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 Cos

 http://functions.wolfram.com/01.07.21.2395.01

 Input Form

 Integrate[1/Sqrt[a Sin[e z]^2 + b Cos[e z]^2], z] == -(2 I (1 + Cos[e z]) Sqrt[(a + b + (-a + b) Cos[2 e z])/(1 + Cos[e z])^2] EllipticF[I ArcSinh[Sqrt[b/(2 a + 2 Sqrt[a] Sqrt[a - b] - b)] Tan[(e z)/2]], (-2 a - 2 Sqrt[a] Sqrt[a - b] + b)/ (-2 a + 2 Sqrt[a] Sqrt[a - b] + b)] Sqrt[1 + (b Tan[(e z)/2]^2)/(2 a + 2 Sqrt[a] Sqrt[a - b] - b)] Sqrt[1 - (b Tan[(e z)/2]^2)/(-2 a + 2 Sqrt[a] Sqrt[a - b] + b)])/ (Sqrt[b/(2 a + 2 Sqrt[a] Sqrt[a - b] - b)] e Sqrt[a + b + (-a + b) Cos[2 e z]] Sqrt[4 a Tan[(e z)/2]^2 + b (-1 + Tan[(e z)/2]^2)^2])

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List["\[Integral]", RowBox[List[FractionBox["1", SqrtBox[RowBox[List[RowBox[List["a", " ", SuperscriptBox[RowBox[List["Sin", "[", RowBox[List["e", " ", "z"]], "]"]], "2"]]], "+", RowBox[List["b", " ", SuperscriptBox[RowBox[List["Cos", "[", RowBox[List["e", " ", "z"]], "]"]], "2"]]]]]]], RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", RowBox[List[RowBox[List["-", RowBox[List["(", RowBox[List["2", " ", "\[ImaginaryI]", " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List["Cos", "[", RowBox[List["e", " ", "z"]], "]"]]]], ")"]], " ", SqrtBox[FractionBox[RowBox[List["a", "+", "b", "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "a"]], "+", "b"]], ")"]], " ", RowBox[List["Cos", "[", RowBox[List["2", " ", "e", " ", "z"]], "]"]]]]]], SuperscriptBox[RowBox[List["(", RowBox[List["1", "+", RowBox[List["Cos", "[", RowBox[List["e", " ", "z"]], "]"]]]], ")"]], "2"]]], " ", RowBox[List["EllipticF", "[", RowBox[List[RowBox[List["\[ImaginaryI]", " ", RowBox[List["ArcSinh", "[", RowBox[List[SqrtBox[FractionBox["b", RowBox[List[RowBox[List["2", " ", "a"]], "+", RowBox[List["2", " ", SqrtBox["a"], " ", SqrtBox[RowBox[List["a", "-", "b"]]]]], "-", "b"]]]], " ", RowBox[List["Tan", "[", FractionBox[RowBox[List["e", " ", "z"]], "2"], "]"]]]], "]"]]]], ",", FractionBox[RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "a"]], "-", RowBox[List["2", " ", SqrtBox["a"], " ", SqrtBox[RowBox[List["a", "-", "b"]]]]], "+", "b"]], RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "a"]], "+", RowBox[List["2", " ", SqrtBox["a"], " ", SqrtBox[RowBox[List["a", "-", "b"]]]]], "+", "b"]]]]], "]"]], " ", SqrtBox[RowBox[List["1", "+", FractionBox[RowBox[List["b", " ", SuperscriptBox[RowBox[List["Tan", "[", FractionBox[RowBox[List["e", " ", "z"]], "2"], "]"]], "2"]]], RowBox[List[RowBox[List["2", " ", "a"]], "+", RowBox[List["2", " ", SqrtBox["a"], " ", SqrtBox[RowBox[List["a", "-", "b"]]]]], "-", "b"]]]]]], " ", SqrtBox[RowBox[List["1", "-", FractionBox[RowBox[List["b", " ", SuperscriptBox[RowBox[List["Tan", "[", FractionBox[RowBox[List["e", " ", "z"]], "2"], "]"]], "2"]]], RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "a"]], "+", RowBox[List["2", " ", SqrtBox["a"], " ", SqrtBox[RowBox[List["a", "-", "b"]]]]], "+", "b"]]]]]]]], ")"]]]], "/", RowBox[List["(", RowBox[List[SqrtBox[FractionBox["b", RowBox[List[RowBox[List["2", " ", "a"]], "+", RowBox[List["2", " ", SqrtBox["a"], " ", SqrtBox[RowBox[List["a", "-", "b"]]]]], "-", "b"]]]], " ", "e", " ", SqrtBox[RowBox[List["a", "+", "b", "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "a"]], "+", "b"]], ")"]], " ", RowBox[List["Cos", "[", RowBox[List["2", " ", "e", " ", "z"]], "]"]]]]]]], " ", SqrtBox[RowBox[List[RowBox[List["4", " ", "a", " ", SuperscriptBox[RowBox[List["Tan", "[", FractionBox[RowBox[List["e", " ", "z"]], "2"], "]"]], "2"]]], "+", RowBox[List["b", " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", SuperscriptBox[RowBox[List["Tan", "[", FractionBox[RowBox[List["e", " ", "z"]], "2"], "]"]], "2"]]], ")"]], "2"]]]]]]]], ")"]]]]]]]]

