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Cosh




Mathematica Notation

Traditional Notation









Elementary Functions > Cosh[z] > Integration > Indefinite integration > Involving functions of the direct function and exponential function > Involving products of powers of two direct functions and exponential function > Involving products of powers of two direct functions and exponential function > Involving ep zcoshmu(c z) coshv(a z+b)





http://functions.wolfram.com/01.20.21.2837.01









  


  








Input Form





Integrate[E^(p z) Cosh[c z]^m Cosh[a z + b]^\[Nu], z] == ((1/(p + a \[Nu])) E^(p z) Binomial[m, m/2] Hypergeometric2F1[ -((p + a \[Nu])/(2 a)), -\[Nu], (1/2) (2 - p/a - \[Nu]), -E^(-2 (b + a z))] (1 - Mod[m, 2]) Cosh[b + a z]^\[Nu])/ (2^m (1 + E^(-2 (b + a z)))^\[Nu]) + (Cosh[b + a z]^\[Nu] Sum[Binomial[m, k] ((E^((2 c k - c m + p) z) Hypergeometric2F1[ -((2 c k - c m + p + a \[Nu])/(2 a)), -\[Nu], -((2 c k - c m + p + a (-2 + \[Nu]))/(2 a)), -E^(-2 (b + a z))])/ (2 c k - c m + p + a \[Nu]) + (E^((c (-2 k + m) + p) z) Hypergeometric2F1[-((c (-2 k + m) + p + a \[Nu])/(2 a)), -\[Nu], -((-2 c k + c m + p + a (-2 + \[Nu]))/(2 a)), -E^(-2 (b + a z))])/ (c (-2 k + m) + p + a \[Nu])), {k, 0, Floor[(1/2) (-1 + m)]}])/ (2^m (1 + E^(-2 (b + a z)))^\[Nu]) /; Element[m, Integers] && m > 0








