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 Tan

 http://functions.wolfram.com/01.08.21.0123.01

 Input Form

 Integrate[Tan[c z] Sqrt[(a + b Tan[c z]^2)^5], z] == (1/(15 c)) (2 Cos[c z]^4 ((46 (a - b)^2 + 22 (a - b) b Sec[c z]^2 + 6 b^2 Sec[c z]^4)/(a + b + (a - b) Cos[2 c z])^2 - (15 Sqrt[2] Csc[2 c z]^2 (Log[-2 (a - b) Sin[c z]^2] - Log[(-a) Sqrt[a + b + (a - b) Cos[2 c z]] + b Sqrt[a + b + (a - b) Cos[2 c z]] + (a - b) Cos[2 c z] Sqrt[a + b + (a - b) Cos[2 c z]] + Sqrt[2] Sqrt[(a - b) Sin[c z]^2] Sqrt[(a - b)^2 Sin[2 c z]^2]]) ((a - b)^2 Sin[2 c z]^2)^(3/2))/ ((a + b + (a - b) Cos[2 c z])^(5/2) Sqrt[(a - b) Sin[c z]^2])) Sqrt[(a + b Tan[c z]^2)^5])

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List["\[Integral]", RowBox[List[RowBox[List["Tan", "[", RowBox[List["c", " ", "z"]], "]"]], " ", SqrtBox[SuperscriptBox[RowBox[List["(", RowBox[List["a", "+", RowBox[List["b", " ", SuperscriptBox[RowBox[List["Tan", "[", RowBox[List["c", " ", "z"]], "]"]], "2"]]]]], ")"]], "5"]], RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", RowBox[List[FractionBox["1", RowBox[List["15", " ", "c"]]], RowBox[List["(", RowBox[List["2", " ", SuperscriptBox[RowBox[List["Cos", "[", RowBox[List["c", " ", "z"]], "]"]], "4"], " ", RowBox[List["(", RowBox[List[FractionBox[RowBox[List[RowBox[List["46", " ", SuperscriptBox[RowBox[List["(", RowBox[List["a", "-", "b"]], ")"]], "2"]]], "+", RowBox[List["22", " ", RowBox[List["(", RowBox[List["a", "-", "b"]], ")"]], " ", "b", " ", SuperscriptBox[RowBox[List["Sec", "[", RowBox[List["c", " ", "z"]], "]"]], "2"]]], "+", RowBox[List["6", " ", SuperscriptBox["b", "2"], " ", SuperscriptBox[RowBox[List["Sec", "[", RowBox[List["c", " ", "z"]], "]"]], "4"]]]]], SuperscriptBox[RowBox[List["(", RowBox[List["a", "+", "b", "+", RowBox[List[RowBox[List["(", RowBox[List["a", "-", "b"]], ")"]], " ", RowBox[List["Cos", "[", RowBox[List["2", " ", "c", " ", "z"]], "]"]]]]]], ")"]], "2"]], "-", RowBox[List[RowBox[List["(", RowBox[List["15", " ", SqrtBox["2"], " ", SuperscriptBox[RowBox[List["Csc", "[", RowBox[List["2", " ", "c", " ", "z"]], "]"]], "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["Log", "[", RowBox[List[RowBox[List["-", "2"]], " ", RowBox[List["(", RowBox[List["a", "-", "b"]], ")"]], " ", SuperscriptBox[RowBox[List["Sin", "[", RowBox[List["c", " ", "z"]], "]"]], "2"]]], "]"]], "-", RowBox[List["Log", "[", RowBox[List[RowBox[List[RowBox[List["-", "a"]], " ", SqrtBox[RowBox[List["a", "+", "b", "+", RowBox[List[RowBox[List["(", RowBox[List["a", "-", "b"]], ")"]], " ", RowBox[List["Cos", "[", RowBox[List["2", " ", "c", " ", "z"]], "]"]]]]]]]]], "+", RowBox[List["b", " ", SqrtBox[RowBox[List["a", "+", "b", "+", RowBox[List[RowBox[List["(", RowBox[List["a", "-", "b"]], ")"]], " ", RowBox[List["Cos", "[", RowBox[List["2", " ", "c", " ", "z"]], "]"]]]]]]]]], "+", RowBox[List[RowBox[List["(", RowBox[List["a", "-", "b"]], ")"]], " ", RowBox[List["Cos", "[", RowBox[List["2", " ", "c", " ", "z"]], "]"]], " ", SqrtBox[RowBox[List["a", "+", "b", "+", RowBox[List[RowBox[List["(", RowBox[List["a", "-", "b"]], ")"]], " ", RowBox[List["Cos", "[", RowBox[List["2", " ", "c", " ", "z"]], "]"]]]]]]]]], "+", RowBox[List[SqrtBox["2"], " ", SqrtBox[RowBox[List[RowBox[List["(", RowBox[List["a", "-", "b"]], ")"]], " ", SuperscriptBox[RowBox[List["Sin", "[", RowBox[List["c", " ", "z"]], "]"]], "2"]]]], " ", SqrtBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["a", "-", "b"]], ")"]], "2"], " ", SuperscriptBox[RowBox[List["Sin", "[", RowBox[List["2", " ", "c", " ", "z"]], "]"]], "2"]]]]]]]], "]"]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["a", "-", "b"]], ")"]], "2"], " ", SuperscriptBox[RowBox[List["Sin", "[", RowBox[List["2", " ", "c", " ", "z"]], "]"]], "2"]]], ")"]], RowBox[List["3", "/", "2"]]]]], ")"]], "/", RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["a", "+", "b", "+", RowBox[List[RowBox[List["(", RowBox[List["a", "-", "b"]], ")"]], " ", RowBox[List["Cos", "[", RowBox[List["2", " ", "c", " ", "z"]], "]"]]]]]], ")"]], RowBox[List["5", "/", "2"]]], " ", SqrtBox[RowBox[List[RowBox[List["(", RowBox[List["a", "-", "b"]], ")"]], " ", SuperscriptBox[RowBox[List["Sin", "[", RowBox[List["c", " ", "z"]], "]"]], "2"]]]]]], ")"]]]]]], ")"]], " ", SqrtBox[SuperscriptBox[RowBox[List["(", RowBox[List["a", "+", RowBox[List["b", " ", SuperscriptBox[RowBox[List["Tan", "[", RowBox[List["c", " ", "z"]], "]"]], "2"]]]]], ")"]], "5"]]]], ")"]]]]]]]]