 MathML Form

 1 a sin 2 ( e z ) + b cos 2 ( e z ) z - ( 2 ( cos ( e z ) + 1 ) a + b + ( b - a ) cos ( 2 e z ) ( cos ( e z ) + 1 ) 2 F ( sinh - 1 ( b 2 a + 2 a - b a - b tan ( e z 2 ) ) - 2 a - 2 a - b a + b - 2 a + 2 a - b a + b ) b tan 2 ( e z 2 ) 2 a + 2 a - b a - b + 1 1 - b tan 2 ( e z 2 ) - 2 a + 2 a - b a + b ) / ( b 2 a + 2 a - b a - b e a + b + ( b - a ) cos ( 2 e z ) 4 a tan 2 ( e z 2 ) + b ( tan 2 ( e z 2 ) - 1 ) 2 ) z 1 a e z 2 b e z 2 1 2 -1 -1 2 e z 1 a b b -1 a 2 e z e z 1 2 -1 1 2 EllipticF b 2 a 2 a -1 b 1 2 a 1 2 -1 b -1 1 2 e z 2 -1 -2 a -1 2 a -1 b 1 2 a 1 2 b -2 a 2 a -1 b 1 2 a 1 2 b -1 b e z 2 -1 2 2 a 2 a -1 b 1 2 a 1 2 -1 b -1 1 1 2 1 -1 b e z 2 -1 2 -2 a 2 a -1 b 1 2 a 1 2 b -1 1 2 b 2 a 2 a -1 b 1 2 a 1 2 -1 b -1 1 2 e a b b -1 a 2 e z 1 2 4 a e z 2 -1 2 b e z 2 -1 2 -1 2 1 2 -1 [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["\[Integral]", RowBox[List[FractionBox["1", SqrtBox[RowBox[List[RowBox[List["a_", " ", SuperscriptBox[RowBox[List["Sin", "[", RowBox[List["e_", " ", "z_"]], "]"]], "2"]]], "+", RowBox[List["b_", " ", SuperscriptBox[RowBox[List["Cos", "[", RowBox[List["e_", " ", "z_"]], "]"]], "2"]]]]]]], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List["-", FractionBox[RowBox[List["2", " ", "\[ImaginaryI]", " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List["Cos", "[", RowBox[List["e", " ", "z"]], "]"]]]], ")"]], " ", SqrtBox[FractionBox[RowBox[List["a", "+", "b", "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "a"]], "+", "b"]], ")"]], " ", RowBox[List["Cos", "[", RowBox[List["2", " ", "e", " ", "z"]], "]"]]]]]], SuperscriptBox[RowBox[List["(", RowBox[List["1", "+", RowBox[List["Cos", "[", RowBox[List["e", " ", "z"]], "]"]]]], ")"]], "2"]]], " ", RowBox[List["EllipticF", "[", RowBox[List[RowBox[List["\[ImaginaryI]", " ", RowBox[List["ArcSinh", "[", RowBox[List[SqrtBox[FractionBox["b", RowBox[List[RowBox[List["2", " ", "a"]], "+", RowBox[List["2", " ", SqrtBox["a"], " ", SqrtBox[RowBox[List["a", "-", "b"]]]]], "-", "b"]]]], " ", RowBox[List["Tan", "[", FractionBox[RowBox[List["e", " ", "z"]], "2"], "]"]]]], "]"]]]], ",", FractionBox[RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "a"]], "-", RowBox[List["2", " ", SqrtBox["a"], " ", SqrtBox[RowBox[List["a", "-", "b"]]]]], "+", "b"]], RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "a"]], "+", RowBox[List["2", " ", SqrtBox["a"], " ", SqrtBox[RowBox[List["a", "-", "b"]]]]], "+", "b"]]]]], "]"]], " ", SqrtBox[RowBox[List["1", "+", FractionBox[RowBox[List["b", " ", SuperscriptBox[RowBox[List["Tan", "[", FractionBox[RowBox[List["e", " ", "z"]], "2"], "]"]], "2"]]], RowBox[List[RowBox[List["2", " ", "a"]], "+", RowBox[List["2", " ", SqrtBox["a"], " ", SqrtBox[RowBox[List["a", "-", "b"]]]]], "-", "b"]]]]]], " ", SqrtBox[RowBox[List["1", "-", FractionBox[RowBox[List["b", " ", SuperscriptBox[RowBox[List["Tan", "[", FractionBox[RowBox[List["e", " ", "z"]], "2"], "]"]], "2"]]], RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "a"]], "+", RowBox[List["2", " ", SqrtBox["a"], " ", SqrtBox[RowBox[List["a", "-", "b"]]]]], "+", "b"]]]]]]]], RowBox[List[SqrtBox[FractionBox["b", RowBox[List[RowBox[List["2", " ", "a"]], "+", RowBox[List["2", " ", SqrtBox["a"], " ", SqrtBox[RowBox[List["a", "-", "b"]]]]], "-", "b"]]]], " ", "e", " ", SqrtBox[RowBox[List["a", "+", "b", "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "a"]], "+", "b"]], ")"]], " ", RowBox[List["Cos", "[", RowBox[List["2", " ", "e", " ", "z"]], "]"]]]]]]], " ", SqrtBox[RowBox[List[RowBox[List["4", " ", "a", " ", SuperscriptBox[RowBox[List["Tan", "[", FractionBox[RowBox[List["e", " ", "z"]], "2"], "]"]], "2"]]], "+", RowBox[List["b", " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", SuperscriptBox[RowBox[List["Tan", "[", FractionBox[RowBox[List["e", " ", "z"]], "2"], "]"]], "2"]]], ")"]], "2"]]]]]]]]]]]]]]]

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

 2002-12-18