Standard Form





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MathML Form







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&quot;, RowBox[List[&quot;(&quot;, RowBox[List[&quot;b&quot;, &quot;+&quot;, RowBox[List[&quot;a&quot;, &quot; &quot;, &quot;z&quot;]]]], &quot;)&quot;]]]]]]], Hypergeometric2F1]]], &quot;)&quot;]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]]], Hypergeometric2F1] </annotation> </semantics> </mrow> <mo> + </mo> <mrow> <mfrac> <msup> <mi> &#8519; </mi> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> c </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> m </mi> <mo> - </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> k </mi> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mi> p </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mi> z </mi> </mrow> </msup> <mrow> <mrow> <mi> c </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> m </mi> <mo> - </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> k </mi> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mi> p </mi> <mo> + </mo> <mrow> <mi> a </mi> <mo> &#8290; </mo> <mi> &#957; </mi> </mrow> </mrow> </mfrac> <mo> &#8290; </mo> <semantics> <mrow> <mrow> <msub> <mo> &#8202; </mo> <mn> 2 </mn> </msub> <msub> <mi> F </mi> <mn> 1 </mn> </msub> </mrow> <mo> &#8289; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mo> - </mo> <mfrac> <mrow> <mrow> <mi> c </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> m </mi> <mo> - </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> k </mi> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mi> p </mi> <mo> + </mo> <mrow> <mi> a </mi> <mo> &#8290; </mo> <mi> &#957; </mi> </mrow> </mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> a </mi> </mrow> </mfrac> </mrow> <mo> , </mo> <mrow> <mo> - </mo> <mi> &#957; </mi> </mrow> </mrow> <mo> ; </mo> <mrow> <mo> - </mo> <mfrac> <mrow> <mrow> <mrow> <mo> - </mo> <mn> 2 </mn> </mrow> <mo> &#8290; </mo> <mi> c </mi> <mo> &#8290; </mo> <mi> k </mi> </mrow> <mo> + </mo> <mrow> <mi> c </mi> <mo> &#8290; </mo> <mi> m </mi> </mrow> <mo> + </mo> <mi> p </mi> <mo> + </mo> <mrow> <mi> a </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> &#957; </mi> <mo> - </mo> <mn> 2 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> a </mi> </mrow> </mfrac> </mrow> <mo> ; </mo> <mrow> <mo> - </mo> <msup> <mi> &#8519; </mi> <mrow> <mrow> <mo> - </mo> <mn> 2 </mn> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> b </mi> <mo> + </mo> <mrow> <mi> a </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </msup> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation encoding='Mathematica'> TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox[&quot;\[InvisiblePrefixScriptBase]&quot;, FormBox[&quot;2&quot;, TraditionalForm]], SubscriptBox[&quot;F&quot;, FormBox[&quot;1&quot;, TraditionalForm]]]], &quot;\[InvisibleApplication]&quot;, RowBox[List[&quot;(&quot;, RowBox[List[TagBox[TagBox[RowBox[List[TagBox[RowBox[List[&quot;-&quot;, FractionBox[RowBox[List[RowBox[List[&quot;c&quot;, &quot; &quot;, RowBox[List[&quot;(&quot;, RowBox[List[&quot;m&quot;, &quot;-&quot;, RowBox[List[&quot;2&quot;, &quot; &quot;, &quot;k&quot;]]]], &quot;)&quot;]]]], &quot;+&quot;, &quot;p&quot;, &quot;+&quot;, RowBox[List[&quot;a&quot;, &quot; &quot;, &quot;\[Nu]&quot;]]]], RowBox[List[&quot;2&quot;, &quot; &quot;, &quot;a&quot;]]]]], Hypergeometric2F1], &quot;,&quot;, TagBox[RowBox[List[&quot;-&quot;, &quot;\[Nu]&quot;]], Hypergeometric2F1]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1], &quot;;&quot;, TagBox[TagBox[TagBox[RowBox[List[&quot;-&quot;, FractionBox[RowBox[List[RowBox[List[RowBox[List[&quot;-&quot;, &quot;2&quot;]], &quot; &quot;, &quot;c&quot;, &quot; &quot;, &quot;k&quot;]], &quot;+&quot;, RowBox[List[&quot;c&quot;, &quot; &quot;, &quot;m&quot;]], &quot;+&quot;, &quot;p&quot;, &quot;+&quot;, RowBox[List[&quot;a&quot;, &quot; &quot;, RowBox[List[&quot;(&quot;, RowBox[List[&quot;\[Nu]&quot;, &quot;-&quot;, &quot;2&quot;]], &quot;)&quot;]]]]]], RowBox[List[&quot;2&quot;, &quot; &quot;, &quot;a&quot;]]]]], Hypergeometric2F1], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1], &quot;;&quot;, TagBox[RowBox[List[&quot;-&quot;, SuperscriptBox[&quot;\[ExponentialE]&quot;, RowBox[List[RowBox[List[&quot;-&quot;, &quot;2&quot;]], &quot; &quot;, RowBox[List[&quot;(&quot;, RowBox[List[&quot;b&quot;, &quot;+&quot;, RowBox[List[&quot;a&quot;, &quot; &quot;, &quot;z&quot;]]]], &quot;)&quot;]]]]]]], Hypergeometric2F1]]], &quot;)&quot;]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]]], Hypergeometric2F1] </annotation> </semantics> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mi> v </mi> <mo> &#8712; </mo> <msup> <mi> &#8469; </mi> <mo> + </mo> </msup> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <int /> <bvar> <ci> z </ci> </bvar> <apply> <times /> <apply> <power /> <exponentiale /> <apply> <times /> <ci> p </ci> <ci> z </ci> </apply> </apply> <apply> <power /> <apply> <cosh /> <apply> <times /> <ci> c </ci> <ci> z </ci> </apply> </apply> <ci> m </ci> </apply> <apply> <power /> <apply> <cosh /> <apply> <plus /> <ci> b </ci> <apply> <times /> <ci> a </ci> <ci> z </ci> </apply> </apply> </apply> <ci> &#957; </ci> </apply> </apply> </apply> <apply> <plus /> <apply> <times /> <apply> <times /> <apply> <power /> <cn type='integer'> 2 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> m </ci> </apply> </apply> <apply> <power /> <exponentiale /> <apply> <times /> <ci> p </ci> <ci> z </ci> </apply> </apply> <apply> <power /> <apply> <plus /> <ci> p </ci> <apply> <times /> <ci> a </ci> <ci> &#957; </ci> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <ci> Binomial </ci> <ci> m </ci> <apply> <times /> <ci> m </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <power /> <apply> <cosh /> <apply> <plus /> <ci> b </ci> <apply> <times /> <ci> a </ci> <ci> z </ci> </apply> </apply> </apply> <ci> &#957; </ci> </apply> <apply> <ci> Hypergeometric2F1 </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <plus /> <ci> p </ci> <apply> <times /> <ci> a </ci> <ci> &#957; </ci> </apply> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> a </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> &#957; </ci> </apply> <apply> <times /> <cn type='rational'> 1 <sep /> 2 </cn> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <ci> p </ci> <apply> <power /> <ci> a </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> &#957; </ci> </apply> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <exponentiale /> <apply> <times /> <cn type='integer'> -2 </cn> <apply> <plus /> <ci> b </ci> <apply> <times /> <ci> a </ci> <ci> z </ci> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <rem /> <ci> $CellContext`m </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <power /> <exponentiale /> <apply> <times /> <cn type='integer'> -2 </cn> <apply> <plus /> <ci> b </ci> <apply> <times /> <ci> a </ci> <ci> z </ci> </apply> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> &#957; 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</ci> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





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





2002-12-18