Secant: Integration (formula 01.11.21.0095)

Sec

$Mathematica Notation$

Elementary Functions

Sec[z]

Integration

Indefinite integration

Involving functions of the direct function

Involving algebraic functions of the direct function

Involving ((a+b sec(c z))ⁿ)^beta

http://functions.wolfram.com/01.11.21.0095.01

Input Form

Integrate[Sec[c z]/Sqrt[(a + b Sec[c z])^3], z] == -(2 Cot[(c z)/2] Sqrt[(a + b Sec[c z])/(a + b)] ((b + a Cos[c z]) EllipticE[ArcSin[Sqrt[1 - Sec[c z]]/Sqrt[2]], (2 b)/(a + b)] Sqrt[1 - Sec[c z]] Sec[c z] + b (-1 + Sec[c z]) Sqrt[1 + Sec[c z]] Sqrt[(a + b Sec[c z])/(a + b)]))/ ((a - b) c Sqrt[1 + Sec[c z]] Sqrt[(a + b Sec[c z])^3])

Standard Form

Cell[BoxData[RowBox[List[RowBox[List["\[Integral]", RowBox[List[FractionBox[RowBox[List["Sec", "[", RowBox[List["c", " ", "z"]], "]"]], SqrtBox[SuperscriptBox[RowBox[List["(", RowBox[List["a", "+", RowBox[List["b", " ", RowBox[List["Sec", "[", RowBox[List["c", " ", "z"]], "]"]]]]]], ")"]], "3"]]], RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", RowBox[List[RowBox[List["-", RowBox[List["(", RowBox[List["2", " ", RowBox[List["Cot", "[", FractionBox[RowBox[List["c", " ", "z"]], "2"], "]"]], " ", SqrtBox[FractionBox[RowBox[List["a", "+", RowBox[List["b", " ", RowBox[List["Sec", "[", RowBox[List["c", " ", "z"]], "]"]]]]]], RowBox[List["a", "+", "b"]]]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["b", "+", RowBox[List["a", " ", RowBox[List["Cos", "[", RowBox[List["c", " ", "z"]], "]"]]]]]], ")"]], " ", RowBox[List["EllipticE", "[", RowBox[List[RowBox[List["ArcSin", "[", FractionBox[SqrtBox[RowBox[List["1", "-", RowBox[List["Sec", "[", RowBox[List["c", " ", "z"]], "]"]]]]], SqrtBox["2"]], "]"]], ",", FractionBox[RowBox[List["2", " ", "b"]], RowBox[List["a", "+", "b"]]]]], "]"]], " ", SqrtBox[RowBox[List["1", "-", RowBox[List["Sec", "[", RowBox[List["c", " ", "z"]], "]"]]]]], " ", RowBox[List["Sec", "[", RowBox[List["c", " ", "z"]], "]"]]]], "+", RowBox[List["b", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", RowBox[List["Sec", "[", RowBox[List["c", " ", "z"]], "]"]]]], ")"]], " ", SqrtBox[RowBox[List["1", "+", RowBox[List["Sec", "[", RowBox[List["c", " ", "z"]], "]"]]]]], " ", SqrtBox[FractionBox[RowBox[List["a", "+", RowBox[List["b", " ", RowBox[List["Sec", "[", RowBox[List["c", " ", "z"]], "]"]]]]]], RowBox[List["a", "+", "b"]]]]]]]], ")"]]]], ")"]]]], "/", RowBox[List["(", RowBox[List[RowBox[List["(", RowBox[List["a", "-", "b"]], ")"]], " ", "c", " ", SqrtBox[RowBox[List["1", "+", RowBox[List["Sec", "[", RowBox[List["c", " ", "z"]], "]"]]]]], " ", SqrtBox[SuperscriptBox[RowBox[List["(", RowBox[List["a", "+", RowBox[List["b", " ", RowBox[List["Sec", "[", RowBox[List["c", " ", "z"]], "]"]]]]]], ")"]], "3"]]]], ")"]]]]]]]]

MathML Form

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Rule Form

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

2002-10-15