html, body, form { margin: 0; padding: 0; width: 100%; } #calculate { position: relative; width: 177px; height: 110px; background: transparent url(/images/alphabox/embed_functions_inside.gif) no-repeat scroll 0 0; } #i { position: relative; left: 18px; top: 44px; width: 133px; border: 0 none; outline: 0; font-size: 11px; } #eq { width: 9px; height: 10px; background: transparent; position: absolute; top: 47px; right: 18px; cursor: pointer; }

 Cot

 http://functions.wolfram.com/01.09.21.0190.01

 Input Form

 Integrate[Sqrt[(a + b Cot[c z]^2)^3], z] == (1/c) (Sqrt[(a + b Cot[c z]^2)^3] Sin[c z] (-((b Cos[c z])/(a + b + (-a + b) Cos[2 c z])) - (I (1 + Cos[c z]) Sqrt[-((a + b + (-a + b) Cos[2 c z])/(1 + Cos[c z])^2)] ((3 a - 2 b) Sqrt[b] Sqrt[-a + b] Log[(1/Sqrt[b]) (-8 I a + 4 I b + 2 Cot[(c z)/2]^2 (-2 I b + Sqrt[2] Sqrt[b] Sqrt[(-a - b + (a - b) Cos[2 c z]) Sec[(c z)/2]^4]))] + 4 (a - b)^2 Log[-((1/(4 Sqrt[-a + b])) (Cos[(c z)/2]^2 (2 I a - 2 I b + Sqrt[2] Sqrt[-a + b] Sqrt[(-a - b + (a - b) Cos[2 c z]) Sec[(c z)/2]^4] - 2 I (a - b) Tan[(c z)/2]^2)))] + Sqrt[b] Sqrt[-a + b] (-3 a + 2 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]^2)/ (Sqrt[2] Sqrt[-a + b] (a + b + (-a + b) Cos[2 c z])^2)))

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List["\[Integral]", RowBox[List[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[FractionBox["1", "c"], RowBox[List["(", RowBox[List[SqrtBox[SuperscriptBox[RowBox[List["(", RowBox[List["a", "+", RowBox[List["b", " ", SuperscriptBox[RowBox[List["Cot", "[", RowBox[List["c", " ", "z"]], "]"]], "2"]]]]], ")"]], "3"]], " ", RowBox[List["Sin", "[", RowBox[List["c", " ", "z"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", FractionBox[RowBox[List["b", " ", RowBox[List["Cos", "[", RowBox[List["c", " ", "z"]], "]"]]]], RowBox[List["a", "+", "b", "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "a"]], "+", "b"]], ")"]], " ", RowBox[List["Cos", "[", RowBox[List["2", " ", "c", " ", "z"]], "]"]]]]]]]]], "-", RowBox[List[RowBox[List["(", RowBox[List["\[ImaginaryI]", " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List["Cos", "[", RowBox[List["c", " ", "z"]], "]"]]]], ")"]], " ", SqrtBox[RowBox[List["-", FractionBox[RowBox[List["a", "+", "b", "+", RowBox[List[RowBox[List["(", 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"]]]]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["3", " ", "a"]], "-", RowBox[List["2", " ", "b"]]]], ")"]], " ", SqrtBox["b"], " ", SqrtBox[RowBox[List[RowBox[List["-", "a"]], "+", "b"]]], " ", RowBox[List["Log", "[", RowBox[List[FractionBox["1", SqrtBox["b"]], RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "8"]], " ", "\[ImaginaryI]", " ", "a"]], "+", RowBox[List["4", " ", "\[ImaginaryI]", " ", "b"]], "+", RowBox[List["2", " ", SuperscriptBox[RowBox[List["Cot", "[", FractionBox[RowBox[List["c", " ", "z"]], "2"], "]"]], "2"], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "\[ImaginaryI]", " ", "b"]], "+", RowBox[List[SqrtBox["2"], " ", 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"]]]]]]]], ")"]]]]]], ")"]]]], "]"]]]], "+", RowBox[List["4", " ", SuperscriptBox[RowBox[List["(", RowBox[List["a", "-", "b"]], ")"]], "2"], " ", RowBox[List["Log", "[", RowBox[List["-", RowBox[List[FractionBox["1", RowBox[List["4", " ", SqrtBox[RowBox[List[RowBox[List["-", "a"]], "+", "b"]]]]]], RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["Cos", "[", FractionBox[RowBox[List["c", " ", "z"]], "2"], "]"]], "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "\[ImaginaryI]", " ", "a"]], "-", RowBox[List["2", " ", "\[ImaginaryI]", " ", "b"]], "+", RowBox[List[SqrtBox["2"], " ", 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"]]]]]], "-", RowBox[List["2", " ", "\[ImaginaryI]", " ", RowBox[List["(", RowBox[List["a", "-", "b"]], ")"]], " ", SuperscriptBox[RowBox[List["Tan", "[", FractionBox[RowBox[List["c", " ", "z"]], "2"], "]"]], "2"]]]]], ")"]]]], ")"]]]]]], "]"]]]], "+", RowBox[List[SqrtBox["b"], " ", SqrtBox[RowBox[List[RowBox[List["-", "a"]], "+", "b"]]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "3"]], " ", "a"]], "+", RowBox[List["2", " ", "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"]]]], "]"]]]]]], ")"]], " ", SuperscriptBox[RowBox[List["Sin", "[", RowBox[List["c", " ", "z"]], "]"]], "2"]]], ")"]], "/", RowBox[List["(", RowBox[List[SqrtBox["2"], " ", SqrtBox[RowBox[List[RowBox[List["-", "a"]], "+", "b"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["a", "+", "b", "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "a"]], "+", "b"]], ")"]], " ", RowBox[List["Cos", "[", RowBox[List["2", " ", "c", " ", "z"]], "]"]]]]]], ")"]], "2"]]], ")"]]]]]], ")"]]]], ")"]]]]]]]]

