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 Cot

 http://functions.wolfram.com/01.09.21.0194.01

 Input Form

 Integrate[Cot[c z] Sqrt[(a + b Cot[c z]^2)^3], z] == -(Sqrt[2] Sqrt[(a + b Cot[c z]^2)^3] Csc[2 c z]^2 (Sqrt[2] Sqrt[(a - b) Cos[c z]^2] Sqrt[-a - b + (a - b) Cos[2 c z]] (-2 a + b + 2 (a - b) Cos[2 c z]) Sin[2 c z]^2 + 12 (a - b) ArcTan[(Sqrt[-a - b + (a - b) Cos[2 c z]] Cot[c z]^2 Sqrt[(a - b)^2 Sin[2 c z]^2])/(2 Sqrt[2] ((a - b) Cos[c z]^2)^ (3/2))] Cos[c z]^2 Sin[c z]^4 Sqrt[(a - b)^2 Sin[2 c z]^2]))/ (3 c Sqrt[(a - b) Cos[c z]^2] (-a - b + (a - b) Cos[2 c z])^(3/2))

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List["\[Integral]", RowBox[List[RowBox[List["Cot", "[", RowBox[List["c", " ", "z"]], "]"]], " ", SqrtBox[SuperscriptBox[RowBox[List["(", RowBox[List["a", "+", RowBox[List["b", " ", SuperscriptBox[RowBox[List["Cot", "[", RowBox[List["c", " ", "z"]], "]"]], "2"]]]]], ")"]], "3"]], RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", RowBox[List[RowBox[List["-", RowBox[List["(", RowBox[List[SqrtBox["2"], " ", SqrtBox[SuperscriptBox[RowBox[List["(", RowBox[List["a", "+", RowBox[List["b", " ", SuperscriptBox[RowBox[List["Cot", "[", RowBox[List["c", " ", "z"]], "]"]], "2"]]]]], ")"]], "3"]], " ", SuperscriptBox[RowBox[List["Csc", "[", RowBox[List["2", " ", "c", " ", "z"]], "]"]], "2"], " ", RowBox[List["(", RowBox[List[RowBox[List[SqrtBox["2"], " ", SqrtBox[RowBox[List[RowBox[List["(", RowBox[List["a", "-", "b"]], ")"]], " ", SuperscriptBox[RowBox[List["Cos", "[", RowBox[List["c", " ", "z"]], "]"]], "2"]]]], " ", SqrtBox[RowBox[List[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[RowBox[List["-", "2"]], " ", "a"]], "+", "b", "+", RowBox[List["2", " ", RowBox[List["(", RowBox[List["a", "-", "b"]], ")"]], " ", RowBox[List["Cos", "[", RowBox[List["2", " ", "c", " ", "z"]], "]"]]]]]], ")"]], " ", SuperscriptBox[RowBox[List["Sin", "[", RowBox[List["2", " ", "c", " ", "z"]], "]"]], "2"]]], "+", RowBox[List["12", " ", RowBox[List["(", RowBox[List["a", "-", "b"]], ")"]], " ", RowBox[List["ArcTan", "[", RowBox[List[RowBox[List["(", RowBox[List[SqrtBox[RowBox[List[RowBox[List["-", "a"]], "-", "b", "+", RowBox[List[RowBox[List["(", RowBox[List["a", "-", "b"]], ")"]], " ", RowBox[List["Cos", "[", RowBox[List["2", " ", "c", " ", "z"]], "]"]]]]]]], " ", SuperscriptBox[RowBox[List["Cot", "[", 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"]]]]]], ")"]], "/", RowBox[List["(", RowBox[List["2", " ", SqrtBox["2"], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["(", RowBox[List["a", "-", "b"]], ")"]], " ", SuperscriptBox[RowBox[List["Cos", "[", RowBox[List["c", " ", "z"]], "]"]], "2"]]], ")"]], RowBox[List["3", "/", "2"]]]]], ")"]]]], "]"]], " ", SuperscriptBox[RowBox[List["Cos", "[", RowBox[List["c", " ", "z"]], "]"]], "2"], " ", SuperscriptBox[RowBox[List["Sin", "[", RowBox[List["c", " ", "z"]], "]"]], "4"], " ", SqrtBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["a", "-", "b"]], ")"]], "2"], " ", SuperscriptBox[RowBox[List["Sin", "[", RowBox[List["2", " ", "c", " ", "z"]], "]"]], "2"]]]]]]]], ")"]]]], ")"]]]], "/", RowBox[List["(", RowBox[List["3", " ", "c", " ", SqrtBox[RowBox[List[RowBox[List["(", RowBox[List["a", "-", "b"]], ")"]], " ", SuperscriptBox[RowBox[List["Cos", "[", RowBox[List["c", " ", "z"]], "]"]], "2"]]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "a"]], "-", "b", "+", RowBox[List[RowBox[List["(", RowBox[List["a", "-", "b"]], ")"]], " ", RowBox[List["Cos", "[", RowBox[List["2", " ", "c", " ", "z"]], "]"]]]]]], ")"]], RowBox[List["3", "/", "2"]]]]], ")"]]]]]]]]

