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 Cos

 http://functions.wolfram.com/01.07.21.2396.01

 Input Form

 Integrate[1/(a Sin[e z]^2 + b Cos[e z]^2)^(3/2), z] == (Sqrt[2] a Sin[2 e z] - Sqrt[2] b Sin[2 e z] + ((1 + Cos[e z]) Sqrt[(a + b + (-a + b) Cos[2 e z])/(1 + Cos[e z])^2] (Sqrt[-((-2 a + 2 Sqrt[a] Sqrt[a - b] + b)/b)] (a + b + (-a + b) Cos[2 e z]) Sec[(e z)/2]^2 Tan[(e z)/2] - 2 I (-2 a + 2 Sqrt[a] Sqrt[a - b] + b) EllipticE[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[(b + (2 a + 2 Sqrt[a] Sqrt[a - b] - b) Tan[(e z)/2]^2)/b] Sqrt[(b - (-2 a + 2 Sqrt[a] Sqrt[a - b] + b) Tan[(e z)/2]^2)/b] + 4 I Sqrt[a] (-Sqrt[a] + Sqrt[a - b]) 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[(b + (2 a + 2 Sqrt[a] Sqrt[a - b] - b) Tan[(e z)/2]^2)/b] Sqrt[(b - (-2 a + 2 Sqrt[a] Sqrt[a - b] + b) Tan[(e z)/2]^2)/b]))/ (Sqrt[-((-2 a + 2 Sqrt[a] Sqrt[a - b] + b)/b)] Sqrt[4 a Tan[(e z)/2]^2 + b (-1 + Tan[(e z)/2]^2)^2]))/ (2 a b e Sqrt[a + b + (-a + b) Cos[2 e z]])

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List["\[Integral]", RowBox[List[FractionBox["1", SuperscriptBox[RowBox[List["(", 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["3", "/", "2"]]]], RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[SqrtBox["2"], " ", "a", " ", RowBox[List["Sin", "[", RowBox[List["2", " ", "e", " ", "z"]], "]"]]]], "-", RowBox[List[SqrtBox["2"], " ", "b", " ", RowBox[List["Sin", "[", RowBox[List["2", " ", "e", " ", "z"]], "]"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List[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"]], 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 MathML Form

 1 ( a sin 2 ( e z ) + b cos 2 ( e z ) ) 3 / 2 z ( 2 a sin ( 2 e z ) - 2 b sin ( 2 e z ) + ( ( cos ( e z ) + 1 ) a + b + ( b - a ) cos ( 2 e z ) ( cos ( e z ) + 1 ) 2 ( - - 2 a + 2 a - b a + b b ( a + b + ( b - a ) cos ( 2 e z ) ) tan ( e z 2 ) sec 2 ( e z 2 ) + 4 a ( a - b - a ) ( 2 a + 2 a - b a - b ) tan 2 ( e z 2 ) + b b b - ( - 2 a + 2 a - b a + b ) tan 2 ( e z 2 ) b 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 ) - 2 ( - 2 a + 2 a - b a + b ) E ( 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 ) ( 2 a + 2 a - b a - b ) tan 2 ( e z 2 ) + b b b - ( - 2 a + 2 a - b a + b ) tan 2 ( e z 2 ) b ) ) / ( - - 2 a + 2 a - b a + b b 4 a tan 2 ( e z 2 ) + b ( tan 2 ( e z 2 ) - 1 ) 2 ) ) / ( 2 a b e a + b + ( b - a ) cos ( 2 e z ) ) z 1 a e z 2 b e z 2 3 2 -1 2 1 2 a 2 e z -1 2 1 2 b 2 e z e z 1 a b b -1 a 2 e z e z 1 2 -1 1 2 -1 -2 a 2 a -1 b 1 2 a 1 2 b b -1 1 2 a b b -1 a 2 e z e z 2 -1 e z 2 -1 2 4 a 1 2 a -1 b 1 2 -1 a 1 2 2 a 2 a -1 b 1 2 a 1 2 -1 b e z 2 -1 2 b b -1 1 2 b -1 -2 a 2 a -1 b 1 2 a 1 2 b e z 2 -1 2 b -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 -1 2 -2 a 2 a -1 b 1 2 a 1 2 b EllipticE 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 2 a 2 a -1 b 1 2 a 1 2 -1 b e z 2 -1 2 b b -1 1 2 b -1 -2 a 2 a -1 b 1 2 a 1 2 b e z 2 -1 2 b -1 1 2 -1 -2 a 2 a -1 b 1 2 a 1 2 b b -1 1 2 4 a e z 2 -1 2 b e z 2 -1 2 -1 2 1 2 -1 2 a b e a b b -1 a 2 e z 1 2 -1 [/itex]

 Rule Form

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

 2002-12-18