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/01.09.21.0193.01

 Input Form

 Integrate[Cot[c z] Sqrt[(a + b Cot[c z]^2)^5], z] == (1/c) (Sqrt[(a + b Cot[c z]^2)^5] Sin[c z]^5 (-((4 Csc[c z] (23 (a - b)^2 + 11 (a - b) b Csc[c z]^2 + 3 b^2 Csc[c z]^4))/(15 (a + b + (-a + b) Cos[2 c z])^2)) + (ArcTan[(Sqrt[-a - b + (a - b) Cos[2 c z]] Cot[c z]^2 Sqrt[(a - b)^2 Sin[2 c z]^2])/(2 Sqrt[2] ((a - b) Cos[c z]^2)^ (3/2))] Csc[c z]^3 Sec[c z]^2 ((a - b)^2 Sin[2 c z]^2)^(3/2))/ (Sqrt[2] Sqrt[(a - b) Cos[c z]^2] (-a - b + (a - b) Cos[2 c z])^(5/2))))

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List["\[Integral]", RowBox[List[RowBox[List["Cot", "[", RowBox[List["c", " ", "z"]], "]"]], " ", SqrtBox[SuperscriptBox[RowBox[List["(", RowBox[List["a", "+", RowBox[List["b", " ", SuperscriptBox[RowBox[List["Cot", "[", RowBox[List["c", " ", "z"]], "]"]], "2"]]]]], ")"]], "5"]], 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"]]]]], ")"]], "5"]], " ", SuperscriptBox[RowBox[List["Sin", "[", RowBox[List["c", " ", "z"]], "]"]], "5"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", RowBox[List[RowBox[List["(", RowBox[List["4", " ", RowBox[List["Csc", "[", RowBox[List["c", " ", "z"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["23", " ", SuperscriptBox[RowBox[List["(", RowBox[List["a", "-", "b"]], ")"]], "2"]]], "+", RowBox[List["11", " ", RowBox[List["(", RowBox[List["a", "-", "b"]], ")"]], " ", "b", " ", SuperscriptBox[RowBox[List["Csc", "[", RowBox[List["c", " ", "z"]], "]"]], "2"]]], "+", RowBox[List["3", " ", SuperscriptBox["b", "2"], " ", SuperscriptBox[RowBox[List["Csc", "[", RowBox[List["c", " ", "z"]], "]"]], "4"]]]]], ")"]]]], ")"]], "/", RowBox[List["(", RowBox[List["15", " ", 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"]]], ")"]]]]]], "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["ArcTan", "[", RowBox[List[RowBox[List["(", RowBox[List[SqrtBox[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["Cot", "[", RowBox[List["c", " ", "z"]], "]"]], "2"], " ", SqrtBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["a", "-", "b"]], ")"]], "2"], " ", SuperscriptBox[RowBox[List["Sin", "[", RowBox[List["2", " ", "c", " ", "z"]], "]"]], "2"]]]]]], ")"]], "/", RowBox[List["(", RowBox[List["2", " ", SqrtBox["2"], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["(", RowBox[List["a", "-", "b"]], ")"]], " ", SuperscriptBox[RowBox[List["Cos", "[", RowBox[List["c", " ", "z"]], "]"]], "2"]]], ")"]], RowBox[List["3", "/", "2"]]]]], ")"]]]], "]"]], " ", SuperscriptBox[RowBox[List["Csc", "[", RowBox[List["c", " ", "z"]], "]"]], "3"], " ", SuperscriptBox[RowBox[List["Sec", "[", RowBox[List["c", " ", "z"]], "]"]], "2"], " ", SuperscriptBox[RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["a", "-", "b"]], ")"]], "2"], " ", SuperscriptBox[RowBox[List["Sin", "[", RowBox[List["2", " ", "c", " ", "z"]], "]"]], "2"]]], ")"]], RowBox[List["3", "/", "2"]]]]], ")"]], "/", RowBox[List["(", RowBox[List[SqrtBox["2"], " ", SqrtBox[RowBox[List[RowBox[List["(", RowBox[List["a", "-", "b"]], ")"]], " ", SuperscriptBox[RowBox[List["Cos", "[", RowBox[List["c", " ", "z"]], "]"]], "2"]]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "a"]], "-", "b", "+", RowBox[List[RowBox[List["(", RowBox[List["a", "-", "b"]], ")"]], " ", RowBox[List["Cos", "[", RowBox[List["2", " ", "c", " ", "z"]], "]"]]]]]], ")"]], RowBox[List["5", "/", "2"]]]]], ")"]]]]]], ")"]]]], ")"]]]]]]]]

