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/01.08.21.0120.01

 Input Form

 Integrate[Sqrt[(a + b Tan[c z]^2)^3], z] == (I Sqrt[(a + b Tan[c z]^2)^3] ((3 a - 2 b) Sqrt[-b] Log[2 (I Sqrt[-b] Tan[c z] + Sqrt[a + b Tan[c z]^2])] + (a - b)^(3/2) (-Log[-((4 (a - I b Tan[c z] + Sqrt[a - b] Sqrt[a + b Tan[c z]^2]))/ ((a - b)^(5/2) (-1 + I Tan[c z])))] + Log[(4 (a + I b Tan[c z] + Sqrt[a - b] Sqrt[a + b Tan[c z]^2]))/ ((a - b)^(5/2) (1 + I Tan[c z]))]) - I b Tan[c z] Sqrt[a + b Tan[c z]^2]))/(2 c (a + b Tan[c z]^2)^(3/2))

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List["\[Integral]", RowBox[List[SqrtBox[SuperscriptBox[RowBox[List["(", RowBox[List["a", "+", RowBox[List["b", " ", SuperscriptBox[RowBox[List["Tan", "[", RowBox[List["c", " ", "z"]], "]"]], "2"]]]]], ")"]], "3"]], RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", RowBox[List[RowBox[List["(", RowBox[List["\[ImaginaryI]", " ", SqrtBox[SuperscriptBox[RowBox[List["(", RowBox[List["a", "+", RowBox[List["b", " ", SuperscriptBox[RowBox[List["Tan", "[", RowBox[List["c", " ", "z"]], "]"]], "2"]]]]], ")"]], "3"]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["3", " ", "a"]], "-", RowBox[List["2", " ", "b"]]]], ")"]], " ", SqrtBox[RowBox[List["-", "b"]]], " ", RowBox[List["Log", "[", RowBox[List["2", " ", RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", SqrtBox[RowBox[List["-", "b"]]], " ", RowBox[List["Tan", "[", RowBox[List["c", " ", "z"]], "]"]]]], "+", SqrtBox[RowBox[List["a", "+", RowBox[List["b", " ", SuperscriptBox[RowBox[List["Tan", "[", RowBox[List["c", " ", "z"]], "]"]], "2"]]]]]]]], ")"]]]], "]"]]]], "+", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["a", "-", "b"]], ")"]], RowBox[List["3", "/", "2"]]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", RowBox[List["Log", "[", RowBox[List["-", FractionBox[RowBox[List["4", " ", RowBox[List["(", RowBox[List["a", "-", RowBox[List["\[ImaginaryI]", " ", "b", " ", RowBox[List["Tan", "[", RowBox[List["c", " ", "z"]], "]"]]]], "+", RowBox[List[SqrtBox[RowBox[List["a", "-", "b"]]], " ", SqrtBox[RowBox[List["a", "+", RowBox[List["b", " ", SuperscriptBox[RowBox[List["Tan", "[", RowBox[List["c", " ", "z"]], "]"]], "2"]]]]]]]]]], ")"]]]], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["a", "-", "b"]], ")"]], RowBox[List["5", "/", "2"]]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", RowBox[List["\[ImaginaryI]", " ", RowBox[List["Tan", "[", RowBox[List["c", " ", "z"]], "]"]]]]]], ")"]]]]]]], "]"]]]], "+", RowBox[List["Log", "[", FractionBox[RowBox[List["4", " ", RowBox[List["(", RowBox[List["a", "+", RowBox[List["\[ImaginaryI]", " ", "b", " ", RowBox[List["Tan", "[", RowBox[List["c", " ", "z"]], "]"]]]], "+", RowBox[List[SqrtBox[RowBox[List["a", "-", "b"]]], " ", SqrtBox[RowBox[List["a", "+", RowBox[List["b", " ", SuperscriptBox[RowBox[List["Tan", "[", RowBox[List["c", " ", "z"]], "]"]], "2"]]]]]]]]]], ")"]]]], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["a", "-", "b"]], ")"]], RowBox[List["5", "/", "2"]]], " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List["\[ImaginaryI]", " ", RowBox[List["Tan", "[", RowBox[List["c", " ", "z"]], "]"]]]]]], ")"]]]]], "]"]]]], ")"]]]], "-", RowBox[List["\[ImaginaryI]", " ", "b", " ", RowBox[List["Tan", "[", RowBox[List["c", " ", "z"]], "]"]], " ", SqrtBox[RowBox[List["a", "+", RowBox[List["b", " ", SuperscriptBox[RowBox[List["Tan", "[", RowBox[List["c", " ", "z"]], "]"]], "2"]]]]]]]]]], ")"]]]], ")"]], "/", RowBox[List["(", RowBox[List["2", " ", "c", " ", SuperscriptBox[RowBox[List["(", RowBox[List["a", "+", RowBox[List["b", " ", SuperscriptBox[RowBox[List["Tan", "[", RowBox[List["c", " ", "z"]], "]"]], "2"]]]]], ")"]], RowBox[List["3", "/", "2"]]]]], ")"]]]]]]]]

