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; }

 Tan

 http://functions.wolfram.com/01.08.21.0126.01

 Input Form

 Integrate[Tan[c z]/Sqrt[(a + b Tan[c z]^2)^5], z] == ((a + b + (a - b) Cos[2 c z]) (6 a^2 - 6 a b + 2 (a - b)^2 Cos[4 c z] - 3 Sqrt[2] a Sqrt[(a - b) Cos[c z]^2] Sqrt[a + b + (a - b) Cos[2 c z]] Log[Sqrt[2] Sqrt[(a - b) Cos[c z]^2] + Sqrt[a + b + (a - b) Cos[2 c z]]] - 3 Sqrt[2] b Sqrt[(a - b) Cos[c z]^2] Sqrt[a + b + (a - b) Cos[2 c z]] Log[Sqrt[2] Sqrt[(a - b) Cos[c z]^2] + Sqrt[a + b + (a - b) Cos[2 c z]]] + (a - b) Cos[2 c z] (8 a - 2 b - 3 Sqrt[2] Sqrt[(a - b) Cos[c z]^2] Sqrt[a + b + (a - b) Cos[2 c z]] Log[Sqrt[2] Sqrt[(a - b) Cos[c z]^2] + Sqrt[a + b + (a - b) Cos[2 c z]]])) Sec[c z]^6)/ (24 (a - b)^3 c Sqrt[(a + b Tan[c z]^2)^5])

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List["\[Integral]", RowBox[List[FractionBox[RowBox[List["Tan", "[", RowBox[List["c", " ", "z"]], "]"]], SqrtBox[SuperscriptBox[RowBox[List["(", RowBox[List["a", "+", RowBox[List["b", " ", SuperscriptBox[RowBox[List["Tan", "[", RowBox[List["c", " ", "z"]], "]"]], "2"]]]]], ")"]], "5"]]], RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", RowBox[List[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"]], "]"]]]]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["6", " ", SuperscriptBox["a", "2"]]], "-", RowBox[List["6", " ", "a", " ", "b"]], "+", RowBox[List["2", " ", SuperscriptBox[RowBox[List["(", RowBox[List["a", "-", "b"]], ")"]], "2"], " ", RowBox[List["Cos", "[", RowBox[List["4", " ", "c", " ", "z"]], "]"]]]], "-", RowBox[List["3", " ", SqrtBox["2"], " ", "a", " ", SqrtBox[RowBox[List[RowBox[List["(", RowBox[List["a", "-", "b"]], ")"]], " ", SuperscriptBox[RowBox[List["Cos", "[", RowBox[List["c", " ", "z"]], "]"]], "2"]]]], " ", SqrtBox[RowBox[List["a", "+", "b", "+", RowBox[List[RowBox[List["(", RowBox[List["a", "-", "b"]], ")"]], " ", RowBox[List["Cos", "[", RowBox[List["2", " ", "c", " ", "z"]], "]"]]]]]]], " ", RowBox[List["Log", "[", RowBox[List[RowBox[List[SqrtBox["2"], " ", SqrtBox[RowBox[List[RowBox[List["(", RowBox[List["a", "-", "b"]], ")"]], " ", SuperscriptBox[RowBox[List["Cos", "[", RowBox[List["c", " ", "z"]], "]"]], "2"]]]]]], "+", SqrtBox[RowBox[List["a", "+", "b", "+", RowBox[List[RowBox[List["(", RowBox[List["a", "-", "b"]], ")"]], " ", RowBox[List["Cos", "[", RowBox[List["2", " ", "c", " ", "z"]], "]"]]]]]]]]], "]"]]]], "-", RowBox[List["3", " ", SqrtBox["2"], " ", "b", " ", SqrtBox[RowBox[List[RowBox[List["(", RowBox[List["a", "-", "b"]], ")"]], " ", SuperscriptBox[RowBox[List["Cos", "[", RowBox[List["c", " ", "z"]], "]"]], "2"]]]], " ", SqrtBox[RowBox[List["a", "+", "b", "+", RowBox[List[RowBox[List["(", RowBox[List["a", "-", "b"]], ")"]], " ", RowBox[List["Cos", "[", RowBox[List["2", " ", "c", " ", "z"]], "]"]]]]]]], " ", RowBox[List["Log", "[", RowBox[List[RowBox[List[SqrtBox["2"], " ", SqrtBox[RowBox[List[RowBox[List["(", RowBox[List["a", "-", "b"]], ")"]], " ", SuperscriptBox[RowBox[List["Cos", "[", RowBox[List["c", " ", "z"]], "]"]], "2"]]]]]], "+", SqrtBox[RowBox[List["a", "+", "b", "+", RowBox[List[RowBox[List["(", RowBox[List["a", "-", "b"]], ")"]], " ", RowBox[List["Cos", "[", RowBox[List["2", " ", "c", " ", "z"]], "]"]]]]]]]]], "]"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List["a", "-", "b"]], ")"]], " ", RowBox[List["Cos", "[", RowBox[List["2", " ", "c", " ", "z"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["8", " ", "a"]], "-", RowBox[List["2", " ", "b"]], "-", RowBox[List["3", " ", SqrtBox["2"], " ", SqrtBox[RowBox[List[RowBox[List["(", RowBox[List["a", "-", "b"]], ")"]], " ", SuperscriptBox[RowBox[List["Cos", "[", RowBox[List["c", " ", "z"]], "]"]], "2"]]]], " ", SqrtBox[RowBox[List["a", "+", "b", "+", RowBox[List[RowBox[List["(", RowBox[List["a", "-", "b"]], ")"]], " ", RowBox[List["Cos", "[", RowBox[List["2", " ", "c", " ", "z"]], "]"]]]]]]], " ", RowBox[List["Log", "[", RowBox[List[RowBox[List[SqrtBox["2"], " ", SqrtBox[RowBox[List[RowBox[List["(", RowBox[List["a", "-", "b"]], ")"]], " ", SuperscriptBox[RowBox[List["Cos", "[", RowBox[List["c", " ", "z"]], "]"]], "2"]]]]]], "+", SqrtBox[RowBox[List["a", "+", "b", "+", RowBox[List[RowBox[List["(", RowBox[List["a", "-", "b"]], ")"]], " ", RowBox[List["Cos", "[", RowBox[List["2", " ", "c", " ", "z"]], "]"]]]]]]]]], "]"]]]]]], ")"]]]]]], ")"]], " ", SuperscriptBox[RowBox[List["Sec", "[", RowBox[List["c", " ", "z"]], "]"]], "6"]]], ")"]], "/", RowBox[List["(", RowBox[List["24", " ", SuperscriptBox[RowBox[List["(", RowBox[List["a", "-", "b"]], ")"]], "3"], " ", "c", " ", SqrtBox[SuperscriptBox[RowBox[List["(", RowBox[List["a", "+", RowBox[List["b", " ", SuperscriptBox[RowBox[List["Tan", "[", RowBox[List["c", " ", "z"]], "]"]], "2"]]]]], ")"]], "5"]]]], ")"]]]]]]]]

