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Sinh






Mathematica Notation

Traditional Notation









Elementary Functions > Sinh[z] > Integration > Indefinite integration > Involving functions of the direct function > Involving algebraic functions of the direct function > Other integrals





http://functions.wolfram.com/01.19.21.1833.01









  


  










Input Form





Integrate[Sqrt[(a + b Sinh[e z]^2)/(c + d Sinh[e z]^2)]/(c + d Sinh[e z]^2), z] == (Sqrt[(a Cosh[e z]^2)/(2 a - b + b Cosh[2 e z])] Sech[e z] ((-I) a Sqrt[1 - b/a] (2 c - d + d Cosh[2 e z]) EllipticE[I ArcSinh[(Sqrt[2 - (2 b)/a] Sinh[e z])/ Sqrt[(2 a - b + b Cosh[2 e z])/a]], (b c - a d)/((-a) c + b c)] + 2 (b c - a d) Sqrt[(a Cosh[e z]^2)/(2 a - b + b Cosh[2 e z])] Sqrt[(2 a - b + b Cosh[2 e z])/a] Sqrt[(a (2 c - d + d Cosh[2 e z]))/ (c (2 a - b + b Cosh[2 e z]))] Sinh[e z]) Sqrt[(a + b Sinh[e z]^2)/(c + d Sinh[e z]^2)])/ (a c (c - d) e Sqrt[(2 a - b + b Cosh[2 e z])/a] Sqrt[(a (2 c - d + d Cosh[2 e z]))/(c (2 a - b + b Cosh[2 e z]))])










Standard Form





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MathML Form







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-1 </cn> <apply> <times /> <ci> a </ci> <ci> d </ci> </apply> </apply> </apply> <apply> <power /> <apply> <times /> <ci> a </ci> <apply> <power /> <apply> <cosh /> <apply> <times /> <ci> e </ci> <ci> z </ci> </apply> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> a </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> b </ci> </apply> <apply> <times /> <ci> b </ci> <apply> <cosh /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> e </ci> <ci> z </ci> </apply> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <times /> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> a </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> b </ci> </apply> <apply> <times /> <ci> b </ci> <apply> <cosh /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> e </ci> <ci> z </ci> 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<apply> <times /> <ci> e </ci> <ci> z </ci> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <imaginaryi /> <ci> a </ci> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <ci> b </ci> <apply> <power /> <ci> a </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> c </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> d </ci> </apply> <apply> <times /> <ci> d </ci> <apply> <cosh /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> e </ci> <ci> z </ci> </apply> </apply> </apply> </apply> <apply> <ci> EllipticE </ci> <apply> <times /> <imaginaryi /> <apply> <arcsinh /> <apply> <times /> <apply> <power /> <apply> <plus /> <cn type='integer'> 2 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <ci> b </ci> <apply> <power /> <ci> a </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <sinh /> <apply> <times /> <ci> e </ci> <ci> z </ci> </apply> </apply> <apply> <power /> <apply> <power /> <apply> <times /> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> a </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> b </ci> </apply> <apply> <times /> <ci> b </ci> <apply> <cosh /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> e </ci> <ci> z </ci> </apply> </apply> </apply> </apply> <apply> <power /> <ci> a </ci> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <times /> <apply> <plus /> <apply> <times /> <ci> b </ci> <ci> c </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <ci> a </ci> <ci> d </ci> </apply> </apply> </apply> <apply> <power /> <apply> <plus /> <apply> <times /> <ci> b </ci> <ci> c </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <ci> a </ci> <ci> c </ci> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <times /> <apply> <plus /> <apply> <times /> <ci> b </ci> <apply> <power /> <apply> <sinh /> <apply> <times /> <ci> e </ci> <ci> z </ci> </apply> </apply> <cn type='integer'> 2 </cn> </apply> </apply> <ci> a </ci> </apply> <apply> <power /> <apply> <plus /> <apply> <times /> <ci> d </ci> <apply> <power /> <apply> <sinh /> <apply> <times /> <ci> e </ci> <ci> z </ci> </apply> </apply> <cn type='integer'> 2 </cn> </apply> </apply> <ci> c </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <apply> <power /> <apply> <times /> <ci> a </ci> <ci> c </ci> <apply> <plus /> <ci> c </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> d </ci> </apply> </apply> <ci> e </ci> <apply> <power /> <apply> <times /> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> a </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> b </ci> </apply> <apply> <times /> <ci> b </ci> <apply> <cosh /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> e </ci> <ci> z </ci> </apply> </apply> </apply> </apply> <apply> <power /> <ci> a </ci> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <times /> <ci> a </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> c </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> d </ci> </apply> <apply> <times /> <ci> d </ci> <apply> <cosh /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> e </ci> <ci> z </ci> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <times /> <ci> c </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> a </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> b </ci> </apply> <apply> <times /> <ci> b </ci> <apply> <cosh /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> e </ci> <ci> z </ci> </apply> </apply> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </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