 MathML Form

 cot ( c z ) ( a + b cot 2 ( c z ) ) 5 z 1 c ( ( b cot 2 ( c z ) + a ) 5 sin 5 ( c z ) ( tan - 1 ( - a - b + ( a - b ) cos ( 2 c z ) cot 2 ( c z ) ( a - b ) 2 sin 2 ( 2 c z ) 2 2 ( ( a - b ) cos 2 ( c z ) ) 3 / 2 ) csc 3 ( c z ) sec 2 ( c z ) ( ( a - b ) 2 sin 2 ( 2 c z ) ) 3 / 2 2 ( a - b ) cos 2 ( c z ) ( - a - b + ( a - b ) cos ( 2 c z ) ) 5 / 2 - 4 csc ( c z ) ( 3 b 2 csc 4 ( c z ) + 11 ( a - b ) b csc 2 ( c z ) + 23 ( a - b ) 2 ) 15 ( a + b + ( b - a ) cos ( 2 c z ) ) 2 ) ) z c z a b c z 2 5 1 2 1 c -1 b c z 2 a 5 1 2 c z 5 -1 a -1 b a -1 b 2 c z 1 2 c z 2 a -1 b 2 2 c z 2 1 2 2 2 1 2 a -1 b c z 2 3 2 -1 c z 3 c z 2 a -1 b 2 2 c z 2 3 2 2 1 2 a -1 b c z 2 1 2 -1 a -1 b a -1 b 2 c z 5 2 -1 -1 4 c z 3 b 2 c z 4 11 a -1 b b c z 2 23 a -1 b 2 15 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[RowBox[List[RowBox[List["Cot", "[", RowBox[List["c_", " ", "z_"]], "]"]], " ", SqrtBox[SuperscriptBox[RowBox[List["(", RowBox[List["a_", "+", RowBox[List["b_", " ", SuperscriptBox[RowBox[List["Cot", "[", RowBox[List["c_", " ", "z_"]], "]"]], "2"]]]]], ")"]], "5"]]]], 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"]]]]], ")"]], "5"]], " ", SuperscriptBox[RowBox[List["Sin", "[", RowBox[List["c", " ", "z"]], "]"]], "5"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", FractionBox[RowBox[List["4", " ", RowBox[List["Csc", "[", RowBox[List["c", " ", "z"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["23", " ", SuperscriptBox[RowBox[List["(", RowBox[List["a", "-", "b"]], ")"]], "2"]]], "+", RowBox[List["11", " ", RowBox[List["(", RowBox[List["a", "-", "b"]], ")"]], " ", "b", " ", SuperscriptBox[RowBox[List["Csc", "[", RowBox[List["c", " ", "z"]], "]"]], "2"]]], "+", RowBox[List["3", " ", SuperscriptBox["b", "2"], " ", SuperscriptBox[RowBox[List["Csc", "[", RowBox[List["c", " ", "z"]], "]"]], "4"]]]]], ")"]]]], RowBox[List["15", " ", 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"]]]]]], "+", FractionBox[RowBox[List[RowBox[List["ArcTan", "[", FractionBox[RowBox[List[SqrtBox[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["Cot", "[", RowBox[List["c", " ", "z"]], "]"]], "2"], " ", SqrtBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["a", "-", "b"]], ")"]], "2"], " ", SuperscriptBox[RowBox[List["Sin", "[", RowBox[List["2", " ", "c", " ", "z"]], "]"]], "2"]]]]]], RowBox[List["2", " ", SqrtBox["2"], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["(", RowBox[List["a", "-", "b"]], ")"]], " ", SuperscriptBox[RowBox[List["Cos", "[", RowBox[List["c", " ", "z"]], "]"]], "2"]]], ")"]], RowBox[List["3", "/", "2"]]]]]], "]"]], " ", SuperscriptBox[RowBox[List["Csc", "[", RowBox[List["c", " ", "z"]], "]"]], "3"], " ", SuperscriptBox[RowBox[List["Sec", "[", RowBox[List["c", " ", "z"]], "]"]], "2"], " ", SuperscriptBox[RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["a", "-", "b"]], ")"]], "2"], " ", SuperscriptBox[RowBox[List["Sin", "[", RowBox[List["2", " ", "c", " ", "z"]], "]"]], "2"]]], ")"]], RowBox[List["3", "/", "2"]]]]], RowBox[List[SqrtBox["2"], " ", SqrtBox[RowBox[List[RowBox[List["(", RowBox[List["a", "-", "b"]], ")"]], " ", SuperscriptBox[RowBox[List["Cos", "[", RowBox[List["c", " ", "z"]], "]"]], "2"]]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "a"]], "-", "b", "+", RowBox[List[RowBox[List["(", RowBox[List["a", "-", "b"]], ")"]], " ", RowBox[List["Cos", "[", RowBox[List["2", " ", "c", " ", "z"]], "]"]]]]]], ")"]], RowBox[List["5", "/", "2"]]]]]]]], ")"]]]], "c"]]]]]

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

 2002-12-18