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 Cot

 http://functions.wolfram.com/01.09.21.0191.01

 Input Form

 Integrate[1/Sqrt[(a + b Cot[c z]^2)^3], z] == (Csc[c z]^2 (-((2 (a - b) b (-a - b + (a - b) Cos[2 c z]) Cot[c z])/a) + (Sqrt[2] (-a - b + (a - b) Cos[2 c z])^(3/2) Csc[2 c z] (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]]) Sqrt[(a - b)^2 Sin[2 c z]^2])/Sqrt[(a - b) Sin[c z]^2]))/ (4 (a - b)^2 c Sqrt[(a + b Cot[c z]^2)^3])

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List["\[Integral]", RowBox[List[FractionBox["1", 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[SuperscriptBox[RowBox[List["Csc", "[", RowBox[List["c", " ", "z"]], "]"]], "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", FractionBox[RowBox[List["2", " ", RowBox[List["(", RowBox[List["a", "-", "b"]], ")"]], " ", "b", " ", 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["Cot", "[", RowBox[List["c", " ", "z"]], "]"]]]], "a"]]], "+", RowBox[List[RowBox[List["(", RowBox[List[SqrtBox["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"]]], " ", RowBox[List["Csc", "[", RowBox[List["2", " ", "c", " ", "z"]], "]"]], " ", 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[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[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[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"]]]]]]]], "]"]]]], ")"]], " ", SqrtBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["a", "-", "b"]], ")"]], "2"], " ", SuperscriptBox[RowBox[List["Sin", "[", RowBox[List["2", " ", "c", " ", "z"]], "]"]], "2"]]]]]], ")"]], "/", RowBox[List["(", SqrtBox[RowBox[List[RowBox[List["(", RowBox[List["a", "-", "b"]], ")"]], " ", SuperscriptBox[RowBox[List["Sin", "[", RowBox[List["c", " ", "z"]], "]"]], "2"]]]], ")"]]]]]], ")"]]]], ")"]], "/", RowBox[List["(", RowBox[List["4", " ", SuperscriptBox[RowBox[List["(", RowBox[List["a", "-", "b"]], ")"]], "2"], " ", "c", " ", SqrtBox[SuperscriptBox[RowBox[List["(", RowBox[List["a", "+", RowBox[List["b", " ", SuperscriptBox[RowBox[List["Cot", "[", RowBox[List["c", " ", "z"]], "]"]], "2"]]]]], ")"]], "3"]]]], ")"]]]]]]]]

 MathML Form

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

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["\[Integral]", RowBox[List[FractionBox["1", 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]", FractionBox[RowBox[List[SuperscriptBox[RowBox[List["Csc", "[", RowBox[List["c", " ", "z"]], "]"]], "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", FractionBox[RowBox[List["2", " ", RowBox[List["(", RowBox[List["a", "-", "b"]], ")"]], " ", "b", " ", 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["Cot", "[", RowBox[List["c", " ", "z"]], "]"]]]], "a"]]], "+", FractionBox[RowBox[List[SqrtBox["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"]]], " ", RowBox[List["Csc", "[", RowBox[List["2", " ", "c", " ", "z"]], "]"]], " ", 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[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[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[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"]]]]]]]], "]"]]]], ")"]], " ", SqrtBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["a", "-", "b"]], ")"]], "2"], " ", SuperscriptBox[RowBox[List["Sin", "[", RowBox[List["2", " ", "c", " ", "z"]], "]"]], "2"]]]]]], SqrtBox[RowBox[List[RowBox[List["(", RowBox[List["a", "-", "b"]], ")"]], " ", SuperscriptBox[RowBox[List["Sin", "[", RowBox[List["c", " ", "z"]], "]"]], "2"]]]]]]], ")"]]]], RowBox[List["4", " ", SuperscriptBox[RowBox[List["(", RowBox[List["a", "-", "b"]], ")"]], "2"], " ", "c", " ", SqrtBox[SuperscriptBox[RowBox[List["(", RowBox[List["a", "+", RowBox[List["b", " ", SuperscriptBox[RowBox[List["Cot", "[", RowBox[List["c", " ", "z"]], "]"]], "2"]]]]], ")"]], "3"]]]]]]]]]

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

 2002-12-18