 MathML Form

 tan ( c z ) ( a + b tan 2 ( c z ) ) 5 z ( ( a + b + ( a - b ) cos ( 2 c z ) ) ( 6 a 2 - 6 b a - 3 2 ( a - b ) cos 2 ( c z ) a + b + ( a - b ) cos ( 2 c z ) log ( 2 ( a - b ) cos 2 ( c z ) + a + b + ( a - b ) cos ( 2 c z ) ) a + 2 ( a - b ) 2 cos ( 4 c z ) - 3 2 b ( a - b ) cos 2 ( c z ) a + b + ( a - b ) cos ( 2 c z ) log ( 2 ( a - b ) cos 2 ( c z ) + a + b + ( a - b ) cos ( 2 c z ) ) + ( a - b ) cos ( 2 c z ) ( 8 a - 2 b - 3 2 ( a - b ) cos 2 ( c z ) a + b + ( a - b ) cos ( 2 c z ) log ( 2 ( a - b ) cos 2 ( c z ) + a + b + ( a - b ) cos ( 2 c z ) ) ) ) sec 6 ( c z ) ) / ( 24 ( a - b ) 3 c ( b tan 2 ( c z ) + a ) 5 ) z c z a b c z 2 5 1 2 -1 a b a -1 b 2 c z 6 a 2 -1 6 b a -1 3 2 1 2 a -1 b c z 2 1 2 a b a -1 b 2 c z 1 2 2 1 2 a -1 b c z 2 1 2 a b a -1 b 2 c z 1 2 a 2 a -1 b 2 4 c z -1 3 2 1 2 b a -1 b c z 2 1 2 a b a -1 b 2 c z 1 2 2 1 2 a -1 b c z 2 1 2 a b a -1 b 2 c z 1 2 a -1 b 2 c z 8 a -1 2 b -1 3 2 1 2 a -1 b c z 2 1 2 a b a -1 b 2 c z 1 2 2 1 2 a -1 b c z 2 1 2 a b a -1 b 2 c z 1 2 c z 6 24 a -1 b 3 c b c z 2 a 5 1 2 -1 [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["\[Integral]", RowBox[List[FractionBox[RowBox[List["Tan", "[", RowBox[List["c_", " ", "z_"]], "]"]], SqrtBox[SuperscriptBox[RowBox[List["(", RowBox[List["a_", "+", RowBox[List["b_", " ", SuperscriptBox[RowBox[List["Tan", "[", RowBox[List["c_", " ", "z_"]], "]"]], "2"]]]]], ")"]], "5"]]], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", FractionBox[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["(", RowBox[List[RowBox[List["6", " ", SuperscriptBox["a", "2"]]], "-", RowBox[List["6", " ", "a", " ", "b"]], "+", RowBox[List["2", " ", SuperscriptBox[RowBox[List["(", RowBox[List["a", "-", "b"]], ")"]], "2"], " ", RowBox[List["Cos", "[", RowBox[List["4", " ", "c", " ", "z"]], "]"]]]], "-", RowBox[List["3", " ", SqrtBox["2"], " ", "a", " ", SqrtBox[RowBox[List[RowBox[List["(", RowBox[List["a", "-", "b"]], ")"]], " ", SuperscriptBox[RowBox[List["Cos", "[", RowBox[List["c", " ", "z"]], "]"]], "2"]]]], " ", SqrtBox[RowBox[List["a", "+", "b", "+", RowBox[List[RowBox[List["(", RowBox[List["a", "-", "b"]], ")"]], " ", RowBox[List["Cos", "[", RowBox[List["2", " ", "c", " ", "z"]], "]"]]]]]]], " ", RowBox[List["Log", "[", RowBox[List[RowBox[List[SqrtBox["2"], " ", SqrtBox[RowBox[List[RowBox[List["(", RowBox[List["a", "-", "b"]], ")"]], " ", SuperscriptBox[RowBox[List["Cos", "[", RowBox[List["c", " ", "z"]], "]"]], "2"]]]]]], "+", SqrtBox[RowBox[List["a", "+", "b", "+", RowBox[List[RowBox[List["(", RowBox[List["a", "-", "b"]], ")"]], " ", RowBox[List["Cos", "[", RowBox[List["2", " ", "c", " ", "z"]], "]"]]]]]]]]], "]"]]]], "-", RowBox[List["3", " ", SqrtBox["2"], " ", "b", " ", SqrtBox[RowBox[List[RowBox[List["(", RowBox[List["a", "-", "b"]], ")"]], " ", SuperscriptBox[RowBox[List["Cos", "[", RowBox[List["c", " ", "z"]], "]"]], "2"]]]], " ", SqrtBox[RowBox[List["a", "+", "b", "+", RowBox[List[RowBox[List["(", RowBox[List["a", "-", "b"]], ")"]], " ", RowBox[List["Cos", "[", RowBox[List["2", " ", "c", " ", "z"]], "]"]]]]]]], " ", RowBox[List["Log", "[", RowBox[List[RowBox[List[SqrtBox["2"], " ", SqrtBox[RowBox[List[RowBox[List["(", RowBox[List["a", "-", "b"]], ")"]], " ", SuperscriptBox[RowBox[List["Cos", "[", RowBox[List["c", " ", "z"]], "]"]], "2"]]]]]], "+", SqrtBox[RowBox[List["a", "+", "b", "+", RowBox[List[RowBox[List["(", RowBox[List["a", "-", "b"]], ")"]], " ", RowBox[List["Cos", "[", RowBox[List["2", " ", "c", " ", "z"]], "]"]]]]]]]]], "]"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List["a", "-", "b"]], ")"]], " ", RowBox[List["Cos", "[", RowBox[List["2", " ", "c", " ", "z"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["8", " ", "a"]], "-", RowBox[List["2", " ", "b"]], "-", RowBox[List["3", " ", SqrtBox["2"], " ", SqrtBox[RowBox[List[RowBox[List["(", RowBox[List["a", "-", "b"]], ")"]], " ", SuperscriptBox[RowBox[List["Cos", "[", RowBox[List["c", " ", "z"]], "]"]], "2"]]]], " ", SqrtBox[RowBox[List["a", "+", "b", "+", RowBox[List[RowBox[List["(", RowBox[List["a", "-", "b"]], ")"]], " ", RowBox[List["Cos", "[", RowBox[List["2", " ", "c", " ", "z"]], "]"]]]]]]], " ", RowBox[List["Log", "[", RowBox[List[RowBox[List[SqrtBox["2"], " ", SqrtBox[RowBox[List[RowBox[List["(", RowBox[List["a", "-", "b"]], ")"]], " ", SuperscriptBox[RowBox[List["Cos", "[", RowBox[List["c", " ", "z"]], "]"]], "2"]]]]]], "+", SqrtBox[RowBox[List["a", "+", "b", "+", RowBox[List[RowBox[List["(", RowBox[List["a", "-", "b"]], ")"]], " ", RowBox[List["Cos", "[", RowBox[List["2", " ", "c", " ", "z"]], "]"]]]]]]]]], "]"]]]]]], ")"]]]]]], ")"]], " ", SuperscriptBox[RowBox[List["Sec", "[", RowBox[List["c", " ", "z"]], "]"]], "6"]]], RowBox[List["24", " ", SuperscriptBox[RowBox[List["(", RowBox[List["a", "-", "b"]], ")"]], "3"], " ", "c", " ", SqrtBox[SuperscriptBox[RowBox[List["(", RowBox[List["a", "+", RowBox[List["b", " ", SuperscriptBox[RowBox[List["Tan", "[", RowBox[List["c", " ", "z"]], "]"]], "2"]]]]], ")"]], "5"]]]]]]]]]

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

 2002-12-18