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 Tan

 http://functions.wolfram.com/01.08.21.0055.01

 Input Form

 Integrate[Cos[2 c z] Sqrt[a + b Cos[2 c z]] Tan[c z], z] == -(a Sqrt[a - b] b Cos[2 c z] + Sqrt[a - b] b^2 Cos[2 c z]^2 + 2 a b ArcTanh[Sqrt[a + b Cos[2 c z]]/Sqrt[a - b]] Sqrt[a + b Cos[2 c z]] + (a - 3 b) Cos[(c z)/2]^2 Sqrt[(a + b Cos[2 c z])/(1 + Cos[c z])^2] (2 b Log[(2 (a - b + Sqrt[a - b] Sqrt[(a + b Cos[2 c z]) Sec[(c z)/2]^4] + (a - b) Tan[(c z)/2]^2))/ (Sqrt[a - b] (-1 + Tan[(c z)/2]^2))] + Sqrt[a - b] Sqrt[(a + b Cos[2 c z]) Sec[(c z)/2]^4] + Sqrt[a - b] Cos[c z] Sqrt[(a + b Cos[2 c z]) Sec[(c z)/2]^4]))/ (3 Sqrt[a - b] b c Sqrt[a + b Cos[2 c z]])

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List["\[Integral]", RowBox[List[RowBox[List["Cos", "[", RowBox[List["2", "c", " ", "z"]], "]"]], " ", SqrtBox[RowBox[List["a", "+", RowBox[List["b", " ", RowBox[List["Cos", "[", RowBox[List["2", "c", " ", "z"]], "]"]]]]]]], " ", RowBox[List["Tan", "[", RowBox[List["c", " ", "z"]], "]"]], RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", RowBox[List[RowBox[List["-", RowBox[List["(", RowBox[List[RowBox[List["a", " ", SqrtBox[RowBox[List["a", "-", "b"]]], " ", "b", " ", RowBox[List["Cos", "[", RowBox[List["2", " ", "c", " ", "z"]], "]"]]]], "+", RowBox[List[SqrtBox[RowBox[List["a", "-", "b"]]], " ", SuperscriptBox["b", "2"], " ", SuperscriptBox[RowBox[List["Cos", "[", RowBox[List["2", " ", "c", " ", "z"]], "]"]], "2"]]], "+", RowBox[List["2", " ", "a", " ", "b", " ", RowBox[List["ArcTanh", "[", FractionBox[SqrtBox[RowBox[List["a", "+", RowBox[List["b", " ", RowBox[List["Cos", "[", RowBox[List["2", " ", "c", " ", "z"]], "]"]]]]]]], SqrtBox[RowBox[List["a", "-", "b"]]]], "]"]], " ", SqrtBox[RowBox[List["a", "+", RowBox[List["b", " ", RowBox[List["Cos", "[", RowBox[List["2", " ", "c", " ", "z"]], "]"]]]]]]]]], "+", RowBox[List[RowBox[List["(", RowBox[List["a", "-", RowBox[List["3", " ", "b"]]]], ")"]], " ", SuperscriptBox[RowBox[List["Cos", "[", FractionBox[RowBox[List["c", " ", "z"]], "2"], "]"]], "2"], " ", SqrtBox[FractionBox[RowBox[List["a", "+", RowBox[List["b", " ", RowBox[List["Cos", "[", RowBox[List["2", " ", "c", " ", "z"]], "]"]]]]]], SuperscriptBox[RowBox[List["(", RowBox[List["1", "+", RowBox[List["Cos", "[", RowBox[List["c", " ", "z"]], "]"]]]], ")"]], "2"]]], " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "b", " ", RowBox[List["Log", "[", RowBox[List[RowBox[List["(", RowBox[List["2", " ", RowBox[List["(", RowBox[List["a", "-", "b", "+", RowBox[List[SqrtBox[RowBox[List["a", "-", "b"]]], " ", SqrtBox[RowBox[List[RowBox[List["(", RowBox[List["a", "+", RowBox[List["b", " ", RowBox[List["Cos", "[", RowBox[List["2", " ", "c", " ", "z"]], "]"]]]]]], ")"]], " ", SuperscriptBox[RowBox[List["Sec", "[", FractionBox[RowBox[List["c", " ", "z"]], "2"], "]"]], "4"]]]]]], "+", RowBox[List[RowBox[List["(", RowBox[List["a", "-", "b"]], ")"]], " ", SuperscriptBox[RowBox[List["Tan", "[", FractionBox[RowBox[List["c", " ", "z"]], "2"], "]"]], "2"]]]]], ")"]]]], ")"]], "/", RowBox[List["(", RowBox[List[SqrtBox[RowBox[List["a", "-", "b"]]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", SuperscriptBox[RowBox[List["Tan", "[", FractionBox[RowBox[List["c", " ", "z"]], "2"], "]"]], "2"]]], ")"]]]], ")"]]]], "]"]]]], "+", RowBox[List[SqrtBox[RowBox[List["a", "-", "b"]]], " ", SqrtBox[RowBox[List[RowBox[List["(", RowBox[List["a", "+", RowBox[List["b", " ", RowBox[List["Cos", "[", RowBox[List["2", " ", "c", " ", "z"]], "]"]]]]]], ")"]], " ", SuperscriptBox[RowBox[List["Sec", "[", FractionBox[RowBox[List["c", " ", "z"]], "2"], "]"]], "4"]]]]]], "+", RowBox[List[SqrtBox[RowBox[List["a", "-", "b"]]], " ", RowBox[List["Cos", "[", RowBox[List["c", " ", "z"]], "]"]], " ", SqrtBox[RowBox[List[RowBox[List["(", RowBox[List["a", "+", RowBox[List["b", " ", RowBox[List["Cos", "[", RowBox[List["2", " ", "c", " ", "z"]], "]"]]]]]], ")"]], " ", SuperscriptBox[RowBox[List["Sec", "[", FractionBox[RowBox[List["c", " ", "z"]], "2"], "]"]], "4"]]]]]]]], ")"]]]]]], ")"]]]], "/", RowBox[List["(", RowBox[List["3", " ", SqrtBox[RowBox[List["a", "-", "b"]]], " ", "b", " ", "c", " ", SqrtBox[RowBox[List["a", "+", RowBox[List["b", " ", RowBox[List["Cos", "[", RowBox[List["2", " ", "c", " ", "z"]], "]"]]]]]]]]], ")"]]]]]]]]

