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 Cot

 http://functions.wolfram.com/01.09.21.0203.01

 Input Form

 Integrate[Cot[c z]/(a + b Sqrt[Cot[c z]])^2, z] == (-(1/(2 (a^4 + b^4)^2 c))) (-2 b (Sqrt[2] a^5 - 3 a^4 b + 2 Sqrt[2] a^3 b^2 - Sqrt[2] a b^4 + b^5) ArcTan[1 - Sqrt[2] Sqrt[Cot[c z]]] - 2 b ((-Sqrt[2]) a^5 - 3 a^4 b - 2 Sqrt[2] a^3 b^2 + Sqrt[2] a b^4 + b^5) ArcTan[1 + Sqrt[2] Sqrt[Cot[c z]]] + (4 a^3 (a^4 + b^4))/ (a + b Sqrt[Cot[c z]]) - 4 (a^6 - 3 a^2 b^4) Log[a + b Sqrt[Cot[c z]]] - Sqrt[2] a b (a^4 - 2 a^2 b^2 - b^4) Log[-1 + Sqrt[2] Sqrt[Cot[c z]] - Cot[c z]] + Sqrt[2] a b (a^4 - 2 a^2 b^2 - b^4) Log[1 + Sqrt[2] Sqrt[Cot[c z]] + Cot[c z]] + (a^6 - 3 a^2 b^4) Log[1 + Cot[c z]^2])

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List["\[Integral]", RowBox[List[FractionBox[RowBox[List["Cot", "[", RowBox[List["c", " ", "z"]], "]"]], SuperscriptBox[RowBox[List["(", RowBox[List["a", "+", RowBox[List["b", SqrtBox[RowBox[List["Cot", "[", RowBox[List["c", " ", "z"]], "]"]]]]]]], ")"]], "2"]], RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", RowBox[List[RowBox[List["-", FractionBox["1", RowBox[List["2", " ", SuperscriptBox[RowBox[List["(", RowBox[List[SuperscriptBox["a", "4"], "+", SuperscriptBox["b", "4"]]], ")"]], "2"], " ", "c"]]]]], RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "b", " ", RowBox[List["(", RowBox[List[RowBox[List[SqrtBox["2"], " ", SuperscriptBox["a", "5"]]], "-", RowBox[List["3", " ", SuperscriptBox["a", "4"], " ", "b"]], "+", RowBox[List["2", " ", SqrtBox["2"], " ", SuperscriptBox["a", "3"], " ", SuperscriptBox["b", "2"]]], "-", RowBox[List[SqrtBox["2"], " ", "a", " ", SuperscriptBox["b", "4"]]], "+", SuperscriptBox["b", "5"]]], ")"]], " ", RowBox[List["ArcTan", "[", RowBox[List["1", "-", RowBox[List[SqrtBox["2"], " ", SqrtBox[RowBox[List["Cot", "[", RowBox[List["c", " ", "z"]], "]"]]]]]]], "]"]]]], "-", RowBox[List["2", " ", "b", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", SqrtBox["2"]]], " ", SuperscriptBox["a", "5"]]], "-", RowBox[List["3", " ", SuperscriptBox["a", "4"], " ", "b"]], "-", RowBox[List["2", " ", SqrtBox["2"], " ", SuperscriptBox["a", "3"], " ", SuperscriptBox["b", "2"]]], "+", RowBox[List[SqrtBox["2"], " ", "a", " ", SuperscriptBox["b", "4"]]], "+", SuperscriptBox["b", "5"]]], ")"]], " ", RowBox[List["ArcTan", "[", RowBox[List["1", "+", RowBox[List[SqrtBox["2"], " ", SqrtBox[RowBox[List["Cot", "[", RowBox[List["c", " ", "z"]], "]"]]]]]]], "]"]]]], "+", FractionBox[RowBox[List["4", " ", SuperscriptBox["a", "3"], " ", RowBox[List["(", RowBox[List[SuperscriptBox["a", "4"], "+", SuperscriptBox["b", "4"]]], ")"]]]], RowBox[List["a", "+", RowBox[List["b", " ", SqrtBox[RowBox[List["Cot", "[", RowBox[List["c", " ", "z"]], "]"]]]]]]]], "-", RowBox[List["4", " ", RowBox[List["(", RowBox[List[SuperscriptBox["a", "6"], "-", RowBox[List["3", " ", SuperscriptBox["a", "2"], " ", SuperscriptBox["b", "4"]]]]], ")"]], " ", RowBox[List["Log", "[", RowBox[List["a", "+", RowBox[List["b", " ", SqrtBox[RowBox[List["Cot", "[", RowBox[List["c", " ", "z"]], "]"]]]]]]], "]"]]]], "-", RowBox[List[SqrtBox["2"], " ", "a", " ", "b", " ", RowBox[List["(", RowBox[List[SuperscriptBox["a", "4"], "-", RowBox[List["2", " ", SuperscriptBox["a", "2"], " ", SuperscriptBox["b", "2"]]], "-", SuperscriptBox["b", "4"]]], ")"]], " ", RowBox[List["Log", "[", RowBox[List[RowBox[List["-", "1"]], "+", RowBox[List[SqrtBox["2"], " ", SqrtBox[RowBox[List["Cot", "[", RowBox[List["c", " ", "z"]], "]"]]]]], "-", RowBox[List["Cot", "[", RowBox[List["c", " ", "z"]], "]"]]]], "]"]]]], "+", RowBox[List[SqrtBox["2"], " ", "a", " ", "b", " ", RowBox[List["(", RowBox[List[SuperscriptBox["a", "4"], "-", RowBox[List["2", " ", SuperscriptBox["a", "2"], " ", SuperscriptBox["b", "2"]]], "-", SuperscriptBox["b", "4"]]], ")"]], " ", RowBox[List["Log", "[", RowBox[List["1", "+", RowBox[List[SqrtBox["2"], " ", SqrtBox[RowBox[List["Cot", "[", RowBox[List["c", " ", "z"]], "]"]]]]], "+", RowBox[List["Cot", "[", RowBox[List["c", " ", "z"]], "]"]]]], "]"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["a", "6"], "-", RowBox[List["3", " ", SuperscriptBox["a", "2"], " ", SuperscriptBox["b", "4"]]]]], ")"]], " ", RowBox[List["Log", "[", RowBox[List["1", "+", SuperscriptBox[RowBox[List["Cot", "[", RowBox[List["c", " ", "z"]], "]"]], "2"]]], "]"]]]]]], ")"]]]]]]]]