 MathML Form

 ( a + b cot 2 ( c z ) ) 3 z 1 c ( ( b cot 2 ( c z ) + a ) 3 sin ( c z ) ( - b cos ( c z ) a + b + ( b - a ) cos ( 2 c z ) - ( ( cos ( c z ) + 1 ) - a + b + ( b - a ) cos ( 2 c z ) ( cos ( c z ) + 1 ) 2 ( 4 log ( - 1 4 b - a ( cos 2 ( c z 2 ) ( - 2 ( a - b ) tan 2 ( c z 2 ) - 2 b + 2 a + 2 b - a ( - a - b + ( a - b ) cos ( 2 c z ) ) sec 4 ( c z 2 ) ) ) ) ( a - b ) 2 + b b - a ( 2 b - 3 a ) 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 ) + b b - a ( 3 a - 2 b ) log ( 1 b ( 2 ( - 2 b + 2 b ( - a - b + ( a - b ) cos ( 2 c z ) ) sec 4 ( c z 2 ) ) cot 2 ( c z 2 ) - 8 a + 4 b ) ) ) sin 2 ( c z ) ) / ( 2 b - a ( a + b + ( b - a ) cos ( 2 c z ) ) 2 ) ) ) z a b c z 2 3 1 2 1 c -1 b c z 2 a 3 1 2 c z -1 b c z a b b -1 a 2 c z -1 -1 c z 1 -1 a b b -1 a 2 c z c z 1 2 -1 1 2 4 -1 1 4 b -1 a 1 2 -1 c z 2 -1 2 -2 a -1 b c z 2 -1 2 -1 2 b 2 a 2 1 2 b -1 a 1 2 -1 a -1 b a -1 b 2 c z c z 2 -1 4 1 2 a -1 b 2 b 1 2 b -1 a 1 2 2 b -1 3 a 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 b 1 2 b -1 a 1 2 3 a -1 2 b 1 b 1 2 -1 2 -2 b 2 1 2 b 1 2 -1 a -1 b a -1 b 2 c z c z 2 -1 4 1 2 c z 2 -1 2 -1 8 a 4 b c z 2 2 1 2 b -1 a 1 2 a b b -1 a 2 c z 2 -1 [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["\[Integral]", RowBox[List[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[SqrtBox[SuperscriptBox[RowBox[List["(", RowBox[List["a", "+", RowBox[List["b", " ", SuperscriptBox[RowBox[List["Cot", "[", RowBox[List["c", " ", "z"]], "]"]], "2"]]]]], ")"]], "3"]], " ", RowBox[List["Sin", "[", RowBox[List["c", " ", "z"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", FractionBox[RowBox[List["b", " ", RowBox[List["Cos", "[", RowBox[List["c", " ", "z"]], "]"]]]], RowBox[List["a", "+", "b", "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "a"]], "+", "b"]], ")"]], " ", RowBox[List["Cos", "[", RowBox[List["2", " ", "c", " ", "z"]], "]"]]]]]]]]], "-", FractionBox[RowBox[List["\[ImaginaryI]", " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List["Cos", "[", RowBox[List["c", " ", "z"]], "]"]]]], ")"]], " ", SqrtBox[RowBox[List["-", FractionBox[RowBox[List["a", "+", "b", "+", RowBox[List[RowBox[List["(", 