 MathML Form

 cos ( 2 c z ) a + b cos ( 2 c z ) tan ( c z ) z - ( ( a - 3 b ) a + b cos ( 2 c z ) ( cos ( c z ) + 1 ) 2 ( a - b ( a + b cos ( 2 c z ) ) sec 4 ( c z 2 ) cos ( c z ) + 2 b log ( 2 ( ( a - b ) tan 2 ( c z 2 ) + a - b + a - b ( a + b cos ( 2 c z ) ) sec 4 ( c z 2 ) ) a - b ( tan 2 ( c z 2 ) - 1 ) ) + a - b ( a + b cos ( 2 c z ) ) sec 4 ( c z 2 ) ) cos 2 ( c z 2 ) + a - b b 2 cos 2 ( 2 c z ) + a a - b b cos ( 2 c z ) + 2 a b tanh - 1 ( a + b cos ( 2 c z ) a - b ) a + b cos ( 2 c z ) ) / ( 3 a - b b c a + b cos ( 2 c z ) ) z 2 c z a b 2 c z 1 2 c z -1 a -1 3 b a b 2 c z c z 1 2 -1 1 2 a -1 b 1 2 a b 2 c z c z 2 -1 4 1 2 c z 2 b 2 a -1 b c z 2 -1 2 a -1 b a -1 b 1 2 a b 2 c z c z 2 -1 4 1 2 a -1 b 1 2 c z 2 -1 2 -1 -1 a -1 b 1 2 a b 2 c z c z 2 -1 4 1 2 c z 2 -1 2 a -1 b 1 2 b 2 2 c z 2 a a -1 b 1 2 b 2 c z 2 a b a b 2 c z 1 2 a -1 b 1 2 -1 a b 2 c z 1 2 3 a -1 b 1 2 b c a b 2 c z 1 2 -1 [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["\[Integral]", RowBox[List[RowBox[List[RowBox[List["Cos", "[", RowBox[List["2", " ", "c_", " ", "z_"]], "]"]], " ", SqrtBox[RowBox[List["a_", "+", RowBox[List["b_", " ", RowBox[List["Cos", "[", RowBox[List["2", " ", "c_", " ", "z_"]], "]"]]]]]]], " ", RowBox[List["Tan", "[", RowBox[List["c_", " ", "z_"]], "]"]]]], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List["-", FractionBox[RowBox[List[RowBox[List["a", " ", SqrtBox[RowBox[List["a", "-", "b"]]], " ", "b", " ", RowBox[List["Cos", "[", RowBox[List["2", " ", "c", " ", "z"]], "]"]]]], "+", RowBox[List[SqrtBox[RowBox[List["a", "-", "b"]]], " ", SuperscriptBox["b", "2"], " ", SuperscriptBox[RowBox[List["Cos", "[", RowBox[List["2", " ", "c", " ", "z"]], "]"]], "2"]]], "+", RowBox[List["2", " ", "a", " ", "b", " ", RowBox[List["ArcTanh", "[", FractionBox[SqrtBox[RowBox[List["a", "+", RowBox[List["b", " ", RowBox[List["Cos", "[", RowBox[List["2", " ", "c", " ", "z"]], "]"]]]]]]], SqrtBox[RowBox[List["a", "-", "b"]]]], "]"]], " ", SqrtBox[RowBox[List["a", "+", RowBox[List["b", " ", RowBox[List["Cos", "[", RowBox[List["2", " ", "c", " ", "z"]], "]"]]]]]]]]], "+", RowBox[List[RowBox[List["(", RowBox[List["a", "-", RowBox[List["3", " ", "b"]]]], ")"]], " ", SuperscriptBox[RowBox[List["Cos", "[", FractionBox[RowBox[