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 Cot

 http://functions.wolfram.com/01.09.21.0167.01

 Input Form

 Integrate[(A + B Cot[z])/(a + b Cot[z])^3, z] == ((A + B Cot[z]) Csc[z]^2 (b Cos[z] + a Sin[z]) (b^2 (a^2 + b^2) (A b - a B) + 2 (a^2 + b^2) (3 a A b - 2 a^2 B + b^2 B) Sin[z] (b Cos[z] + a Sin[z]) + 2 (a^3 A - 3 a A b^2 + 3 a^2 b B - b^3 B) z (b Cos[z] + a Sin[z])^2 + 2 (-3 a^2 A b + A b^3 + a^3 B - 3 a b^2 B) Log[b Cos[z] + a Sin[z]] (b Cos[z] + a Sin[z])^2))/ (2 (a^2 + b^2)^3 (a + b Cot[z])^3 (B Cos[z] + A Sin[z]))

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List["\[Integral]", RowBox[List[FractionBox[RowBox[List["A", "+", RowBox[List["B", " ", RowBox[List["Cot", "[", "z", "]"]]]]]], SuperscriptBox[RowBox[List["(", RowBox[List["a", "+", RowBox[List["b", " ", RowBox[List["Cot", "[", "z", "]"]]]]]], ")"]], "3"]], RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["(", RowBox[List["A", "+", RowBox[List["B", " ", RowBox[List["Cot", "[", "z", "]"]]]]]], ")"]], " ", SuperscriptBox[RowBox[List["Csc", "[", "z", "]"]], "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["b", " ", RowBox[List["Cos", "[", "z", "]"]]]], "+", RowBox[List["a", " ", RowBox[List["Sin", "[", "z", "]"]]]]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox["b", "2"], " ", RowBox[List["(", RowBox[List[SuperscriptBox["a", "2"], "+", SuperscriptBox["b", "2"]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["A", " ", "b"]], "-", RowBox[List["a", " ", "B"]]]], ")"]]]], "+", RowBox[List["2", " ", RowBox[List["(", RowBox[List[SuperscriptBox["a", "2"], "+", SuperscriptBox["b", "2"]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["3", " ", "a", " ", "A", " ", "b"]], "-", RowBox[List["2", " ", SuperscriptBox["a", "2"], " ", "B"]], "+", RowBox[List[SuperscriptBox["b", "2"], " ", "B"]]]], ")"]], " ", RowBox[List["Sin", "[", "z", "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["b", " ", RowBox[List["Cos", "[", "z", "]"]]]], "+", RowBox[List["a", " ", RowBox[List["Sin", "[", "z", "]"]]]]]], ")"]]]], "+", RowBox[List["2", " ", RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox["a", "3"], " ", "A"]], "-", RowBox[List["3", " ", "a", " ", "A", " ", SuperscriptBox["b", "2"]]], "+", RowBox[List["3", " ", SuperscriptBox["a", "2"], " ", "b", " ", "B"]], "-", RowBox[List[SuperscriptBox["b", "3"], " ", "B"]]]], ")"]], " ", "z", " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["b", " ", RowBox[List["Cos", "[", "z", "]"]]]], "+", RowBox[List["a", " ", RowBox[List["Sin", "[", "z", "]"]]]]]], ")"]], "2"]]], "+", RowBox[List["2", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "3"]], " ", SuperscriptBox["a", "2"], " ", "A", " ", "b"]], "+", RowBox[List["A", " ", SuperscriptBox["b", "3"]]], "+", RowBox[List[SuperscriptBox["a", "3"], " ", "B"]], "-", RowBox[List["3", " ", "a", " ", SuperscriptBox["b", "2"], " ", "B"]]]], ")"]], " ", RowBox[List["Log", "[", RowBox[List[RowBox[List["b", " ", RowBox[List["Cos", "[", "z", "]"]]]], "+", RowBox[List["a", " ", RowBox[List["Sin", "[", "z", "]"]]]]]], "]"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["b", " ", RowBox[List["Cos", "[", "z", "]"]]]], "+", RowBox[List["a", " ", RowBox[List["Sin", "[", "z", "]"]]]]]], ")"]], "2"]]]]], ")"]]]], ")"]], "/", RowBox[List["(", RowBox[List["2", " ", SuperscriptBox[RowBox[List["(", RowBox[List[SuperscriptBox["a", "2"], "+", SuperscriptBox["b", "2"]]], ")"]], "3"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["a", "+", RowBox[List["b", " ", RowBox[List["Cot", "[", "z", "]"]]]]]], ")"]], "3"], " ", RowBox[List["(", RowBox[List[RowBox[List["B", " ", RowBox[List["Cos", "[", "z", "]"]]]], "+", RowBox[List["A", " ", RowBox[List["Sin", "[", "z", "]"]]]]]], ")"]]]], ")"]]]]]]]]