 MathML Form

 tan ( c z ) ( a + b tan 2 ( c z ) ) 5 z 1 15 c ( 2 cos 4 ( c z ) ( 6 b 2 sec 4 ( c z ) + 22 ( a - b ) b sec 2 ( c z ) + 46 ( a - b ) 2 ( a + b + ( a - b ) cos ( 2 c z ) ) 2 - ( 15 2 csc 2 ( 2 c z ) ( log ( - 2 ( a - b ) sin 2 ( c z ) ) - log ( - a a + b + ( a - b ) cos ( 2 c z ) + b a + b + ( a - b ) cos ( 2 c z ) + ( a - b ) cos ( 2 c z ) a + b + ( a - b ) cos ( 2 c z ) + 2 ( a - b ) sin 2 ( c z ) ( a - b ) 2 sin 2 ( 2 c z ) ) ) ( ( a - b ) 2 sin 2 ( 2 c z ) ) 3 / 2 ) / ( ( a + b + ( a - b ) cos ( 2 c z ) ) 5 / 2 ( a - b ) sin 2 ( c z ) ) ) ( b tan 2 ( c z ) + a ) 5 ) z c z a b c z 2 5 1 2 1 15 c -1 2 c z 4 6 b 2 c z 4 22 a -1 b b c z 2 46 a -1 b 2 a b a -1 b 2 c z 2 -1 -1 15 2 1 2 2 c z 2 -2 a -1 b c z 2 -1 -1 a a b a -1 b 2 c z 1 2 b a b a -1 b 2 c z 1 2 a -1 b 2 c z a b a -1 b 2 c z 1 2 2 1 2 a -1 b c z 2 1 2 a -1 b 2 2 c z 2 1 2 a -1 b 2 2 c z 2 3 2 a b a -1 b 2 c z 5 2 a -1 b c z 2 1 2 -1 b c z 2 a 5 1 2 [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["\[Integral]", RowBox[List[RowBox[List[RowBox[List["Tan", "[", RowBox[List["c_", " ", "z_"]], "]"]], " ", SqrtBox[SuperscriptBox[RowBox[List["(", RowBox[List["a_", "+", RowBox[List["b_", " ", SuperscriptBox[RowBox[List["Tan", "[", RowBox[List["c_", " ", "z_"]], "]"]], "2"]]]]], ")"]], "5"]]]], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", FractionBox[RowBox[List["2", " ", SuperscriptBox[RowBox[List["Cos", "[", RowBox[List["c", " ", "z"]], "]"]], "4"], " ", RowBox[List["(", RowBox[List[FractionBox[RowBox[List[RowBox[List["46", " ", SuperscriptBox[RowBox[List["(", RowBox[List["a", "-", "b"]], ")"]], "2"]]], "+", RowBox[List["22", " ", RowBox[List["(", RowBox[List["a", "-", "b"]], ")"]], " ", "b", " ", SuperscriptBox[RowBox[List["Sec", "[", RowBox[List["c", " ", "z"]], "]"]], "2"]]], "+", RowBox[List["6", " ", SuperscriptBox["b", "2"], " ", SuperscriptBox[RowBox[List["Sec", "[", RowBox[List["c", " ", "z"]], "]"]], "4"]]]]], SuperscriptBox[RowBox[List["(", RowBox[List["a", "+", "b", "+", RowBox[List[RowBox[List["(", RowBox[List["a", "-", "b"]], ")"]], " ", RowBox[List["Cos", "[", RowBox[List["2", " ", "c", " ", "z"]], "]"]]]]]], ")"]], "2"]], "-", FractionBox[RowBox[List["15", " ", SqrtBox["2"], " ", SuperscriptBox[RowBox[List["Csc", "[", RowBox[List["2", " ", "c", " ", "z"]], "]"]], "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["Log