 MathML Form

 cot ( c z ) ( a + b cot 2 ( c z ) ) 3 z - ( 2 ( b cot 2 ( c z ) + a ) 3 csc 2 ( 2 c z ) ( 12 ( a - b ) tan - 1 ( - a - b + ( a - b ) cos ( 2 c z ) cot 2 ( c z ) ( a - b ) 2 sin 2 ( 2 c z ) 2 2 ( ( a - b ) cos 2 ( c z ) ) 3 / 2 ) cos 2 ( c z ) ( a - b ) 2 sin 2 ( 2 c z ) sin 4 ( c z ) + 2 ( a - b ) cos 2 ( c z ) - a - b + ( a - b ) cos ( 2 c z ) ( - 2 a + b + 2 ( a - b ) cos ( 2 c z ) ) sin 2 ( 2 c z ) ) ) / ( 3 c ( a - b ) cos 2 ( c z ) ( - a - b + ( a - b ) cos ( 2 c z ) ) 3 / 2 ) z c z a b c z 2 3 1 2 -1 2 1 2 b c z 2 a 3 1 2 2 c z 2 12 a -1 b -1 a -1 b a -1 b 2 c z 1 2 c z 2 a -1 b 2 2 c z 2 1 2 2 2 1 2 a -1 b c z 2 3 2 -1 c z 2 a -1 b 2 2 c z 2 1 2 c z 4 2 1 2 a -1 b c z 2 1 2 -1 a -1 b a -1 b 2 c z 1 2 -2 a b 2 a -1 b 2 c z 2 c z 2 3 c a -1 b c z 2 1 2 -1 a -1 b a -1 b 2 c z 3 2 -1 [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["\[Integral]", RowBox[List[RowBox[List[RowBox[List["Cot", "[", RowBox[List["c_", " ", "z_"]], "]"]], " ", SqrtBox[SuperscriptBox[RowBox[List["(", RowBox[List["a_", "+", RowBox[List["b_", " ", SuperscriptBox[RowBox[List["Cot", "[", RowBox[List["c_", " ", "z_"]], "]"]], "2"]]]]], ")"]], "3"]]]], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List["-", FractionBox[RowBox[List[SqrtBox["2"], " ", SqrtBox[SuperscriptBox[RowBox[List["(", RowBox[List["a", "+", RowBox[List["b", " ", SuperscriptBox[RowBox[List["Cot", "[", RowBox[List["c", " ", "z"]], "]"]], "2"]]]]], ")"]], "3"]], " ", SuperscriptBox[RowBox[List["Csc", "[", RowBox[List["2", " ", "c", " ", "z"]], "]"]], "2"], " ", RowBox[List["(", RowBox[List[RowBox[List[SqrtBox["2"], " ", SqrtBox[RowBox[List[RowBox[List["(", RowBox[List["a", "-", "b"]], ")"]], " ", SuperscriptBox[RowBox[List["Cos", "[", RowBox[List["c", " ", "z"]], "]"]], "2"]]]], " ", SqrtBox[RowBox[List[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[RowBox[List["-", "2"]], " ", "a"]], "+", "b", "+", RowBox[List["2", " ", RowBox[List["(", RowBox[List["a", "-", "b"]], ")"]], " ", RowBox[List["Cos", "[", RowBox[List["2", " ", "c", " ", "z"]], "]"]]]]]], ")"]], " ", SuperscriptBox[RowBox[List["Sin", "[", RowBox[List["2", " ", "c", " ", "z"]], "]"]], "2"]]], "+", RowBox[List["12", " ", RowBox[List["(", RowBox[List["a", "-", "b"]], ")"]], " ", RowBox[List["ArcTan", "[", FractionBox[RowBox[List[SqrtBox[RowBox[List[RowBox[List["-", "a"]], "-", "b", "+", RowBox[List[RowBox[List["(", RowBox[List["a", "-", "b"]], ")"]], " ", RowBox[List["Cos", "[", RowBox[List["2", " ", "c", " ", "z"]], "]"]]]]]]], " ", SuperscriptBox[RowBox[List["Cot", "[", 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"]]]]]], RowBox[List["2", " ", SqrtBox["2"], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["(", RowBox[List["a", "-", "b"]], ")"]], " ", SuperscriptBox[RowBox[List["Cos", "[", RowBox[List["c", " ", "z"]], "]"]], "2"]]], ")"]], RowBox[List["3", "/", "2"]]]]]], "]"]], " ", SuperscriptBox[RowBox[List["Cos", "[", RowBox[List["c", " ", "z"]], "]"]], "2"], " ", SuperscriptBox[RowBox[List["Sin", "[", RowBox[List["c", " ", "z"]], "]"]], "4"], " ", SqrtBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["a", "-", "b"]], ")"]], "2"], " ", SuperscriptBox[RowBox[List["Sin", "[", RowBox[List["2", " ", "c", " ", "z"]], "]"]], "2"]]]]]]]], ")"]]]], RowBox[List["3", " ", "c", " ", SqrtBox[RowBox[List[RowBox[List["(", RowBox[List["a", "-", "b"]], ")"]], " ", SuperscriptBox[RowBox[List["Cos", "[", RowBox[List["c", " ", "z"]], "]"]], "2"]]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "a"]], "-", "b", "+", RowBox[List[RowBox[List["(", RowBox[List["a", "-", "b"]], ")"]], " ", RowBox[List["Cos", "[", RowBox[List["2", " ", "c", " ", "z"]], "]"]]]]]], ")"]], RowBox[List["3", "/", "2"]]]]]]]]]]]]

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

 2002-12-18