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Cos






Mathematica Notation

Traditional Notation









Elementary Functions > Cos[z] > Integration > Indefinite integration > Involving functions of the direct function > Involving algebraic functions of the direct function > Involving (a+b sin2(c z))betaand rational function of sin(c z)





http://functions.wolfram.com/01.07.21.1420.01









  


  










Input Form





Integrate[1/((d + e Cos[c z]^2)^2 Sqrt[a + b Cos[c z]^2]), z] == (Sin[c z] ((I b Sqrt[1 + (b Cos[c z]^2)/a] EllipticF[I ArcSinh[Sqrt[b/a] Cos[c z]], -(a/b)])/ ((-b) c d^2 + a c d e) + (I (a e (2 d + e) - b d (3 d + 2 e)) Sqrt[1 + (b Cos[c z]^2)/a] EllipticPi[(a e)/(b d), I ArcSinh[Sqrt[b/a] Cos[c z]], -(a/b)])/(c d^2 (d + e) ((-b) d + a e)) - (e (I b Sqrt[1 + (b Cos[c z]^2)/a] (d + e Cos[c z]^2) EllipticE[I ArcSinh[Sqrt[b/a] Cos[c z]], -(a/b)] + Sqrt[b/a] e Cos[c z] (a + b Cos[c z]^2) Sqrt[Sin[c z]^2]))/ (c d (d + e) ((-b) d + a e) (d + e Cos[c z]^2))))/ (2 Sqrt[b/a] Sqrt[a + b Cos[c z]^2] Sqrt[Sin[c z]^2])










Standard Form





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MathML Form







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-1 </cn> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <plus /> <apply> <times /> <ci> a </ci> <ci> c </ci> <ci> d </ci> <ci> e </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <ci> b </ci> <ci> c </ci> <apply> <power /> <ci> d </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <apply> <times /> <imaginaryi /> <apply> <plus /> <apply> <times /> <ci> a </ci> <ci> e </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> d </ci> </apply> <ci> e </ci> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <ci> b </ci> <ci> d </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> 3 </cn> <ci> d </ci> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <ci> e </ci> </apply> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <plus /> <apply> <times /> <ci> b </ci> <apply> <power /> <apply> <cos /> 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<power /> <apply> <times /> <ci> c </ci> <apply> <power /> <ci> d </ci> <cn type='integer'> 2 </cn> </apply> <apply> <plus /> <ci> d </ci> <ci> e </ci> </apply> <apply> <plus /> <apply> <times /> <ci> a </ci> <ci> e </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <ci> b </ci> <ci> d </ci> </apply> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <times /> <ci> e </ci> <apply> <plus /> <apply> <times /> <apply> <power /> <apply> <times /> <ci> b </ci> <apply> <power /> <ci> a </ci> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <ci> e </ci> <apply> <cos /> <apply> <times /> <ci> c </ci> <ci> z </ci> </apply> </apply> <apply> <power /> <apply> <power /> <apply> <sin /> <apply> <times /> <ci> c </ci> <ci> z </ci> </apply> </apply> <cn type='integer'> 2 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <plus /> <apply> <times /> <ci> b </ci> <apply> <power /> <apply> <cos /> <apply> <times /> <ci> c </ci> <ci> z </ci> </apply> </apply> <cn type='integer'> 2 </cn> </apply> </apply> <ci> a </ci> </apply> </apply> <apply> <times /> <apply> <power /> <apply> <plus /> <apply> <times /> <ci> b </ci> <apply> <power /> <apply> <cos /> <apply> <times /> <ci> c </ci> <ci> z </ci> </apply> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <ci> a </ci> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='integer'> 1 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <ci> b </ci> <imaginaryi /> <apply> <plus /> <apply> <times /> <ci> e </ci> <apply> <power /> <apply> <cos /> <apply> <times /> <ci> c </ci> <ci> z </ci> </apply> </apply> <cn type='integer'> 2 </cn> </apply> </apply> <ci> d </ci> </apply> <apply> <ci> EllipticE </ci> <apply> <times /> <imaginaryi /> <apply> <arcsinh /> <apply> <times /> <apply> <power /> <apply> <times /> <ci> b </ci> <apply> <power /> <ci> a </ci> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <cos /> <apply> <times /> <ci> c </ci> <ci> z </ci> </apply> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <ci> a </ci> <apply> <power /> <ci> b </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <times /> <ci> c </ci> <ci> d </ci> <apply> <plus /> <ci> d </ci> <ci> e </ci> </apply> <apply> <plus /> <apply> <times /> <ci> a </ci> <ci> e </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <ci> b </ci> <ci> d </ci> </apply> </apply> </apply> <apply> <plus /> <apply> <times /> <ci> e </ci> <apply> <power /> <apply> <cos /> <apply> <times /> <ci> c </ci> <ci> z </ci> </apply> </apply> <cn type='integer'> 2 </cn> </apply> </apply> <ci> d </ci> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <apply> <times /> <ci> b </ci> <apply> <power /> <ci> a </ci> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <plus /> <apply> <times /> <ci> b </ci> <apply> <power /> <apply> <cos /> <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 /> <apply> <power /> <apply> <sin /> <apply> <times /> <ci> c </ci> <ci> z </ci> </apply> </apply> <cn type='integer'> 2 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





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





2002-12-18