Secant: Integration (formula 01.11.21.0091)

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))^beta

http://functions.wolfram.com/01.11.21.0091.01

Input Form

Integrate[Sec[c z] Sqrt[a + b Sec[c z]], z] == -(2 I Cot[(c z)/2] (EllipticE[I ArcSinh[Sqrt[-(1/(a + b))] Sqrt[a + b Sec[c z]]], (a + b)/(a - b)] - EllipticF[ I ArcSinh[Sqrt[-(1/(a + b))] Sqrt[a + b Sec[c z]]], (a + b)/(a - b)]) Sqrt[(b - b Sec[c z])/(a + b)])/(Sqrt[-(1/(a + b))] c Sqrt[(b (1 + Sec[c z]))/(-a + b)])

Standard Form

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

MathML Form

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<apply> <power /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <plus /> <ci> a </ci> <ci> b </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <ci> c </ci> <apply> <power /> <apply> <times /> <ci> b </ci> <apply> <plus /> <apply> <sec /> <apply> <times /> <ci> c </ci> <ci> z </ci> </apply> </apply> <cn type='integer'> 1 </cn> </apply> <apply> <power /> <apply> <plus /> <ci> b </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> a </ci> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <imaginaryi /> <apply> <cot /> <apply> <times /> <ci> c </ci> <ci> z </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <plus /> <apply> <ci> EllipticE </ci> <apply> <times /> <imaginaryi /> <apply> <arcsinh /> <apply> <times /> <apply> <power /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <plus /> <ci> a </ci> <ci> b </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <plus /> <ci> a </ci> <apply> <times /> <ci> b </ci> <apply> <sec /> <apply> <times /> <ci> c </ci> <ci> z </ci> </apply> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> </apply> <apply> <times /> <apply> <plus /> <ci> a </ci> <ci> b </ci> </apply> <apply> <power /> <apply> <plus /> <ci> a </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> b </ci> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> EllipticF </ci> <apply> <times /> 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Rule Form

Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["\[Integral]", RowBox[List[RowBox[List[RowBox[List["Sec", "[", RowBox[List["c_", " ", "z_"]], "]"]], " ", SqrtBox[RowBox[List["a_", "+", RowBox[List["b_", " ", RowBox[List["Sec", "[", RowBox[List["c_", " ", "z_"]], "]"]]]]]]]]], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List["-", FractionBox[RowBox[List["2", " ", "\[ImaginaryI]", " ", RowBox[List["Cot", "[", FractionBox[RowBox[List["c", " ", "z"]], "2"], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["EllipticE", "[", RowBox[List[RowBox[List["\[ImaginaryI]", " ", RowBox[List["ArcSinh", "[", RowBox[List[SqrtBox[RowBox[List["-", FractionBox["1", RowBox[List["a", "+", "b"]]]]]], " ", SqrtBox[RowBox[List["a", "+", RowBox[List["b", " ", RowBox[List["Sec", "[", RowBox[List["c", " ", "z"]], "]"]]]]]]]]], "]"]]]], ",", FractionBox[RowBox[List["a", "+", "b"]], RowBox[List["a", "-", "b"]]]]], "]"]], "-", RowBox[List["EllipticF", "[", RowBox[List[RowBox[List["\[ImaginaryI]", " ", RowBox[List["ArcSinh", "[", RowBox[List[SqrtBox[RowBox[List["-", FractionBox["1", RowBox[List["a", "+", "b"]]]]]], " ", SqrtBox[RowBox[List["a", "+", RowBox[List["b", " ", RowBox[List["Sec", "[", RowBox[List["c", " ", "z"]], "]"]]]]]]]]], "]"]]]], ",", FractionBox[RowBox[List["a", "+", "b"]], RowBox[List["a", "-", "b"]]]]], "]"]]]], ")"]], " ", SqrtBox[FractionBox[RowBox[List["b", "-", RowBox[List["b", " ", RowBox[List["Sec", "[", RowBox[List["c", " ", "z"]], "]"]]]]]], RowBox[List["a", "+", "b"]]]]]], RowBox[List[SqrtBox[RowBox[List["-", FractionBox["1", RowBox[List["a", "+", "b"]]]]]], " ", "c", " ", SqrtBox[FractionBox[RowBox[List["b", " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List["Sec", "[", RowBox[List["c", " ", "z"]], "]"]]]], ")"]]]], RowBox[List[RowBox[List["-", "a"]], "+", "b"]]]]]]]]]]]]]

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

2002-10-15