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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 and trigonometric functions > Involving powers of cos and powers of sinh > Involving cosm(a z) sinhu(b z) coth(c z)





http://functions.wolfram.com/01.22.21.0179.01









  


  










Input Form





Integrate[Cos[a z]^m Sinh[c z]^\[Mu] Coth[c z], z] == ((-(1/(c (-2 + \[Mu]) \[Mu]))) 2^(-m - \[Mu]) (-E^((-c) z) + E^(c z))^\[Mu] Binomial[m, m/2] (E^(2 c z) \[Mu] Hypergeometric2F1[1 - \[Mu]/2, 1 - \[Mu], 2 - \[Mu]/2, E^(2 c z)] + (-2 + \[Mu]) Hypergeometric2F1[-(\[Mu]/2), 1 - \[Mu], 1 - \[Mu]/2, E^(2 c z)]) (-1 + Mod[m, 2]))/(1 - E^(2 c z))^\[Mu] - (2^(-m - \[Mu]) (-E^((-c) z) + E^(c z))^\[Mu] Sum[Binomial[m, k] (Hypergeometric2F1[((-I) a (-2 k + m) - c \[Mu])/ (2 c), 1 - \[Mu], ((-I) a (-2 k + m) - c (\[Mu] - 2))/(2 c), E^(2 c z)]/E^(I a (-2 k + m) z)/((-I) a (-2 k + m) - c \[Mu]) + (E^((2 c - I a (-2 k + m)) z) Hypergeometric2F1[ ((-I) a (-2 k + m) - c (\[Mu] - 2))/(2 c), 1 - \[Mu], ((-I) a (-2 k + m) - c (\[Mu] - 4))/(2 c), E^(2 c z)])/ (I a (2 k - m) - c (\[Mu] - 2)) + (E^(I a (-2 k + m) z) Hypergeometric2F1[(I a (-2 k + m) - c \[Mu])/(2 c), 1 - \[Mu], (I a (-2 k + m) - c (\[Mu] - 2))/(2 c), E^(2 c z)])/ (I a (-2 k + m) - c \[Mu]) + (E^((2 c + I a (-2 k + m)) z) Hypergeometric2F1[(I a (-2 k + m) - c (\[Mu] - 2))/(2 c), 1 - \[Mu], (I a (-2 k + m) - c (\[Mu] - 4))/(2 c), E^(2 c z)])/ (I a (-2 k + m) - c (\[Mu] - 2))), {k, 0, Floor[(1/2) (-1 + m)]}])/ (1 - E^(2 c z))^\[Mu] /; Element[m, Integers] && m > 0










Standard Form





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MathML Form







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</mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mfrac> <mrow> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> a </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> <mrow> <mi> c </mi> <mo> &#8290; </mo> <mi> &#956; </mi> </mrow> </mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> c </mi> </mrow> </mfrac> <mo> , </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> &#956; </mi> </mrow> </mrow> <mo> ; </mo> <mfrac> <mrow> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> a </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> <mrow> <mi> c </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> &#956; </mi> <mo> - </mo> <mn> 2 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> c </mi> </mrow> </mfrac> <mo> ; </mo> <msup> <mi> &#8519; </mi> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> c </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> </msup> </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[FractionBox[RowBox[List[RowBox[List[&quot;\[ImaginaryI]&quot;, &quot; &quot;, &quot;a&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;, RowBox[List[&quot;c&quot;, &quot; &quot;, &quot;\[Mu]&quot;]]]], RowBox[List[&quot;2&quot;, &quot; &quot;, &quot;c&quot;]]], Hypergeometric2F1], &quot;,&quot;, TagBox[RowBox[List[&quot;1&quot;, &quot;-&quot;, &quot;\[Mu]&quot;]], Hypergeometric2F1]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1], &quot;;&quot;, TagBox[TagBox[TagBox[FractionBox[RowBox[List[RowBox[List[&quot;\[ImaginaryI]&quot;, &quot; 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</mo> <mi> k </mi> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <mi> c </mi> <mo> &#8290; </mo> <mi> &#956; </mi> </mrow> </mrow> </mfrac> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mi> m </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 /> <apply> <cos /> <apply> <times /> <ci> a </ci> <ci> z </ci> </apply> </apply> <ci> m </ci> </apply> <apply> <power /> <apply> <sinh /> <apply> <times /> <ci> c </ci> <ci> z </ci> </apply> </apply> <ci> &#956; </ci> </apply> <apply> <coth /> <apply> <times /> <ci> c </ci> <ci> z </ci> </apply> </apply> </apply> </apply> <apply> <plus /> <apply> <times /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <times /> <ci> c </ci> <apply> <plus /> <ci> &#956; </ci> <cn type='integer'> -2 </cn> </apply> <ci> &#956; </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <cn type='integer'> 2 </cn> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> m </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> &#956; </ci> </apply> </apply> </apply> <apply> <power /> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <exponentiale /> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> c </ci> </apply> <ci> z </ci> </apply> </apply> </apply> <apply> <power /> <exponentiale /> <apply> <times /> <ci> c </ci> <ci> z </ci> </apply> </apply> </apply> <ci> &#956; </ci> </apply> <cn type='integer'> -1 </cn> <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> <plus /> <apply> <rem /> <ci> $CellContext`m </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <exponentiale /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> c </ci> <ci> z </ci> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> &#956; 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</ci> </apply> </apply> <apply> <times /> <apply> <plus /> <apply> <times /> <imaginaryi /> <ci> a </ci> <apply> <plus /> <ci> m </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <ci> k </ci> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <ci> c </ci> <apply> <plus /> <ci> &#956; </ci> <cn type='integer'> -2 </cn> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> c </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <exponentiale /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> c </ci> <ci> z </ci> </apply> </apply> </apply> <apply> <power /> <apply> <plus /> <apply> <times /> <imaginaryi /> <ci> a </ci> <apply> <plus /> <ci> m </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <ci> k </ci> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <ci> c </ci> <ci> &#956; </ci> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <in /> <ci> m </ci> <apply> <ci> SuperPlus </ci> <ci> &#8469; </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