List["c", " ", "z"]], "2"], "]"]], "2"], " ", SqrtBox[FractionBox[RowBox[List["a", "+", RowBox[List["b", " ", RowBox[List["Cos", "[", RowBox[List["2", " ", "c", " ", "z"]], "]"]]]]]], SuperscriptBox[RowBox[List["(", RowBox[List["1", "+", RowBox[List["Cos", "[", RowBox[List["c", " ", "z"]], "]"]]]], ")"]], "2"]]], " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "b", " ", RowBox[List["Log", "[", FractionBox[RowBox[List["2", " ", RowBox[List["(", RowBox[List["a", "-", "b", "+", RowBox[List[SqrtBox[RowBox[List["a", "-", "b"]]], " ", SqrtBox[RowBox[List[RowBox[List["(", RowBox[List["a", "+", RowBox[List["b", " ", RowBox[List["Cos", "[", RowBox[List["2", " ", "c", " ", "z"]], "]"]]]]]], ")"]], " ", SuperscriptBox[RowBox[List["Sec", "[", FractionBox[RowBox[List["c", " ", "z"]], "2"], "]"]], "4"]]]]]], "+", RowBox[List[RowBox[List["(", RowBox[List["a", "-", "b"]], ")"]], " ", SuperscriptBox[RowBox[List["Tan", "[", FractionBox[RowBox[List["c", " ", "z"]], "2"], "]"]], "2"]]]]], ")"]]]], RowBox[List[SqrtBox[RowBox[List["a", "-", "b"]]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", SuperscriptBox[RowBox[List["Tan", "[", FractionBox[RowBox[List["c", " ", "z"]], "2"], "]"]], "2"]]], ")"]]]]], "]"]]]], "+", RowBox[List[SqrtBox[RowBox[List["a", "-", "b"]]], " ", SqrtBox[RowBox[List[RowBox[List["(", RowBox[List["a", "+", RowBox[List["b", " ", RowBox[List["Cos", "[", RowBox[List["2", " ", "c", " ", "z"]], "]"]]]]]], ")"]], " ", SuperscriptBox[RowBox[List["Sec", "[", FractionBox[RowBox[List["c", " ", "z"]], "2"], "]"]], "4"]]]]]], "+", RowBox[List[SqrtBox[RowBox[List["a", "-", "b"]]], " ", RowBox[List["Cos", "[", RowBox[List["c", " ", "z"]], "]"]], " ", SqrtBox[RowBox[List[RowBox[List["(", RowBox[List["a", "+", RowBox[List["b", " ", RowBox[List["Cos", "[", RowBox[List["2", " ", "c", " ", "z"]], "]"]]]]]], ")"]], " ", SuperscriptBox[RowBox[List["Sec", "[", FractionBox[RowBox[List["c", " ", "z"]], "2"], "]"]], "4"]]]]]]]], ")"]]]]]], RowBox[List["3", " ", SqrtBox[RowBox[List["a", "-", "b"]]], " ", "b", " ", "c", " ", SqrtBox[RowBox[List["a", "+", RowBox[List["b", " ", RowBox[List["Cos", "[", RowBox[List["2", " ", "c", " ", "z"]], "]"]]]]]]]]]]]]]]]]

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

 2002-12-18