 MathML Form

 cot ( c z ) ( a + b cot 1 2 ( c z ) ) 2 z - 1 2 ( a 4 + b 4 ) 2 c ( 4 ( a 4 + b 4 ) a 3 a + b cot 1 2 ( c z ) - 2 b ( a 4 - 2 b 2 a 2 - b 4 ) log ( - cot ( c z ) + 2 cot 1 2 ( c z ) - 1 ) a + 2 b ( a 4 - 2 b 2 a 2 - b 4 ) log ( cot ( c z ) + 2 cot 1 2 ( c z ) + 1 ) a - 2 b ( - 2 a 5 - 3 b a 4 - 2 2 b 2 a 3 + 2 b 4 a + b 5 ) tan - 1 ( 2 cot 1 2 ( c z ) + 1 ) - 2 b ( 2 a 5 - 3 b a 4 + 2 2 b 2 a 3 - 2 b 4 a + b 5 ) tan - 1 ( 1 - 2 cot 1 2 ( c z ) ) - 4 ( a 6 - 3 a 2 b 4 ) log ( a + b cot 1 2 ( c z ) ) + ( a 6 - 3 a 2 b 4 ) log ( cot 2 ( c z ) + 1 ) ) z c z a b c z 1 2 2 -1 -1 1 2 a 4 b 4 2 c -1 4 a 4 b 4 a 3 a b c z 1 2 -1 -1 2 1 2 b a 4 -1 2 b 2 a 2 -1 b 4 -1 c z 2 1 2 c z 1 2 -1 a 2 1 2 b a 4 -1 2 b 2 a 2 -1 b 4 c z 2 1 2 c z 1 2 1 a -1 2 b -1 2 1 2 a 5 -1 3 b a 4 -1 2 2 1 2 b 2 a 3 2 1 2 b 4 a b 5 2 1 2 c z 1 2 1 -1 2 b 2 1 2 a 5 -1 3 b a 4 2 2 1 2 b 2 a 3 -1 2 1 2 b 4 a b 5 1 -1 2 1 2 c z 1 2 -1 4 a 6 -1 3 a 2 b 4 a b c z 1 2 a 6 -1 3 a 2 b 4 c z 2 1 [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["\[Integral]", RowBox[List[FractionBox[RowBox[List["Cot", "[", RowBox[List["c_", " ", "z_"]], "]"]], SuperscriptBox[RowBox[List["(", RowBox[List["a_", "+", RowBox[List["b_", " ", SqrtBox[RowBox[List["Cot", "[", RowBox[List["c_", " ", "z_"]], "]"]]]]]]], ")"]], "2"]], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List["-", FractionBox[RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "b", " ", RowBox[List["(", RowBox[List[RowBox[List[SqrtBox["2"], " ", SuperscriptBox["a", "5"]]], "-", RowBox[List["3", " ", SuperscriptBox["a", "4"], " ", "b"]], "+", RowBox[List["2", " ", SqrtBox["2"], " ", SuperscriptBox["a", "3"], " ", SuperscriptBox["b", "2"]]], "-", RowBox[List[SqrtBox["2"], " ", "a", " ", SuperscriptBox["b", "4"]]], "+", SuperscriptBox["b", "5"]]], ")"]], " ", RowBox[List["ArcTan", "[", RowBox[List["1", "-", RowBox[List[SqrtBox["2"], " ", SqrtBox[RowBox[List["Cot", "[", RowBox[List["c", " ", "z"]], "]"]]]]]]], "]"]]]], "-", RowBox[List["2", " ", "b", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", SqrtBox["2"]]], " ", SuperscriptBox["a", "5"]]], "-", RowBox[List["3", " ", SuperscriptBox["a", "4"], " ", "b"]], "-", RowBox[List["2", " ", SqrtBox["2"], " ", SuperscriptBox["a", "3"], " ", SuperscriptBox["b", "2"]]], "+", RowBox[List[SqrtBox["2"], " ", "a", " ", SuperscriptBox["b", "4"]]], "+", SuperscriptBox["b", "5"]]], ")"]], " ", RowBox[List["ArcTan", "[", RowBox[List["1", "+", RowBox[List[SqrtBox["2"], " ", SqrtBox[RowBox[List["Cot", "[", RowBox[List["c", " ", "z"]], "]"]]]]]]], "]"]]]], "+", FractionBox[RowBox[List["4", " ", SuperscriptBox["a", "3"], " ", RowBox[List["(", RowBox[List[SuperscriptBox["a", "4"], "+", SuperscriptBox["b", "4"]]], ")"]]]], RowBox[List["a", "+", RowBox[List["b", " ", SqrtBox[RowBox[List["Cot", "[", RowBox[List["c", " ", "z"]], "]"]]]]]]]], "-", RowBox[List["4", " ", RowBox[List["(", RowBox[List[SuperscriptBox["a", "6"], "-", RowBox[List["3", " ", SuperscriptBox["a", "2"], " ", SuperscriptBox["b", "4"]]]]], ")"]], " ", RowBox[List["Log", "[", RowBox[List["a", "+", RowBox[List["b", " ", SqrtBox[RowBox[List["Cot", "[", RowBox[List["c", " ", "z"]], "]"]]]]]]], "]"]]]], "-", RowBox[List[SqrtBox["2"], " ", "a", " ", "b", " ", RowBox[List["(", RowBox[List[SuperscriptBox["a", "4"], "-", RowBox[List["2", " ", SuperscriptBox["a", "2"], " ", SuperscriptBox["b", "2"]]], "-", SuperscriptBox["b", "4"]]], ")"]], " ", RowBox[List["Log", "[", RowBox[List[RowBox[List["-", "1"]], "+", RowBox[List[SqrtBox["2"], " ", SqrtBox[RowBox[List["Cot", "[", RowBox[List["c", " ", "z"]], "]"]]]]], "-", RowBox[List["Cot", "[", RowBox[List["c", " ", "z"]], "]"]]]], "]"]]]], "+", RowBox[List[SqrtBox["2"], " ", "a", " ", "b", " ", RowBox[List["(", RowBox[List[SuperscriptBox["a", "4"], "-", RowBox[List["2", " ", SuperscriptBox["a", "2"], " ", SuperscriptBox["b", "2"]]], "-", SuperscriptBox["b", "4"]]], ")"]], " ", RowBox[List["Log", "[", RowBox[List["1", "+", RowBox[List[SqrtBox["2"], " ", SqrtBox[RowBox[List["Cot", "[", RowBox[List["c", " ", "z"]], "]"]]]]], "+", RowBox[List["Cot", "[", RowBox[List["c", " ", "z"]], "]"]]]], "]"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["a", "6"], "-", RowBox[List["3", " ", SuperscriptBox["a", "2"], " ", SuperscriptBox["b", "4"]]]]], ")"]], " ", RowBox[List["Log", "[", RowBox[List["1", "+", SuperscriptBox[RowBox[List["Cot", "[", RowBox[List["c", " ", "z"]], "]"]], "2"]]], "]"]]]]]], RowBox[List["2", " ", SuperscriptBox[RowBox[List["(", RowBox[List[SuperscriptBox["a", "4"], "+", SuperscriptBox["b", "4"]]], ")"]], "2"], " ", "c"]]]]]]]]]

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

 2002-12-18