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 Tan

 http://functions.wolfram.com/01.08.21.0131.01

 Input Form

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

 Standard Form

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

 MathML Form

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

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["\[Integral]", RowBox[List[RowBox[List[RowBox[List["Tan", "[", RowBox[List["c_", " ", "z_"]], "]"]], " ", SqrtBox[RowBox[List["a_", "+", RowBox[List["b_", " ", SqrtBox[RowBox[List["Tan", "[", RowBox[List["c_", " ", "z_"]], "]"]]]]]]]]]], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", FractionBox[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["ArcTanh", "[", FractionBox[SqrtBox[RowBox[List["a", "+", RowBox[List["b", " ", SqrtBox[RowBox[List["Tan", "[", RowBox[List["c", " ", "z"]], "]"]]]]]]]], SqrtBox[RowBox[List["a", "-", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["1", "/", "4"]]], " ", "b"]]]]]], "]"]]]], "+", 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[SqrtBox[RowBox[List["a", "-", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["3", "/", "4"]]], " ", "b"]]]]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", RowBox[List["(", RowBox[List["a", "+", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["3", "/", "4"]]], " ", "b"]]]], ")"]]]], " ", RowBox[List["ArcTanh", "[", FractionBox[SqrtBox[RowBox[List["a", "+", RowBox[List["b", " ", SqrtBox[RowBox[List["Tan", "[", RowBox[List["c", " ", "z"]], "]"]]]]]]]], SqrtBox[RowBox[List["a", "+", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["3", "/", "4"]]], " ", "b"]]]]]], "]"]], " ", SqrtBox[RowBox[List["a", "+", RowBox[List["b", " ", SqrtBox[RowBox[List["Tan", "[", RowBox[List["c", " ", "z"]], "]"]]]]]]]]]], "+", RowBox[List["4", " ", SqrtBox[RowBox[List["a", "+", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["3", "/", "4"]]], " ", "b"]]]]], " ", RowBox[List["(", RowBox[List["a", "+", RowBox[List["b", " ", SqrtBox[RowBox[List["Tan", "[", RowBox[List["c", " ", "z"]], "]"]]]]]]], ")"]]]]]], ")"]]]], "-", RowBox[List[RowBox[List["(", RowBox[List["a", "-", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["3", "/", "4"]]], " ", "b"]]]], ")"]], " ", SqrtBox[RowBox[List["a", "+", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["3", "/", "4"]]], " ", "b"]]]]], " ", RowBox[List["ArcTanh", "[", FractionBox[SqrtBox[RowBox[List["a", "+", RowBox[List["b", " ", SqrtBox[RowBox[List["Tan", "[", RowBox[List["c", " ", "z"]], "]"]]]]]]]], SqrtBox[RowBox[List["a", "-", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["3", "/", "4"]]], " ", "b"]]]]]], "]"]], " ", SqrtBox[RowBox[List["a", "+", RowBox[List["b", " ", SqrtBox[RowBox[List["Tan", "[", RowBox[List["c", " ", "z"]], "]"]]]]]]]]]]]], ")"]]]], "-", RowBox[List[RowBox[List["(", RowBox[List["a", "+", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["1", "/", "4"]]], " ", "b"]]]], ")"]], " ", SqrtBox[RowBox[List["a", "-", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["3", "/", "4"]]], " ", "b"]]]]], " ", SqrtBox[RowBox[List["a", "+", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["3", "/", "4"]]], " ", "b"]]]]], " ", RowBox[List["ArcTanh", "[", FractionBox[SqrtBox[RowBox[List["a", "+", RowBox[List["b", " ", SqrtBox[RowBox[List["Tan", "[", RowBox[List["c", " ", "z"]], "]"]]]]]]]], SqrtBox[RowBox[List["a", "+", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["1", "/", "4"]]], " ", "b"]]]]]], "]"]], " ", SqrtBox[RowBox[List["a", "+", RowBox[List["b", " ", SqrtBox[RowBox[List["Tan", "[", RowBox[List["c", " ", "z"]], "]"]]]]]]]]]]]], RowBox[List[SqrtBox[RowBox[List["a", "+", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["1", "/", "4"]]], " ", "b"]]]]], " ", SqrtBox[RowBox[List["a", "-", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["3", "/", "4"]]], " ", "b"]]]]], " ", SqrtBox[RowBox[List["a", "+", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["3", "/", "4"]]], " ", "b"]]]]], " ", SqrtBox[RowBox[List["a", "+", RowBox[List["b", " ", SqrtBox[RowBox[List["Tan", "[", RowBox[List["c", " ", "z"]], "]"]]]]]]]]]]]]], "c"]]]]]

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

 2002-12-18