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Sech






Mathematica Notation

Traditional Notation









Elementary Functions > Sech[z] > Integration > Indefinite integration > Involving one direct function and elementary functions > Involving hyperbolic, trigonometric and a power functions > Involving sin, tanh and power > Involving znsin(a z)tanh(c z) sech( c z)





http://functions.wolfram.com/01.24.21.0275.01









  


  










Input Form





Integrate[z^n Sin[a z] Tanh[c z] Sech[c z], z] == E^(2 c z) n! (E^((I Pi)/2 + (I a - c) z) Sum[(1/(-j + n)!) (-1)^j z^(-j + n) (I a + c)^(-1 - j) HypergeometricPFQ[{Subscript[a, 1], \[Ellipsis], Subscript[a, j + 1], 2}, {1 + Subscript[a, 1], \[Ellipsis], 1 + Subscript[a, j + 1]}, -E^(2 c z)], {j, 0, n}] - E^((I Pi)/2 + ((-I) a - c) z) Sum[(1/(-j + n)!) (-1)^j z^(-j + n) ((-I) a + c)^(-1 - j) HypergeometricPFQ[{Subscript[b, 1], \[Ellipsis], Subscript[b, j + 1], 2}, {1 + Subscript[b, 1], \[Ellipsis], 1 + Subscript[b, j + 1]}, -E^(2 c z)], {j, 0, n}] + E^(-((I Pi)/2) + (I a + c) z) Sum[(1/(-j + n)!) (-1)^j z^(-j + n) (I a + 3 c)^(-1 - j) HypergeometricPFQ[{Subscript[c, 1], \[Ellipsis], Subscript[c, j + 1], 2}, {1 + Subscript[c, 1], \[Ellipsis], 1 + Subscript[c, j + 1]}, -E^(2 c z)], {j, 0, n}] - E^(-((I Pi)/2) + ((-I) a + c) z) Sum[(1/(-j + n)!) (-1)^j z^(-j + n) ((-I) a + 3 c)^(-1 - j) HypergeometricPFQ[{Subscript[d, 1], \[Ellipsis], Subscript[d, j + 1], 2}, {1 + Subscript[d, 1], \[Ellipsis], 1 + Subscript[d, j + 1]}, -E^(2 c z)], {j, 0, n}]) /; Subscript[a, 1] == Subscript[a, 2] == \[Ellipsis] == Subscript[a, n + 1] == (I a + c)/(2 c) && Subscript[b, 1] == Subscript[b, 2] == \[Ellipsis] == Subscript[b, n + 1] == ((-I) a + c)/(2 c) && Subscript[c, 1] == Subscript[c, 2] == \[Ellipsis] == Subscript[c, n + 1] == (I a + 3 c)/(2 c) && Subscript[d, 1] == Subscript[d, 2] == \[Ellipsis] == Subscript[d, n + 1] == ((-I) a + 3 c)/(2 c) && Element[n, Integers] && n >= 0










Standard Form





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MathML Form







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</mo> <mi> c </mi> </mrow> </mfrac> <mo> , </mo> <mn> 2 </mn> </mrow> <mo> ; </mo> <mrow> <mrow> <mfrac> <mrow> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> a </mi> </mrow> <mo> + </mo> <mi> c </mi> </mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> c </mi> </mrow> </mfrac> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> , </mo> <mo> &#8230; </mo> <mo> , </mo> <mrow> <mfrac> <mrow> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> a </mi> </mrow> <mo> + </mo> <mi> c </mi> </mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> c </mi> </mrow> </mfrac> <mo> + </mo> <mn> 1 </mn> </mrow> </mrow> <mo> ; </mo> <mrow> <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> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation encoding='Mathematica'> TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox[&quot;\[InvisiblePrefixScriptBase]&quot;, FormBox[RowBox[List[&quot;j&quot;, &quot;+&quot;, &quot;2&quot;]], TraditionalForm]], SubscriptBox[&quot;F&quot;, FormBox[RowBox[List[&quot;j&quot;, &quot;+&quot;, &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; 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Rule Form





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





2002-12-18