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 | | http://functions.wolfram.com/01.20.21.4411.01 | 
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 | | Integrate[Sinh[d z] (a + b Sinh[e z] + c Cosh[e z])^\[Beta], z] == 
 (1/2) ((1/(d + e \[Beta])) 
    ((((c + 2 a E^(e z) + c E^(2 e z) + b (-1 + E^(2 e z)))/E^(e z))^\[Beta] 
      AppellF1[-(d/e) - \[Beta], -\[Beta], -\[Beta], 1 - d/e - \[Beta], 
       -(((b + c) E^(e z))/(a + Sqrt[a^2 + b^2 - c^2])), 
       ((b + c) E^(e z))/(-a + Sqrt[a^2 + b^2 - c^2])])/
     (2^\[Beta] E^(d z) (1 - ((b + c) E^(e z))/(-a + Sqrt[a^2 + b^2 - c^2]))^
       \[Beta] (1 + ((b + c) E^(e z))/(a + Sqrt[a^2 + b^2 - c^2]))^
       \[Beta])) - (1/(-d + e \[Beta])) 
    ((E^(d z) ((c + 2 a E^(e z) + c E^(2 e z) + b (-1 + E^(2 e z)))/E^(e z))^
       \[Beta] AppellF1[d/e - \[Beta], -\[Beta], -\[Beta], 1 + d/e - \[Beta], 
       -(((b + c) E^(e z))/(a + Sqrt[a^2 + b^2 - c^2])), 
       ((b + c) E^(e z))/(-a + Sqrt[a^2 + b^2 - c^2])])/
     (2^\[Beta] (1 - ((b + c) E^(e z))/(-a + Sqrt[a^2 + b^2 - c^2]))^\[Beta] 
      (1 + ((b + c) E^(e z))/(a + Sqrt[a^2 + b^2 - c^2]))^\[Beta]))) | 
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 | | Cell[BoxData[RowBox[List[RowBox[List["\[Integral]", RowBox[List[RowBox[List["Sinh", "[", RowBox[List["d", " ", "z"]], "]"]], SuperscriptBox[RowBox[List["(", RowBox[List["a", "+", RowBox[List["b", " ", RowBox[List["Sinh", "[", RowBox[List["e", " ", "z"]], "]"]]]], "+", RowBox[List["c", " ", RowBox[List["Cosh", "[", RowBox[List["e", " ", "z"]], "]"]]]]]], ")"]], "\[Beta]"], RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List[FractionBox["1", RowBox[List["d", "+", RowBox[List["e", " ", "\[Beta]"]]]]], RowBox[List["(", RowBox[List[SuperscriptBox["2", RowBox[List["-", "\[Beta]"]]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "d"]], " ", "z"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["b", "+", "c"]], ")"]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["e", " ", "z"]]]]], RowBox[List[RowBox[List["-", 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SuperscriptBox["\[ExponentialE]", RowBox[List["e", " ", "z"]]]]], RowBox[List["a", "+", SqrtBox[RowBox[List[SuperscriptBox["a", "2"], "+", SuperscriptBox["b", "2"], "-", SuperscriptBox["c", "2"]]]]]]]]], ",", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["b", "+", "c"]], ")"]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["e", " ", "z"]]]]], RowBox[List[RowBox[List["-", "a"]], "+", SqrtBox[RowBox[List[SuperscriptBox["a", "2"], "+", SuperscriptBox["b", "2"], "-", SuperscriptBox["c", "2"]]]]]]]]], "]"]]]], ")"]]]]]], ")"]]]]]]]] | 
