Wolfram Researchfunctions.wolfram.comOther Wolfram Sites
Search Site
Function CategoriesGraphics GalleryNotationsGeneral IdentitiesAbout This Site Email Comments

View Related Information In
The Documentation Center
MathWorld

Download All Formulas For This Function
Mathematica Notebook
PDF File

Download All Introductions For This Function
Mathematica Notebook
PDF File

 

Developed with Mathematica -- Download a Free Trial Version
 











Tan




Mathematica Notation

Traditional Notation









Elementary Functions > Tan[z] > Integration > Indefinite integration > Involving functions of the direct function > Involving algebraic functions of the direct function > Involving (a+b tan2(c z))beta





http://functions.wolfram.com/01.08.21.0117.01









  


  








Input Form





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








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", " ", SuperscriptBox[RowBox[List["Tan", "[", RowBox[List["c", " ", "z"]], "]"]], "2"]]]]]], RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", RowBox[List[RowBox[List["-", FractionBox["1", "c"]]], RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", RowBox[List[RowBox[List["(", RowBox[List[SqrtBox["2"], " ", SqrtBox[RowBox[List[RowBox[List["(", RowBox[List["a", "-", "b"]], ")"]], " ", SuperscriptBox[RowBox[List["Cos", "[", RowBox[List["c", " ", "z"]], "]"]], "2"]]]], " ", 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["(", SqrtBox[RowBox[List["a", "+", "b", "+", RowBox[List[RowBox[List["(", RowBox[List["a", "-", "b"]], ")"]], " ", RowBox[List["Cos", "[", RowBox[List["2", " ", "c", " ", "z"]], "]"]]]]]]], ")"]]]]]], ")"]], " ", SqrtBox[RowBox[List["a", "+", RowBox[List["b", " ", SuperscriptBox[RowBox[List["Tan", "[", RowBox[List["c", " ", "z"]], "]"]], "2"]]]]]]]]]]]]








MathML Form







<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mo> &#8747; </mo> <mrow> <mrow> <mrow> <mi> tan </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> c </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msqrt> <mrow> <mi> a </mi> <mo> + </mo> <mrow> <mi> b </mi> <mo> &#8290; </mo> <mrow> <msup> <mi> tan </mi> <mn> 2 </mn> </msup> <mo> ( </mo> <mrow> <mi> c </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </msqrt> </mrow> <mo> &#8290; </mo> <mrow> <mo> &#8518; </mo> <mi> z </mi> </mrow> </mrow> </mrow> <mo> &#10869; </mo> <mrow> <mrow> <mo> - </mo> <mfrac> <msqrt> <mrow> <mrow> <mi> b </mi> <mo> &#8290; </mo> <mrow> <msup> <mi> tan </mi> <mn> 2 </mn> </msup> <mo> ( </mo> <mrow> <mi> c </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mi> a </mi> </mrow> </msqrt> <mi> c </mi> </mfrac> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mfrac> <mrow> <msqrt> <mn> 2 </mn> </msqrt> <mo> &#8290; </mo> <msqrt> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> a </mi> <mo> - </mo> <mi> b </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <msup> <mi> cos </mi> <mn> 2 </mn> </msup> <mo> ( </mo> <mrow> <mi> c </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </msqrt> <mo> &#8290; </mo> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mrow> <msqrt> <mn> 2 </mn> </msqrt> <mo> &#8290; </mo> <msqrt> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> a </mi> <mo> - </mo> <mi> b </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <msup> <mi> cos </mi> <mn> 2 </mn> </msup> <mo> ( </mo> <mrow> <mi> c </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </msqrt> </mrow> <mo> + </mo> <msqrt> <mrow> <mi> a </mi> <mo> + </mo> <mi> b </mi> <mo> + </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> a </mi> <mo> - </mo> <mi> b </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> cos </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> c </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </msqrt> </mrow> <mo> ) </mo> </mrow> </mrow> <msqrt> <mrow> <mi> a </mi> <mo> + </mo> <mi> b </mi> <mo> + </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> a </mi> <mo> - </mo> <mi> b </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> cos </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> c </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </msqrt> </mfrac> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mtext> </mtext> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <int /> <bvar> <ci> z </ci> </bvar> <apply> <times /> <apply> <tan /> <apply> <times /> <ci> c </ci> <ci> z </ci> </apply> </apply> <apply> <power /> <apply> <plus /> <ci> a </ci> <apply> <times /> <ci> b </ci> <apply> <power /> <apply> <tan /> <apply> <times /> <ci> c </ci> <ci> z </ci> </apply> </apply> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <power /> <apply> <plus /> <apply> <times /> <ci> b </ci> <apply> <power /> <apply> <tan /> <apply> <times /> <ci> c </ci> <ci> z </ci> </apply> </apply> <cn type='integer'> 2 </cn> </apply> </apply> <ci> a </ci> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <ci> c </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <plus /> <apply> <times /> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <times /> <apply> <plus /> <ci> a </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> b </ci> </apply> </apply> <apply> <power /> <apply> <cos /> <apply> <times /> <ci> c </ci> <ci> z </ci> </apply> </apply> <cn type='integer'> 2 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <ln /> <apply> <plus /> <apply> <times /> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <times /> <apply> <plus /> <ci> a </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> b </ci> </apply> </apply> <apply> <power /> <apply> <cos /> <apply> <times /> <ci> c </ci> <ci> z </ci> </apply> </apply> <cn type='integer'> 2 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <apply> <power /> <apply> <plus /> <ci> a </ci> <ci> b </ci> <apply> <times /> <apply> <plus /> <ci> a </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> b </ci> </apply> </apply> <apply> <cos /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> c </ci> <ci> z </ci> </apply> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> <apply> <power /> <apply> <power /> <apply> <plus /> <ci> a </ci> <ci> b </ci> <apply> <times /> <apply> <plus /> <ci> a </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> b </ci> </apply> </apply> <apply> <cos /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> c </ci> <ci> z </ci> </apply> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> -1 </cn> </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["Tan", "[", RowBox[List["c_", " ", "z_"]], "]"]], " ", SqrtBox[RowBox[List["a_", "+", RowBox[List["b_", " ", SuperscriptBox[RowBox[List["Tan", "[", RowBox[List["c_", " ", "z_"]], "]"]], "2"]]]]]]]], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List["-", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", FractionBox[RowBox[List[SqrtBox["2"], " ", SqrtBox[RowBox[List[RowBox[List["(", RowBox[List["a", "-", "b"]], ")"]], " ", SuperscriptBox[RowBox[List["Cos", "[", RowBox[List["c", " ", "z"]], "]"]], "2"]]]], " ", 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"]], "]"]]]]]]]]], "]"]]]], SqrtBox[RowBox[List["a", "+", "b", "+", RowBox[List[RowBox[List["(", RowBox[List["a", "-", "b"]], ")"]], " ", RowBox[List["Cos", "[", RowBox[List["2", " ", "c", " ", "z"]], "]"]]]]]]]]]], ")"]], " ", SqrtBox[RowBox[List["a", "+", RowBox[List["b", " ", SuperscriptBox[RowBox[List["Tan", "[", RowBox[List["c", " ", "z"]], "]"]], "2"]]]]]]]], "c"]]]]]]]


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





2002-12-18