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 Tan

 http://functions.wolfram.com/01.08.21.0128.01

 Input Form

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

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List["\[Integral]", RowBox[List[SqrtBox[RowBox[List["a", "+", RowBox[List["b", " ", SqrtBox[RowBox[List["Tan", "[", RowBox[List["c", " ", "z"]], "]"]]]]]]]], RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", 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["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"]]]]]], "]"]]]], "+", 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["\[ImaginaryI]", " ", "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"]]]]]], "]"]]]], "+", 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[RowBox[List["-", "\[ImaginaryI]"]], " ", 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"]]]]]], "]"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "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"]]]]], " ", 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"]]]]]], "]"]]]]]], ")"]]]]]], ")"]]]]]], ")"]], "/", 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["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"]]]]], " ", "c"]], ")"]]]]]]]]

 MathML Form

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

 Rule Form

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

 2002-12-18