Tangent: Integration (formula 01.08.21.0209)

Tan

$Mathematica Notation$

Elementary Functions

Tan[z]

Integration

Indefinite integration

Involving functions of the direct function and trigonometric functions

Involving algebraic functions of the direct function and trigonometric functions

Involving cos

Involving cos(c z)(a+b tan(c z))^beta

http://functions.wolfram.com/01.08.21.0209.01

Input Form

Integrate[Cos[c z] Sqrt[a + b Tan[c z]], z] == (Cos[c z] ((-((-I) a + b)) EllipticE[ I ArcSinh[Sqrt[-a - I b]/Sqrt[a + b Tan[c z]]], (a - I b)/(a + I b)] Sqrt[(b (-I + Tan[c z]))/(a + b Tan[c z])] Sqrt[(b (I + Tan[c z]))/(a + b Tan[c z])] (a + b Tan[c z])^2 + b (Sqrt[-a - I b] (-b + a Tan[c z]) Sqrt[a + b Tan[c z]] + EllipticF[I ArcSinh[Sqrt[-a - I b]/Sqrt[a + b Tan[c z]]], (a - I b)/(a + I b)] Sqrt[(b (-I + Tan[c z]))/(a + b Tan[c z])] Sqrt[(b (I + Tan[c z]))/(a + b Tan[c z])] (a + b Tan[c z])^2)))/ (Sqrt[-a - I b] b c (a + b Tan[c z]))

Standard Form

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MathML Form

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<mi> c </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mrow> <mi> a </mi> <mo> + </mo> <mrow> <mi> b </mi> <mo> ⁢ </mo> <mrow> <mi> tan </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> c </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mfrac> </msqrt> <mo> ⁢ </mo> <msqrt> <mfrac> <mrow> <mi> b </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> ⅈ </mi> <mo> + </mo> <mrow> <mi> tan </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> c </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mrow> <mi> a </mi> <mo> + </mo> <mrow> <mi> b </mi> <mo> ⁢ </mo> <mrow> <mi> tan </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> c </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mfrac> </msqrt> <mo> ⁢ </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mi> a </mi> <mo> + </mo> <mrow> <mi> b </mi> <mo> ⁢ </mo> <mrow> <mi> tan </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> c </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> / </mo> <mrow> <mo> ( </mo> <mrow> <msqrt> <mrow> <mrow> <mo> - </mo> <mi> a </mi> </mrow> <mo> - </mo> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> b </mi> </mrow> </mrow> </msqrt> <mo> ⁢ </mo> <mi> b </mi> <mo> ⁢ </mo> <mi> c </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> a </mi> <mo> + </mo> <mrow> <mi> b </mi> <mo> ⁢ </mo> <mrow> <mi> tan </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> c </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <int /> <bvar> <ci> z </ci> </bvar> <apply> <times /> <apply> <cos /> <apply> <times /> <ci> c </ci> <ci> z </ci> </apply> </apply> <apply> <power /> <apply> <plus /> <ci> a </ci> <apply> <times /> <ci> b </ci> <apply> 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</ci> <ci> c </ci> <apply> <plus /> <ci> a </ci> <apply> <times /> <ci> b </ci> <apply> <tan /> <apply> <times /> <ci> c </ci> <ci> z </ci> </apply> </apply> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