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<mo> ( </mo>  <mrow>  <mrow>  <mfrac>  <mn> 1 </mn>  <mrow>  <mi> d </mi>  <mo> + </mo>  <mrow>  <mi> e </mi>  <mo> ⁢ </mo>  <mi> β </mi>  </mrow>  </mrow>  </mfrac>  <mo> ⁢ </mo>  <mrow>  <mo> ( </mo>  <mrow>  <msup>  <mn> 2 </mn>  <mrow>  <mo> - </mo>  <mi> β </mi>  </mrow>  </msup>  <mo> ⁢ </mo>  <msup>  <mi> ⅇ </mi>  <mrow>  <mrow>  <mo> - </mo>  <mi> d </mi>  </mrow>  <mo> ⁢ </mo>  <mi> z </mi>  </mrow>  </msup>  <mo> ⁢ </mo>  <msup>  <mrow>  <mo> ( </mo>  <mrow>  <mn> 1 </mn>  <mo> - </mo>  <mfrac>  <mrow>  <mrow>  <mo> ( </mo>  <mrow>  <mi> b </mi>  <mo> + </mo>  <mi> c </mi>  </mrow>  <mo> ) </mo>  </mrow>  <mo> ⁢ </mo>  <msup>  <mi> ⅇ </mi>  <mrow>  <mi> e </mi>  <mo> ⁢ </mo>  <mi> z </mi>  </mrow>  </msup>  </mrow>  <mrow>  <msqrt>  <mrow>  <msup>  <mi> a </mi>  <mn> 2 </mn>  </msup>  <mo> + </mo>  <msup>  <mi> b </mi>  <mn> 2 </mn>  </msup>  <mo> - </mo>  <msup>  <mi> c </mi>  <mn> 2 </mn>  </msup>  </mrow>  </msqrt>  <mo> - </mo>  <mi> a </mi>  </mrow>  </mfrac>  </mrow>  <mo> ) </mo>  </mrow>  <mrow>  <mo> - </mo>  <mi> β </mi>  </mrow>  </msup>  <mo> ⁢ </mo>  <msup>  <mrow>  <mo> ( </mo>  <mrow>  <mfrac>  <mrow>  <msup>  <mi> ⅇ </mi>  <mrow>  <mi> e </mi>  <mo> ⁢ </mo>  <mi> z </mi>  </mrow>  </msup>  <mo> ⁢ </mo>  <mrow>  <mo> ( </mo>  <mrow>  <mi> b </mi>  <mo> + </mo>  <mi> c </mi>  </mrow>  <mo> ) </mo>  </mrow>  </mrow>  <mrow>  <mi> a </mi>  <mo> + </mo>  <msqrt>  <mrow>  <msup>  <mi> a </mi>  <mn> 2 </mn>  </msup>  <mo> + </mo>  <msup>  <mi> b </mi>  <mn> 2 </mn>  </msup>  <mo> - </mo>  <msup>  <mi> c </mi>  <mn> 2 </mn>  </msup>  </mrow>  </msqrt>  </mrow>  </mfrac>  <mo> + </mo>  <mn> 1 </mn>  </mrow>  <mo> ) </mo>  </mrow>  <mrow>  <mo> - </mo>  <mi> β </mi>  </mrow>  </msup>  <mo> ⁢ </mo>  <msup>  <mrow>  <mo> ( </mo>  <mrow>  <msup>  <mi> ⅇ </mi>  <mrow>  <mrow>  <mo> - </mo>  <mi> e </mi>  </mrow>  <mo> ⁢ </mo>  <mi> z </mi>  </mrow>  </msup>  <mo> ⁢ </mo>  <mrow>  <mo> ( </mo>  <mrow>  <mrow>  <mn> 2 </mn>  <mo> ⁢ </mo>  <msup>  <mi> ⅇ </mi>  <mrow>  <mi> e </mi>  <mo> ⁢ </mo>  <mi> z </mi>  </mrow>  </msup>  <mo> ⁢ </mo>  <mi> a </mi>  </mrow>  <mo> + </mo>  <mrow>  <mi> c </mi>  <mo> ⁢ </mo>  <msup>  <mi> ⅇ </mi>  <mrow>  <mn> 2 </mn>  <mo> ⁢ </mo>  <mi> e </mi>  <mo> ⁢ </mo>  <mi> z </mi>  </mrow>  </msup>  </mrow>  <mo> + </mo>  <mi> c </mi>  <mo> + </mo>  <mrow>  <mi> b </mi>  <mo> ⁢ </mo>  <mrow>  <mo> ( </mo>  <mrow>  <mrow>  <mo> - </mo>  <mn> 1 </mn>  </mrow>  <mo> + </mo>  <msup>  <mi> ⅇ </mi>  <mrow>  <mn> 2 </mn>  <mo> ⁢ </mo>  <mi> e </mi>  <mo> ⁢ </mo>  <mi> z </mi>  </mrow>  </msup>  </mrow>  <mo> ) </mo>  </mrow>  </mrow>  </mrow>  <mo> ) </mo>  </mrow>  </mrow>  <mo> ) </mo>  </mrow>  <mi> β </mi>  </msup>  <mo> ⁢ </mo>  <mrow>  <semantics>  <msub>  <mi> F </mi>  <mn> 1 </mn>  </msub>  <annotation-xml encoding='MathML-Content'>  <ci> AppellF1 </ci>  </annotation-xml>  </semantics>  <mo> ( </mo>  <mrow>  <mrow>  <mrow>  <mrow>  <mo> - </mo>  <mfrac>  <mi> d </mi>  <mi> e </mi>  </mfrac>  </mrow>  <mo> - </mo>  <mi> β </mi>  </mrow>  <mo> ; </mo>  <mrow>  <mo> - </mo>  <mi> β </mi>  </mrow>  </mrow>  <mo> , </mo>  <mrow>  <mrow>  <mo> - </mo>  <mi> β </mi>  </mrow>  <mo> ; </mo>  <mrow>  <mrow>  <mo> - </mo>  <mfrac>  <mi> d </mi>  <mi> e </mi>  </mfrac>  </mrow>  <mo> - </mo>  <mi> β </mi>  <mo> + </mo>  <mn> 1 </mn>  </mrow>  <mo> ; </mo>  <mrow>  <mo> - </mo>  <mfrac>  <mrow>  <mrow>  <mo> ( </mo>  <mrow>  <mi> b </mi>  <mo> + </mo>  <mi> c </mi>  </mrow>  <mo> ) </mo>  </mrow>  <mo> ⁢ </mo>  <msup>  <mi> ⅇ </mi>  <mrow>  <mi> e </mi>  <mo> ⁢ </mo>  <mi> z </mi>  </mrow>  </msup>  </mrow>  <mrow>  <mi> a </mi>  <mo> + </mo>  <msqrt>  <mrow>  <msup>  <mi> a </mi>  <mn> 2 </mn>  </msup>  <mo> + </mo>  <msup>  <mi> b </mi>  <mn> 2 </mn>  </msup>  <mo> - </mo>  <msup>  <mi> c </mi>  <mn> 2 </mn>  </msup>  </mrow>  </msqrt>  </mrow>  </mfrac>  </mrow>  </mrow>  <mo> , </mo>  <mfrac>  <mrow>  <mrow>  <mo> ( </mo>  <mrow>  <mi> b </mi>  <mo> + </mo>  <mi> c </mi>  </mrow>  <mo> ) </mo>  </mrow>  <mo> ⁢ </mo>  <msup>  <mi> ⅇ </mi>  <mrow>  <mi> e </mi>  <mo> ⁢ </mo>  <mi> z </mi>  </mrow>  </msup>  </mrow>  <mrow>  <msqrt>  <mrow>  <msup>  <mi> a </mi>  <mn> 2 </mn>  </msup>  <mo> + </mo>  <msup>  <mi> b </mi>  <mn> 2 </mn>  </msup>  <mo> - </mo>  <msup>  <mi> c </mi>  <mn> 2 </mn>  </msup>  </mrow>  </msqrt>  <mo> - </mo>  <mi> a </mi>  </mrow>  </mfrac>  </mrow>  <mo> ) </mo>  </mrow>  </mrow>  <mo> ) </mo>  </mrow>  </mrow>  <mo> - </mo>  <mrow>  <mfrac>  <mn> 1 </mn>  <mrow>  <mrow>  <mi> e </mi>  <mo> ⁢ </mo>  <mi> β </mi>  </mrow>  <mo> - </mo>  <mi> d </mi>  </mrow>  </mfrac>  <mo> ⁢ </mo>  <mrow>  <mo> ( </mo>  <mrow>  <msup>  <mn> 2 </mn>  <mrow>  <mo> - </mo>  <mi> β </mi>  </mrow>  </msup>  <mo> ⁢ </mo>  <msup>  <mi> ⅇ </mi>  <mrow>  <mi> d </mi>  <mo> ⁢ </mo>  <mi> z </mi>  </mrow>  </msup>  <mo> ⁢ </mo>  <msup>  <mrow>  <mo> ( </mo>  <mrow>  <mn> 1 </mn>  <mo> - </mo>  <mfrac>  <mrow>  <mrow>  <mo> ( </mo>  <mrow>  <mi> b </mi>  <mo> + </mo>  <mi> c </mi>  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