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Coth




Mathematica Notation

Traditional Notation









Elementary Functions > Coth[z] > Integration > Indefinite integration > Involving one direct function and elementary functions > Involving hyperbolic functions > Involving sinh and cosh > Other integrals





http://functions.wolfram.com/01.22.21.0155.01









  


  








Input Form





Integrate[Coth[e z]/Sqrt[a Cosh[e z]^2 + b Cosh[e z] Sinh[e z] + c Sinh[e z]^2], z] == ((b - 2 c - Sqrt[b^2 - 4 a c]) (EllipticF[ArcSin[Sqrt[((-b - 2 c + Sqrt[b^2 - 4 a c]) (1 + Tanh[e z]))/ ((-b + 2 c + Sqrt[b^2 - 4 a c]) (-1 + Tanh[e z]))]], -((-a + c + Sqrt[b^2 - 4 a c])/(a - c + Sqrt[b^2 - 4 a c]))] - 2 EllipticPi[(-b + 2 c + Sqrt[b^2 - 4 a c])/ (b + 2 c - Sqrt[b^2 - 4 a c]), ArcSin[Sqrt[((-b - 2 c + Sqrt[b^2 - 4 a c]) (1 + Tanh[e z]))/ ((-b + 2 c + Sqrt[b^2 - 4 a c]) (-1 + Tanh[e z]))]], -((-a + c + Sqrt[b^2 - 4 a c])/(a - c + Sqrt[b^2 - 4 a c]))]) (-1 + Tanh[e z]) Sqrt[((-b - 2 c + Sqrt[b^2 - 4 a c]) (1 + Tanh[e z]))/ ((-b + 2 c + Sqrt[b^2 - 4 a c]) (-1 + Tanh[e z]))] Sqrt[-((2 a - b + Sqrt[b^2 - 4 a c] + (b - 2 c + Sqrt[b^2 - 4 a c]) Tanh[e z])/((a - b + c) (-1 + Tanh[e z])))] Sqrt[(-2 a + b + Sqrt[b^2 - 4 a c] + (-b + 2 c + Sqrt[b^2 - 4 a c]) Tanh[e z])/((a - b + c) (-1 + Tanh[e z]))])/ ((b + 2 c - Sqrt[b^2 - 4 a c]) e (1 + Tanh[e z]) Sqrt[a Cosh[e z]^2 + b Cosh[e z] Sinh[e z] + c Sinh[e z]^2])








Standard Form





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MathML Form







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type='integer'> 2 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 4 </cn> <ci> a </ci> <ci> c </ci> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <apply> <tanh /> <apply> <times /> <ci> e </ci> <ci> z </ci> </apply> </apply> </apply> <apply> <power /> <apply> <plus /> <apply> <power /> <ci> b </ci> <cn type='integer'> 2 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 4 </cn> <ci> a </ci> <ci> c </ci> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <apply> <power /> <apply> <times /> <apply> <plus /> <ci> a </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> b </ci> </apply> <ci> c </ci> </apply> <apply> <plus /> <apply> <tanh /> <apply> <times /> <ci> e </ci> <ci> z </ci> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <apply> <power /> <apply> <times /> <apply> <plus /> <ci> b </ci> <apply> <times /> <cn type='integer'> 2 </cn> <ci> c </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <apply> <plus /> <apply> <power /> <ci> b </ci> <cn type='integer'> 2 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 4 </cn> <ci> a </ci> <ci> c </ci> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> <ci> e </ci> <apply> <plus /> <apply> <tanh /> <apply> <times /> <ci> e </ci> <ci> z </ci> </apply> </apply> <cn type='integer'> 1 </cn> </apply> <apply> <power /> <apply> <plus /> <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> <apply> <times /> <ci> b </ci> <apply> <sinh /> <apply> <times /> <ci> e </ci> <ci> z </ci> </apply> </apply> <apply> <cosh /> <apply> <times /> <ci> e </ci> <ci> z </ci> </apply> </apply> </apply> <apply> <times /> <ci> c </ci> <apply> <power /> <apply> <sinh /> <apply> <times /> <ci> e </ci> <ci> z </ci> </apply> </apply> <cn type='integer'> 2 </cn> </apply> </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