 MathML Form

 A + B cot ( z ) ( a + b cot ( z ) ) 3 z ( ( A + B cot ( z ) ) csc 2 ( z ) ( b cos ( z ) + a sin ( z ) ) ( ( a 2 + b 2 ) ( A b - a B ) b 2 + 2 ( A a 3 + 3 b B a 2 - 3 A b 2 a - b 3 B ) z ( b cos ( z ) + a sin ( z ) ) 2 + 2 ( B a 3 - 3 A b a 2 - 3 b 2 B a + A b 3 ) log ( b cos ( z ) + a sin ( z ) ) ( b cos ( z ) + a sin ( z ) ) 2 + 2 ( a 2 + b 2 ) ( - 2 B a 2 + 3 A b a + b 2 B ) sin ( z ) ( b cos ( z ) + a sin ( z ) ) ) ) / ( 2 ( a 2 + b 2 ) 3 ( a + b cot ( z ) ) 3 ( B cos ( z ) + A sin ( z ) ) ) z A B z a b z 3 -1 A B z z 2 b z a z a 2 b 2 A b -1 a B b 2 2 A a 3 3 b B a 2 -1 3 A b 2 a -1 b 3 B z b z a z 2 2 B a 3 -1 3 A b a 2 -1 3 b 2 B a A b 3 b z a z b z a z 2 2 a 2 b 2 -2 B a 2 3 A b a b 2 B z b z a z 2 a 2 b 2 3 a b z 3 B z A z -1 [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["\[Integral]", RowBox[List[FractionBox[RowBox[List["A_", "+", RowBox[List["B_", " ", RowBox[List["Cot", "[", "z_", "]"]]]]]], SuperscriptBox[RowBox[List["(", RowBox[List["a_", "+", RowBox[List["b_", " ", RowBox[List["Cot", "[", "z_", "]"]]]]]], ")"]], "3"]], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["A", "+", RowBox[List["B", " ", RowBox[List["Cot", "[", "z", "]"]]]]]], ")"]], " ", SuperscriptBox[RowBox[List["Csc", "[", "z", "]"]], "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["b", " ", RowBox[List["Cos", "[", "z", "]"]]]], "+", RowBox[List["a", " ", RowBox[List["Sin", "[", "z", "]"]]]]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox["b", "2"], " ", RowBox[List["(", RowBox[List[SuperscriptBox["a", "2"], "+", SuperscriptBox["b", "2"]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["A", " ", "b"]], "-", RowBox[List["a", " ", "B"]]]], ")"]]]], "+", RowBox[List["2", " ", RowBox[List["(", RowBox[List[SuperscriptBox["a", "2"], "+", SuperscriptBox["b", "2"]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["3", " ", "a", " ", "A", " ", "b"]], "-", RowBox[List["2", " ", SuperscriptBox["a", "2"], " ", "B"]], "+", RowBox[List[SuperscriptBox["b", "2"], " ", "B"]]]], ")"]], " ", RowBox[List["Sin", "[", "z", "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["b", " ", RowBox[List["Cos", "[", "z", "]"]]]], "+", RowBox[List["a", " ", RowBox[List["Sin", "[", "z", "]"]]]]]], ")"]]]], "+", RowBox[List["2", " ", RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox["a", "3"], " ", "A"]], "-", RowBox[List["3", " ", "a", " ", "A", " ", SuperscriptBox["b", "2"]]], "+", RowBox[List["3", " ", SuperscriptBox["a", "2"], " ", "b", " ", "B"]], "-", RowBox[List[SuperscriptBox["b", "3"], " ", "B"]]]], ")"]], " ", "z", " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["b", " ", RowBox[List["Cos", "[", "z", "]"]]]], "+", RowBox[List["a", " ", RowBox[List["Sin", "[", "z", "]"]]]]]], ")"]], "2"]]], "+", RowBox[List["2", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "3"]], " ", SuperscriptBox["a", "2"], " ", "A", " ", "b"]], "+", RowBox[List["A", " ", SuperscriptBox["b", "3"]]], "+", RowBox[List[SuperscriptBox["a", "3"], " ", "B"]], "-", RowBox[List["3", " ", "a", " ", SuperscriptBox["b", "2"], " ", "B"]]]], ")"]], " ", RowBox[List["Log", "[", RowBox[List[RowBox[List["b", " ", RowBox[List["Cos", "[", "z", "]"]]]], "+", RowBox[List["a", " ", RowBox[List["Sin", "[", "z", "]"]]]]]], "]"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["b", " ", RowBox[List["Cos", "[", "z", "]"]]]], "+", RowBox[List["a", " ", RowBox[List["Sin", "[", "z", "]"]]]]]], ")"]], "2"]]]]], ")"]]]], RowBox[List["2", " ", SuperscriptBox[RowBox[List["(", RowBox[List[SuperscriptBox["a", "2"], "+", SuperscriptBox["b", "2"]]], ")"]], "3"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["a", "+", RowBox[List["b", " ", RowBox[List["Cot", "[", "z", "]"]]]]]], ")"]], "3"], " ", RowBox[List["(", RowBox[List[RowBox[List["B", " ", RowBox[List["Cos", "[", "z", "]"]]]], "+", RowBox[List["A", " ", RowBox[List["Sin", "[", "z", "]"]]]]]], ")"]]]]]]]]]

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

 2002-12-18