 MathML Form

 ( a + b tan 2 ( c z ) ) 3 z 1 2 c ( b tan 2 ( c z ) + a ) 3 / 2 ( ( b tan 2 ( c z ) + a ) 3 ( ( log ( 4 ( a + b tan ( c z ) + a - b b tan 2 ( c z ) + a ) ( a - b ) 5 / 2 ( tan ( c z ) + 1 ) ) - log ( - 4 ( a - b tan ( c z ) + a - b b tan 2 ( c z ) + a ) ( a - b ) 5 / 2 ( tan ( c z ) - 1 ) ) ) ( a - b ) 3 / 2 + - b ( 3 a - 2 b ) log ( 2 ( - b tan ( c z ) + b tan 2 ( c z ) + a ) ) - b tan ( c z ) b tan 2 ( c z ) + a ) ) z a b c z 2 3 1 2 1 2 c b c z 2 a 3 2 -1 b c z 2 a 3 1 2 4 a b c z a -1 b 1 2 b c z 2 a 1 2 a -1 b 5 2 c z 1 -1 -1 -1 4 a -1 b c z a -1 b 1 2 b c z 2 a 1 2 a -1 b 5 2 c z -1 -1 a -1 b 3 2 -1 b 1 2 3 a -1 2 b 2 -1 b 1 2 c z b c z 2 a 1 2 -1 b c z b c z 2 a 1 2 [/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["Tan", "[", RowBox[List["c_", " ", "z_"]], "]"]], "2"]]]]], ")"]], "3"]], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", FractionBox[RowBox[List["\[ImaginaryI]", " ", SqrtBox[SuperscriptBox[RowBox[List["(", RowBox[List["a", "+", RowBox[List["b", " ", SuperscriptBox[RowBox[List["Tan", "[", RowBox[List["c", " ", "z"]], "]"]], "2"]]]]], ")"]], "3"]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["3", " ", "a"]], "-", RowBox[List["2", " ", "b"]]]], ")"]], " ", SqrtBox[RowBox[List["-", "b"]]], " ", RowBox[List["Log", "[", RowBox[List["2", " ", RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", SqrtBox[RowBox[List["-", "b"]]], " ", RowBox[List["Tan", "[", RowBox[List["c", " ", "z"]], "]"]]]], "+", SqrtBox[RowBox[List["a", "+", RowBox[List["b", " ", SuperscriptBox[RowBox[List["Tan", "[", RowBox[List["c", " ", "z"]], "]"]], "2"]]]]]]]], ")"]]]], "]"]]]], "+", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["a", "-", "b"]], ")"]], RowBox[List["3", "/", "2"]]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", RowBox[List["Log", "[", RowBox[List["-", FractionBox[RowBox[List["4", " ", RowBox[List["(", RowBox[List["a", "-", RowBox[List["\[ImaginaryI]", " ", "b", " ", RowBox[List["Tan", "[", RowBox[List["c", " ", "z"]], "]"]]]], "+", RowBox[List[SqrtBox[RowBox[List["a", "-", "b"]]], " ", SqrtBox[RowBox[List["a", "+", RowBox[List["b", " ", SuperscriptBox[RowBox[List["Tan", "[", RowBox[List["c", " ", "z"]], "]"]], "2"]]]]]]]]]], ")"]]]], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["a", "-", "b"]], ")"]], RowBox[List["5", "/", "2"]]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", RowBox[List["\[ImaginaryI]", " ", RowBox[List["Tan", "[", RowBox[List["c", " ", "z"]], "]"]]]]]], ")"]]]]]]], "]"]]]], "+", RowBox[List["Log", "[", FractionBox[RowBox[List["4", " ", RowBox[List["(", RowBox[List["a", "+", RowBox[List["\[ImaginaryI]", " ", "b", " ", RowBox[List["Tan", "[", RowBox[List["c", " ", "z"]], "]"]]]], "+", RowBox[List[SqrtBox[RowBox[List["a", "-", "b"]]], " ", SqrtBox[RowBox[List["a", "+", RowBox[List["b", " ", SuperscriptBox[RowBox[List["Tan", "[", RowBox[List["c", " ", "z"]], "]"]], "2"]]]]]]]]]], ")"]]]], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["a", "-", "b"]], ")"]], RowBox[List["5", "/", "2"]]], " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List["\[ImaginaryI]", " ", RowBox[List["Tan", "[", RowBox[List["c", " ", "z"]], "]"]]]]]], ")"]]]]], "]"]]]], ")"]]]], "-", RowBox[List["\[ImaginaryI]", " ", "b", " ", RowBox[List["Tan", "[", RowBox[List["c", " ", "z"]], "]"]], " ", SqrtBox[RowBox[List["a", "+", RowBox[List["b", " ", SuperscriptBox[RowBox[List["Tan", "[", RowBox[List["c", " ", "z"]], "]"]], "2"]]]]]]]]]], ")"]]]], RowBox[List["2", " ", "c", " ", SuperscriptBox[RowBox[List["(", RowBox[List["a", "+", RowBox[List["b", " ", SuperscriptBox[RowBox[List["Tan", "[", RowBox[List["c", " ", "z"]], "]"]], "2"]]]]], ")"]], RowBox[List["3", "/", "2"]]]]]]]]]]

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

 2002-12-18