", "[", RowBox[List[RowBox[List["-", "2"]], " ", RowBox[List["(", RowBox[List["a", "-", "b"]], ")"]], " ", SuperscriptBox[RowBox[List["Sin", "[", RowBox[List["c", " ", "z"]], "]"]], "2"]]], "]"]], "-", RowBox[List["Log", "[", RowBox[List[RowBox[List[RowBox[List["-", "a"]], " ", SqrtBox[RowBox[List["a", "+", "b", "+", RowBox[List[RowBox[List["(", RowBox[List["a", "-", "b"]], ")"]], " ", RowBox[List["Cos", "[", RowBox[List["2", " ", "c", " ", "z"]], "]"]]]]]]]]], "+", RowBox[List["b", " ", SqrtBox[RowBox[List["a", "+", "b", "+", RowBox[List[RowBox[List["(", RowBox[List["a", "-", "b"]], ")"]], " ", RowBox[List["Cos", "[", RowBox[List["2", " ", "c", " ", "z"]], "]"]]]]]]]]], "+", RowBox[List[RowBox[List["(", RowBox[List["a", "-", "b"]], ")"]], " ", RowBox[List["Cos", "[", RowBox[List["2", " ", "c", " ", "z"]], "]"]], " ", SqrtBox[RowBox[List["a", "+", "b", "+", RowBox[List[RowBox[List["(", RowBox[List["a", "-", "b"]], ")"]], " ", RowBox[List["Cos", "[", RowBox[List["2", " ", "c", " ", "z"]], "]"]]]]]]]]], "+", RowBox[List[SqrtBox["2"], " ", SqrtBox[RowBox[List[RowBox[List["(", RowBox[List["a", "-", "b"]], ")"]], " ", SuperscriptBox[RowBox[List["Sin", "[", RowBox[List["c", " ", "z"]], "]"]], "2"]]]], " ", SqrtBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["a", "-", "b"]], ")"]], "2"], " ", SuperscriptBox[RowBox[List["Sin", "[", RowBox[List["2", " ", "c", " ", "z"]], "]"]], "2"]]]]]]]], "]"]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["a", "-", "b"]], ")"]], "2"], " ", SuperscriptBox[RowBox[List["Sin", "[", RowBox[List["2", " ", "c", " ", "z"]], "]"]], "2"]]], ")"]], RowBox[List["3", "/", "2"]]]]], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["a", "+", "b", "+", RowBox[List[RowBox[List["(", RowBox[List["a", "-", "b"]], ")"]], " ", RowBox[List["Cos", "[", RowBox[List["2", " ", "c", " ", "z"]], "]"]]]]]], ")"]], RowBox[List["5", "/", "2"]]], " ", SqrtBox[RowBox[List[RowBox[List["(", RowBox[List["a", "-", "b"]], ")"]], " ", SuperscriptBox[RowBox[List["Sin", "[", RowBox[List["c", " ", "z"]], "]"]], "2"]]]]]]]]], ")"]], " ", SqrtBox[SuperscriptBox[RowBox[List["(", RowBox[List["a", "+", RowBox[List["b", " ", SuperscriptBox[RowBox[List["Tan", "[", RowBox[List["c", " ", "z"]], "]"]], "2"]]]]], ")"]], "5"]]]], RowBox[List["15", " ", "c"]]]]]]]

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

 2002-12-18