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 Cot

 http://functions.wolfram.com/01.09.21.0199.01

 Input Form

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

 Standard Form

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

 MathML Form

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

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["\[Integral]", RowBox[List[FractionBox["1", SqrtBox[RowBox[List["a_", "+", RowBox[List["b_", " ", SqrtBox[RowBox[List["Cot", "[", RowBox[List["c_", " ", "z_"]], "]"]]]]]]]]], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", FractionBox[RowBox[List[RowBox[List[SqrtBox[RowBox[List["a", "-", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["1", "/", "4"]]], " ", "b"]]]]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", SuperscriptBox["a", "3"]]], "-", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["3", "/", "4"]]], " ", SuperscriptBox["a", "2"], " ", "b"]], "+", RowBox[List["a", " ", SuperscriptBox["b", "2"]]], "+", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["1", "/", "4"]]], " ", SuperscriptBox["b", "3"]]]]], ")"]], " ", 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 Date Added to functions.wolfram.com (modification date)

 2002-12-18