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variants of this functions
ArcTan




Mathematica Notation

Traditional Notation









Elementary Functions > ArcTan[z] > Integration > Indefinite integration > For the direct function itself





http://functions.wolfram.com/01.14.21.0012.01









  


  








Input Form





Integrate[ArcTan[a Tan[z]], z] == (1/4) ((2 ArcTan[a Tan[z]]^2)/a - (Sqrt[1 - a^2] ArcTan[a Tan[z]]^2)/ (E^ArcTanh[a] a) - (Sqrt[1 - a^2] E^ArcTanh[a] ArcTan[a Tan[z]]^2)/a + 4 I ArcTan[a Tan[z]] ArcTanh[a] + 2 I ArcTan[a Tan[z]] Log[1 - E^(2 I ArcTan[a Tan[z]] - 2 ArcTanh[a])] - 2 ArcTanh[a] Log[1 - E^(2 I ArcTan[a Tan[z]] - 2 ArcTanh[a])] - 2 I ArcTan[a Tan[z]] Log[1 - E^(2 I ArcTan[a Tan[z]] + 2 ArcTanh[a])] - 2 ArcTanh[a] Log[1 - E^(2 I ArcTan[a Tan[z]] + 2 ArcTanh[a])] + 2 ArcTanh[a] Log[Sin[ArcTan[a Tan[z]] - I ArcTanh[a]]] + 2 ArcTanh[a] Log[Sin[ArcTan[a Tan[z]] + I ArcTanh[a]]] + PolyLog[2, E^(2 I ArcTan[a Tan[z]] - 2 ArcTanh[a])] - PolyLog[2, E^(2 I ArcTan[a Tan[z]] + 2 ArcTanh[a])])








Standard Form





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MathML Form







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<mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mrow> <msup> <mi> tanh </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mi> a </mi> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ) </mo> </mrow> <mo> + </mo> <mrow> <msub> <semantics> <mi> Li </mi> <annotation-xml encoding='MathML-Content'> <ci> PolyLog </ci> </annotation-xml> </semantics> <mn> 2 </mn> </msub> <mo> ( </mo> <mrow> <mi> exp </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> &#8520; </mi> <mo> &#8290; </mo> <mrow> <msup> <mi> tan </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mrow> <mi> a </mi> <mo> &#8290; </mo> <mrow> <mi> tan </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mrow> <msup> <mi> tanh </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mi> a </mi> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </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> <arctan /> <apply> <times /> <ci> a </ci> <apply> <tan /> <ci> z </ci> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='rational'> 1 <sep /> 4 </cn> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <ci> a </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <exponentiale /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <arctanh /> <ci> a </ci> </apply> </apply> </apply> <apply> <power /> <apply> <arctan /> <apply> <times /> <ci> a </ci> <apply> <tan /> <ci> z </ci> </apply> </apply> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <ci> a </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <ci> a </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <exponentiale /> <apply> <arctanh /> <ci> a </ci> </apply> </apply> <apply> <power /> <apply> <arctan /> <apply> <times /> <ci> a </ci> <apply> <tan /> <ci> z </ci> </apply> </apply> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <ci> a </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <apply> <arctan /> <apply> <times /> <ci> a </ci> <apply> <tan /> <ci> z </ci> </apply> </apply> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <ci> a </ci> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 4 </cn> <imaginaryi /> <apply> <arctanh /> <ci> a </ci> </apply> <apply> <arctan /> <apply> <times /> <ci> a </ci> <apply> <tan /> <ci> z </ci> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <imaginaryi /> <apply> <ln /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <exp /> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <imaginaryi /> <apply> <arctan /> <apply> <times /> <ci> a </ci> <apply> <tan /> <ci> z </ci> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <arctanh /> <ci> a </ci> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <arctan /> <apply> <times /> <ci> a </ci> <apply> <tan /> <ci> z </ci> </apply> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <imaginaryi /> <apply> <ln /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <exp /> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <imaginaryi /> <apply> <arctan /> <apply> <times /> <ci> a </ci> <apply> <tan /> <ci> z </ci> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <arctanh /> <ci> a </ci> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <arctan /> <apply> <times /> <ci> a </ci> <apply> <tan /> <ci> z </ci> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <arctanh /> <ci> a </ci> </apply> <apply> <ln /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <exp /> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <imaginaryi /> <apply> <arctan /> <apply> <times /> <ci> a </ci> <apply> <tan /> <ci> z </ci> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <arctanh /> <ci> a </ci> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <arctanh /> <ci> a </ci> </apply> <apply> <ln /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <exp /> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <imaginaryi /> <apply> <arctan /> <apply> <times /> <ci> a </ci> <apply> <tan /> <ci> z </ci> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <arctanh /> <ci> a </ci> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <arctanh /> <ci> a </ci> </apply> <apply> <ln /> <apply> <sin /> <apply> <plus /> <apply> <arctan /> <apply> <times /> <ci> a </ci> <apply> <tan /> <ci> z </ci> </apply> </apply> </apply> <apply> <times /> <imaginaryi /> <apply> <arctanh /> <ci> a </ci> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <arctanh /> <ci> a </ci> </apply> <apply> <ln /> <apply> <sin /> <apply> <plus /> <apply> <arctan /> <apply> <times /> <ci> a </ci> <apply> <tan /> <ci> z </ci> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <imaginaryi /> <apply> <arctanh /> <ci> a </ci> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> PolyLog </ci> <cn type='integer'> 2 </cn> <apply> <exp /> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <imaginaryi /> <apply> <arctan /> <apply> <times /> <ci> a </ci> <apply> <tan /> <ci> z </ci> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <arctanh /> <ci> a </ci> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <ci> PolyLog </ci> <cn type='integer'> 2 </cn> <apply> <exp /> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <imaginaryi /> <apply> <arctan /> <apply> <times /> <ci> a </ci> <apply> <tan /> <ci> z </ci> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <arctanh /> <ci> a </ci> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





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





2001-10-29