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"]]]]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["3", " ", "a"]], "-", RowBox[List["2", " ", "b"]]]], ")"]], " ", SqrtBox["b"], " ", SqrtBox[RowBox[List[RowBox[List["-", "a"]], "+", "b"]]], " ", RowBox[List["Log", "[", FractionBox[RowBox[List[RowBox[List[RowBox[List["-", "8"]], " ", "\[ImaginaryI]", " ", "a"]], "+", RowBox[List["4", " ", "\[ImaginaryI]", " ", "b"]], "+", RowBox[List["2", " ", SuperscriptBox[RowBox[List["Cot", "[", FractionBox[RowBox[List["c", " ", "z"]], "2"], "]"]], "2"], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "\[ImaginaryI]", " ", "b"]], "+", RowBox[List[SqrtBox["2"], " ", 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["b"]], "]"]]]], "+", RowBox[List["4", " ", SuperscriptBox[RowBox[List["(", RowBox[List["a", "-", "b"]], ")"]], "2"], " ", RowBox[List["Log", "[", RowBox[List["-", FractionBox[RowBox[List[SuperscriptBox[RowBox[List["Cos", "[", FractionBox[RowBox[List["c", " ", "z"]], "2"], "]"]], "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "\[ImaginaryI]", " ", "a"]], "-", RowBox[List["2", " ", "\[ImaginaryI]", " ", "b"]], "+", RowBox[List[SqrtBox["2"], " ", 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"]]]]]], "-", RowBox[List["2", " ", "\[ImaginaryI]", " ", RowBox[List["(", RowBox[List["a", "-", "b"]], ")"]], " ", SuperscriptBox[RowBox[List["Tan", "[", FractionBox[RowBox[List["c", " ", "z"]], "2"], "]"]], "2"]]]]], ")"]]]], RowBox[List["4", " ", SqrtBox[RowBox[List[RowBox[List["-", "a"]], "+", "b"]]]]]]]], "]"]]]], "+", RowBox[List[SqrtBox["b"], " ", SqrtBox[RowBox[List[RowBox[List["-", "a"]], "+", "b"]]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "3"]], " ", "a"]], "+", RowBox[List["2", " ", "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"]]]], "]"]]]]]], ")"]], " ", SuperscriptBox[RowBox[List["Sin", "[", RowBox[List["c", " ", "z"]], "]"]], "2"]]], RowBox[List[SqrtBox["2"], " ", SqrtBox[RowBox[List[RowBox[List["-", "a"]], "+", "b"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["a", "+", "b", "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "a"]], "+", "b"]], ")"]], " ", RowBox[List["Cos", "[", RowBox[List["2", " ", "c", " ", "z"]], "]"]]]]]], ")"]], "2"]]]]]], ")"]]]], "c"]]]]]

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

 2002-12-18