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 Cot

 http://functions.wolfram.com/01.09.21.0184.01

 Input Form

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

 Standard Form

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 MathML Form

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

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["\[Integral]", RowBox[List[SqrtBox[RowBox[List["a_", "+", RowBox[List["b_", " ", SuperscriptBox[RowBox[List["Cot", "[", RowBox[List["c_", " ", "z_"]], "]"]], "2"]]]]]], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", FractionBox[RowBox[List["\[ImaginaryI]", " ", SqrtBox[RowBox[List["a", "+", RowBox[List["b", " ", SuperscriptBox[RowBox[List["Cot", "[", RowBox[List["c", " ", "z"]], "]"]], "2"]]]]]], " ", RowBox[List["(", RowBox[List[RowBox[List[SqrtBox["b"], " ", RowBox[List["Log", "[", FractionBox[RowBox[List["4", " ", "\[ImaginaryI]", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "a"]], "-", "b", "+", RowBox[List[SuperscriptBox[RowBox[List["Cot", "[", FractionBox[RowBox[List["c", " ", "z"]], "2"], "]"]], "2"], " ", RowBox[List["(", RowBox[List["b", "+", FractionBox[RowBox[List["\[ImaginaryI]", " ", SqrtBox["b"], " ", SqrtBox[RowBox[List[RowBox[List["(", 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["Sec", "[", FractionBox[RowBox[List["c", " ", "z"]], "2"], "]"]], "4"]]]]]], SqrtBox["2"]]]], ")"]]]]]], ")"]]]], SuperscriptBox["b", RowBox[List["3", "/", "2"]]]], "]"]]]], "-", RowBox[List["2", " ", SqrtBox[RowBox[List[RowBox[List["-", "a"]], "+", "b"]]], " ", RowBox[List["Log", "[", FractionBox[RowBox[List[SuperscriptBox[RowBox[List["Cos", "[", FractionBox[RowBox[List["c", " ", "z"]], "2"], "]"]], "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "a"]], "-", RowBox[List["\[ImaginaryI]", " ", "b"]], "+", FractionBox[RowBox[List[SqrtBox[RowBox[List[RowBox[List["-", "a"]], "+", "b"]]], " ", SqrtBox[RowBox[List[RowBox[List["(", 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["Sec", "[", FractionBox[RowBox[List["c", " ", "z"]], "2"], "]"]], "4"]]]]]], SqrtBox["2"]], "-", RowBox[List["\[ImaginaryI]", " ", RowBox[List["(", RowBox[List["a", "-", "b"]], ")"]], " ", SuperscriptBox[RowBox[List["Tan", "[", FractionBox[RowBox[List["c", " ", "z"]], "2"], "]"]], "2"]]]]], ")"]]]], SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "a"]], "+", "b"]], ")"]], RowBox[List["3", "/", "2"]]]], "]"]]]], "-", RowBox[List[SqrtBox["b"], " ", RowBox[List["Log", "[", RowBox[List[RowBox[List[SqrtBox["2"], " ", SqrtBox[RowBox[List[RowBox[List["(", 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["Sec", "[", FractionBox[RowBox[List["c", " ", "z"]], "2"], "]"]], "4"]]]]]], "-", FractionBox[RowBox[List["2", " ", "\[ImaginaryI]", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "a"]], "-", "b", "+", RowBox[List["b", " ", SuperscriptBox[RowBox[List["Tan", "[", FractionBox[RowBox[List["c", " ", "z"]], "2"], "]"]], "2"]]]]], ")"]]]], SqrtBox["b"]]]], "]"]]]]]], ")"]], " ", RowBox[List["Sin", "[", RowBox[List["c", " ", "z"]], "]"]]]], RowBox[List["c", " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List["Cos", "[", RowBox[List["c", " ", "z"]], "]"]]]], ")"]], " ", SqrtBox[FractionBox[RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", RowBox[List["(", RowBox[List["a", "+", "b"]], ")"]]]], "+", RowBox[List["2", " ", RowBox[List["(", RowBox[List["a", "-", "b"]], ")"]], " ", RowBox[List["Cos", "[", RowBox[List["2", " ", "c", " ", "z"]], "]"]]]]]], SuperscriptBox[RowBox[List["(", RowBox[List["1", "+", RowBox[List["Cos", "[", RowBox[List["c", " ", "z"]], "]"]]]], ")"]], "2"]]]]]]]]]]

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

 2002-12-18