Rule Form

Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["\[Integral]", RowBox[List[RowBox[List[RowBox[List["Cos", "[", RowBox[List["c_", " ", "z_"]], "]"]], " ", SqrtBox[RowBox[List["a_", "+", RowBox[List["b_", " ", RowBox[List["Tan", "[", RowBox[List["c_", " ", "z_"]], "]"]]]]]]]]], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", FractionBox[RowBox[List[RowBox[List["Cos", "[", RowBox[List["c", " ", "z"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "a"]], "+", "b"]], ")"]]]], " ", RowBox[List["EllipticE", "[", RowBox[List[RowBox[List["\[ImaginaryI]", " ", RowBox[List["ArcSinh", "[", FractionBox[SqrtBox[RowBox[List[RowBox[List["-", "a"]], "-", RowBox[List["\[ImaginaryI]", " ", "b"]]]]], SqrtBox[RowBox[List["a", "+", RowBox[List["b", " ", RowBox[List["Tan", "[", RowBox[List["c", " ", "z"]], "]"]]]]]]]], "]"]]]], ",", FractionBox[RowBox[List["a", "-", RowBox[List["\[ImaginaryI]", " ", "b"]]]], RowBox[List["a", "+", RowBox[List["\[ImaginaryI]", " ", "b"]]]]]]], "]"]], " ", SqrtBox[FractionBox[RowBox[List["b", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], "+", RowBox[List["Tan", "[", RowBox[List["c", " ", "z"]], "]"]]]], ")"]]]], RowBox[List["a", "+", RowBox[List["b", " ", RowBox[List["Tan", "[", RowBox[List["c", " ", "z"]], "]"]]]]]]]], " ", SqrtBox[FractionBox[RowBox[List["b", " ", RowBox[List["(", RowBox[List["\[ImaginaryI]", "+", RowBox[List["Tan", "[", RowBox[List["c", " ", "z"]], "]"]]]], ")"]]]], RowBox[List["a", "+", RowBox[List["b", " ", RowBox[List["Tan", "[", RowBox[List["c", " ", "z"]], "]"]]]]]]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["a", "+", RowBox[List["b", " ", RowBox[List["Tan", "[", RowBox[List["c", " ", "z"]], "]"]]]]]], ")"]], "2"]]], "+", RowBox[List["b", " ", RowBox[List["(", RowBox[List[RowBox[List[SqrtBox[RowBox[List[RowBox[List["-", "a"]], "-", RowBox[List["\[ImaginaryI]", " ", "b"]]]]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "b"]], "+", RowBox[List["a", " ", RowBox[List["Tan", "[", RowBox[List["c", " ", "z"]], "]"]]]]]], ")"]], " ", SqrtBox[RowBox[List["a", "+", RowBox[List["b", " ", RowBox[List["Tan", "[", RowBox[List["c", " ", "z"]], "]"]]]]]]]]], "+", RowBox[List[RowBox[List["EllipticF", "[", RowBox[List[RowBox[List["\[ImaginaryI]", " ", RowBox[List["ArcSinh", "[", FractionBox[SqrtBox[RowBox[List[RowBox[List["-", "a"]], "-", RowBox[List["\[ImaginaryI]", " ", "b"]]]]], SqrtBox[RowBox[List["a", "+", RowBox[List["b", " ", RowBox[List["Tan", "[", RowBox[List["c", " ", "z"]], "]"]]]]]]]], "]"]]]], ",", FractionBox[RowBox[List["a", "-", RowBox[List["\[ImaginaryI]", " ", "b"]]]], RowBox[List["a", "+", RowBox[List["\[ImaginaryI]", " ", "b"]]]]]]], "]"]], " ", SqrtBox[FractionBox[RowBox[List["b", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], "+", RowBox[List["Tan", "[", RowBox[List["c", " ", "z"]], "]"]]]], ")"]]]], RowBox[List["a", "+", RowBox[List["b", " ", RowBox[List["Tan", "[", RowBox[List["c", " ", "z"]], "]"]]]]]]]], " ", SqrtBox[FractionBox[RowBox[List["b", " ", RowBox[List["(", RowBox[List["\[ImaginaryI]", "+", RowBox[List["Tan", "[", RowBox[List["c", " ", "z"]], "]"]]]], ")"]]]], RowBox[List["a", "+", RowBox[List["b", " ", RowBox[List["Tan", "[", RowBox[List["c", " ", "z"]], "]"]]]]]]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["a", "+", RowBox[List["b", " ", RowBox[List["Tan", "[", RowBox[List["c", " ", "z"]], "]"]]]]]], ")"]], "2"]]]]], ")"]]]]]], ")"]]]], RowBox[List[SqrtBox[RowBox[List[RowBox[List["-", "a"]], "-", RowBox[List["\[ImaginaryI]", " ", "b"]]]]], " ", "b", " ", "c", " ", RowBox[List["(", RowBox[List["a", "+", RowBox[List["b", " ", RowBox[List["Tan", "[", RowBox[List["c", " ", "z"]], "]"]]]]]], ")"]]]]]]]]]

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

2002-12-18