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 Tan

 http://functions.wolfram.com/01.08.21.0054.01

 Input Form

 Integrate[Cos[2 c z] (a + b Cos[2 c z])^\[Beta] Tan[c z], z] == (1/(2 b c \[Beta] (1 + \[Beta]))) (((a + b Cos[2 c z])^\[Beta] (b (1 + \[Beta]) ((a + b Cos[2 c z])/(a - b))^ \[Beta] Hypergeometric2F1[-\[Beta], -\[Beta], 1 - \[Beta], -(((a - b) Sec[c z]^2)/(2 b))] - \[Beta] (b + b Cos[2 c z] ((a + b Cos[2 c z])/(a - b))^\[Beta] + a (-1 + ((a + b Cos[2 c z])/(a - b))^\[Beta])) (1 + ((a - b) Sec[c z]^2)/(2 b))^\[Beta]))/ (((a + b Cos[2 c z])/(a - b))^\[Beta] (1 + ((a - b) Sec[c z]^2)/(2 b))^ \[Beta]))

 Standard Form

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 MathML Form

 cos ( 2 c z ) ( a + b cos ( 2 c z ) ) β tan ( c z ) z 1 2 b c β ( β + 1 ) ( ( a + b cos ( 2 c z ) ) β ( a + b cos ( 2 c z ) a - b ) - β ( ( a - b ) sec 2 ( c z ) 2 b + 1 ) - β ( b ( β + 1 ) ( a + b cos ( 2 c z ) a - b ) β 2 F 1 ( - β , - β ; 1 - β ; - ( a - b ) sec 2 ( c z ) 2 b ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["2", TraditionalForm]], SubscriptBox["F", FormBox["1", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox[RowBox[List["-", "\[Beta]"]], Hypergeometric2F1], ",", TagBox[RowBox[List["-", "\[Beta]"]], Hypergeometric2F1]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1], ";", TagBox[TagBox[TagBox[RowBox[List["1", "-", "\[Beta]"]], Hypergeometric2F1], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1], ";", TagBox[RowBox[List["-", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["a", "-", "b"]], ")"]], " ", RowBox[List[SuperscriptBox["sec", "2"], "(", RowBox[List["c", " ", "z"]], ")"]]]], RowBox[List["2", " ", "b"]]]]], Hypergeometric2F1]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]]], Hypergeometric2F1] - β ( b cos ( 2 c z ) ( a + b cos ( 2 c z ) a - b ) β + a ( ( a + b cos ( 2 c z ) a - b ) β - 1 ) + b ) ( ( a - b ) sec 2 ( c z ) 2 b + 1 ) β ) ) z 2 c z a b 2 c z β c z 1 2 b c β β 1 -1 a b 2 c z β a b 2 c z a -1 b -1 -1 β a -1 b c z 2 2 b -1 1 -1 β b β 1 a b 2 c z a -1 b -1 β Hypergeometric2F1 -1 β -1 β 1 -1 β -1 a -1 b c z 2 2 b -1 -1 β b 2 c z a b 2 c z a -1 b -1 β a a b 2 c z a -1 b -1 β -1 b a -1 b c z 2 2 b -1 1 β [/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_"]], "]"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["a_", "+", RowBox[List["b_", " ", RowBox[List["Cos", "[", RowBox[List["2", " ", "c_", " ", "z_"]], "]"]]]]]], ")"]], "\[Beta]_"], " ", RowBox[List["Tan", "[", RowBox[List["c_", " ", "z_"]], "]"]]]], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["a", "+", RowBox[List["b", " ", RowBox[List["Cos", "[", RowBox[List["2", " ", "c", " ", "z"]], "]"]]]]]], ")"]], "\[Beta]"], " ", SuperscriptBox[RowBox[List["(", FractionBox[RowBox[List["a", "+", RowBox[List["b", " ", RowBox[List["Cos", "[", RowBox[List["2", " ", "c", " ", "z"]], "]"]]]]]], RowBox[List["a", "-", "b"]]], ")"]], RowBox[List["-", "\[Beta]"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["a", "-", "b"]], ")"]], " ", SuperscriptBox[RowBox[List["Sec", "[", RowBox[List["c", " ", "z"]], "]"]], "2"]]], RowBox[List["2", " ", "b"]]]]], ")"]], RowBox[List["-", "\[Beta]"]]], " ", RowBox[List["(", RowBox[List[RowBox[List["b", " ", RowBox[List["(", RowBox[List["1", "+", "\[Beta]"]], ")"]], " ", SuperscriptBox[RowBox[List["(", FractionBox[RowBox[List["a", "+", RowBox[List["b", " ", RowBox[List["Cos", "[", RowBox[List["2", " ", "c", " ", "z"]], "]"]]]]]], RowBox[List["a", "-", "b"]]], ")"]], "\[Beta]"], " ", RowBox[List["Hypergeometric2F1", "[", RowBox[List[RowBox[List["-", "\[Beta]"]], ",", RowBox[List["-", "\[Beta]"]], ",", RowBox[List["1", "-", "\[Beta]"]], ",", RowBox[List["-", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["a", "-", "b"]], ")"]], " ", SuperscriptBox[RowBox[List["Sec", "[", RowBox[List["c", " ", "z"]], "]"]], "2"]]], RowBox[List["2", " ", "b"]]]]]]], "]"]]]], "-", RowBox[List["\[Beta]", " ", RowBox[List["(", RowBox[List["b", "+", RowBox[List["b", " ", RowBox[List["Cos", "[", RowBox[List["2", " ", "c", " ", "z"]], "]"]], " ", SuperscriptBox[RowBox[List["(", FractionBox[RowBox[List["a", "+", RowBox[List["b", " ", RowBox[List["Cos", "[", RowBox[List["2", " ", "c", " ", "z"]], "]"]]]]]], RowBox[List["a", "-", "b"]]], ")"]], "\[Beta]"]]], "+", RowBox[List["a", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", SuperscriptBox[RowBox[List["(", FractionBox[RowBox[List["a", "+", RowBox[List["b", " ", RowBox[List["Cos", "[", RowBox[List["2", " ", "c", " ", "z"]], "]"]]]]]], RowBox[List["a", "-", "b"]]], ")"]], "\[Beta]"]]], ")"]]]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["a", "-", "b"]], ")"]], " ", SuperscriptBox[RowBox[List["Sec", "[", RowBox[List["c", " ", "z"]], "]"]], "2"]]], RowBox[List["2", " ", "b"]]]]], ")"]], "\[Beta]"]]]]], ")"]]]], RowBox[List["2", " ", "b", " ", "c", " ", "\[Beta]", " ", RowBox[List["(", RowBox[List["1", "+", "\[Beta]"]], ")"]]]]]]]]]

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

 2002-12-18