqrtBox[RowBox[List["a", "+", RowBox[List["b", " ", SuperscriptBox[RowBox[List["Sech", "[", RowBox[List["c", " ", "z"]], "]"]], "2"]]]]]], " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "\[ImaginaryI]", " ", SqrtBox["2"], " ", RowBox[List["(", RowBox[List["a", "+", "b"]], ")"]], " ", RowBox[List["Cosh", "[", RowBox[List["c", " ", "z"]], "]"]], " ", 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["2", " ", "\[ImaginaryI]", " ", SqrtBox["2"], " ", RowBox[List["(", RowBox[List["a", "+", "b"]], ")"]], " ", RowBox[List["Cosh", "[", RowBox[List["c", " ", "z"]], "]"]], " ", 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["2", " ", RowBox[List["(", RowBox[List["a", "+", RowBox[List["2", " ", "b"]], "+", RowBox[List["a", " ", RowBox[List["Cosh", "[", RowBox[List["2", " ", "c", " ", "z"]], "]"]]]]]], ")"]], " ", RowBox[List["Sinh", "[", RowBox[List["c", " ", "z"]], "]"]]]]]], ")"]]]], ")"]], "/", RowBox[List["(", RowBox[List["2", " ", "c", " ", RowBox[List["(", RowBox[List["a", "+", RowBox[List["2", " ", "b"]], "+", RowBox[List["a", " ", RowBox[List["Cosh", "[", RowBox[List["2", " ", "c", " ", "z"]], "]"]]]]]], ")"]]]], ")"]]]]]]]]










MathML Form







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Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["\[Integral]", RowBox[List[RowBox[List[RowBox[List["Sech", "[", RowBox[List["c_", " ", "z_"]], "]"]], " ", SqrtBox[RowBox[List["a_", "+", RowBox[List["b_", " ", SuperscriptBox[RowBox[List["Sech", "[", RowBox[List["c_", " ", "z_"]], "]"]], "2"]]]]]]]], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", FractionBox[RowBox[List[SqrtBox[RowBox[List["a", "+", RowBox[List["b", " ", SuperscriptBox[RowBox[List["Sech", "[", RowBox[List["c", " ", "z"]], "]"]], "2"]]]]]], " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "\[ImaginaryI]", " ", SqrtBox["2"], " ", RowBox[List["(", RowBox[List["a", "+", "b"]], ")"]], " ", RowBox[List["Cosh", "[", RowBox[List["c", " ", "z"]], "]"]], " ", 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["2", " ", "\[ImaginaryI]", " ", SqrtBox["2"], " ", RowBox[List["(", RowBox[List["a", "+", "b"]], ")"]], " ", RowBox[List["Cosh", "[", RowBox[List["c", " ", "z"]], "]"]], " ", 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["2", " ", RowBox[List["(", RowBox[List["a", "+", RowBox[List["2", " ", "b"]], "+", RowBox[List["a", " ", RowBox[List["Cosh", "[", RowBox[List["2", " ", "c", " ", "z"]], "]"]]]]]], ")"]], " ", RowBox[List["Sinh", "[", RowBox[List["c", " ", "z"]], "]"]]]]]], ")"]]]], RowBox[List["2", " ", "c", " ", RowBox[List["(", RowBox[List["a", "+", RowBox[List["2", " ", "b"]], "+", RowBox[List["a", " ", RowBox[List["Cosh", "[", RowBox[List["2", " ", "c", " ", "z"]], "]"]]]]]], ")"]]]]]]]]]










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





2002-12-18





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