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Cosh






Mathematica Notation

Traditional Notation









Elementary Functions > Cosh[z] > Integration > Indefinite integration > Involving one direct function and elementary functions > Involving rational functions > Involving (a z2+b)-n





http://functions.wolfram.com/01.20.21.0152.01









  


  










Input Form





Integrate[Cosh[c z]/(a z^2 + b)^2, z] == (1/(4 b^(3/2))) ((2 Sqrt[b] z Cosh[c z])/(b + a z^2) + (1/a) (I CosIntegral[-((Sqrt[b] c)/Sqrt[a]) + I c z] (Sqrt[a] Cos[(Sqrt[b] c)/Sqrt[a]] + Sqrt[b] c Sin[(Sqrt[b] c)/Sqrt[a]])) - (1/a) (I CosIntegral[(Sqrt[b] c)/Sqrt[a] + I c z] (Sqrt[a] Cos[(Sqrt[b] c)/Sqrt[a]] + Sqrt[b] c Sin[(Sqrt[b] c)/Sqrt[a]])) - (I Sqrt[b] c Cos[(Sqrt[b] c)/Sqrt[a]] SinIntegral[(Sqrt[b] c)/Sqrt[a] - I c z])/a + (I Sin[(Sqrt[b] c)/Sqrt[a]] SinIntegral[(Sqrt[b] c)/Sqrt[a] - I c z])/ Sqrt[a] + (I Sqrt[b] c Cos[(Sqrt[b] c)/Sqrt[a]] SinIntegral[(Sqrt[b] c)/Sqrt[a] + I c z])/a - (I Sin[(Sqrt[b] c)/Sqrt[a]] SinIntegral[(Sqrt[b] c)/Sqrt[a] + I c z])/ Sqrt[a])










Standard Form





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MathML Form







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</mi> </mrow> <msqrt> <mi> a </mi> </msqrt> </mfrac> </mrow> <mo> ) </mo> </mrow> </mrow> <mi> a </mi> </mfrac> <mo> + </mo> <mfrac> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mrow> <mi> sin </mi> <mo> &#8289; </mo> <mo> ( </mo> <mfrac> <mrow> <msqrt> <mi> b </mi> </msqrt> <mo> &#8290; </mo> <mi> c </mi> </mrow> <msqrt> <mi> a </mi> </msqrt> </mfrac> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> Si </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mfrac> <mrow> <msqrt> <mi> b </mi> </msqrt> <mo> &#8290; </mo> <mi> c </mi> </mrow> <msqrt> <mi> a </mi> </msqrt> </mfrac> <mo> - </mo> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> c </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <msqrt> <mi> a </mi> </msqrt> </mfrac> <mo> - </mo> <mfrac> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mrow> <mi> sin </mi> <mo> &#8289; </mo> <mo> ( </mo> <mfrac> <mrow> <msqrt> <mi> b </mi> </msqrt> <mo> &#8290; </mo> <mi> c </mi> </mrow> <msqrt> <mi> a </mi> 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<mi> &#8520; </mi> <mo> &#8290; </mo> <mi> c </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mi> a </mi> </mfrac> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <int /> <bvar> <ci> z </ci> </bvar> <apply> <times /> <apply> <cosh /> <apply> <times /> <ci> c </ci> <ci> z </ci> </apply> </apply> <apply> <power /> <apply> <power /> <apply> <plus /> <apply> <times /> <ci> a </ci> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> <ci> b </ci> </apply> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <times /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <times /> <cn type='integer'> 4 </cn> <apply> <power /> <ci> b </ci> <cn type='rational'> 3 <sep /> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <ci> b </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <ci> z </ci> <apply> <cosh /> <apply> <times /> <ci> c </ci> <ci> z </ci> </apply> </apply> <apply> <power /> <apply> <plus /> <apply> <times /> <ci> a </ci> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> <ci> b </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <imaginaryi /> <apply> <ci> CosIntegral </ci> <apply> <plus /> <apply> <times /> <imaginaryi /> <ci> z </ci> <ci> c </ci> </apply> <apply> <times /> <apply> <power /> <ci> b </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <ci> c </ci> <apply> <power /> <apply> <power /> <ci> a </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <plus /> <apply> <times /> <apply> <power /> <ci> a </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <cos /> <apply> <times /> <apply> <power /> <ci> b </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <ci> c </ci> <apply> <power /> <apply> <power /> <ci> a </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <times /> <apply> <power /> <ci> b </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <ci> c </ci> <apply> <sin /> <apply> <times /> <apply> <power /> <ci> b </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <ci> c </ci> <apply> <power /> <apply> <power /> <ci> a </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <power /> <ci> a </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <times /> <imaginaryi /> <apply> <ci> CosIntegral </ci> <apply> <plus /> <apply> <times /> <imaginaryi /> <ci> c </ci> <ci> z </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <power /> <ci> b </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <ci> c </ci> <apply> <power /> <apply> <power /> <ci> a </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <plus /> <apply> <times /> <apply> <power /> <ci> a </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <cos /> <apply> <times /> <apply> <power /> <ci> b </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <ci> c </ci> <apply> <power /> <apply> <power /> <ci> a </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <times /> <apply> <power /> <ci> b </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <ci> c </ci> <apply> <sin /> <apply> <times /> <apply> <power /> <ci> b </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <ci> c </ci> <apply> <power /> <apply> <power /> <ci> a </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <power /> <ci> a </ci> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <apply> <power /> <ci> b </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <ci> c </ci> <imaginaryi /> <apply> <cos /> <apply> <times /> <apply> <power /> <ci> b </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <ci> c </ci> <apply> <power /> <apply> <power /> <ci> a </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <ci> SinIntegral </ci> <apply> <plus /> <apply> <times /> <imaginaryi /> <ci> z </ci> <ci> c </ci> </apply> <apply> <times /> <apply> <power /> <ci> b </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <ci> c </ci> <apply> <power /> <apply> <power /> <ci> a </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <power /> <ci> a </ci> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times 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<apply> <power /> <ci> b </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <ci> c </ci> <apply> <power /> <apply> <power /> <ci> a </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <ci> SinIntegral </ci> <apply> <plus /> <apply> <times /> <imaginaryi /> <ci> z </ci> <ci> c </ci> </apply> <apply> <times /> <apply> <power /> <ci> b </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <ci> c </ci> <apply> <power /> <apply> <power /> <ci> a </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <power /> <ci> a </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <imaginaryi /> <apply> <power /> <ci> b </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <ci> c </ci> <apply> <cos /> <apply> <times /> <apply> <power /> <ci> b </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <ci> c </ci> <apply> <power /> <apply> <power /> <ci> a </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <ci> SinIntegral </ci> <apply> <plus /> <apply> <times /> <apply> <power /> <ci> b </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <ci> c </ci> <apply> <power /> <apply> <power /> <ci> a </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <imaginaryi /> <ci> c </ci> <ci> z </ci> </apply> </apply> </apply> </apply> <apply> <power /> <ci> a </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





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





2002-12-18