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http://functions.wolfram.com/01.20.21.4146.01
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Integrate[Sinh[d + c z]^\[Mu] Cosh[b + a z]^v, z] ==
(1/(c \[Mu])) ((Sinh[d + c z]^\[Mu]
(Binomial[v, v/2] Hypergeometric2F1[-(\[Mu]/2), -\[Mu], 1 - \[Mu]/2,
E^(2 (d + c z))] (-1 + Mod[v, 2]) +
c \[Mu] Sum[E^(2 b s - b v) Binomial[v, s]
(-((E^((-2 s + v) (2 b + a z)) Hypergeometric2F1[
-((2 a s - a v + c \[Mu])/(2 c)), -\[Mu], (1/2) (2 +
(a (-2 s + v))/c - \[Mu]), E^(2 (d + c z))])/
(2 a s - a v + c \[Mu])) + (E^(a (2 s - v) z) Hypergeometric2F1[
-((-2 a s + a v + c \[Mu])/(2 c)), -\[Mu],
-((a (-2 s + v) + c (-2 + \[Mu]))/(2 c)), E^(2 (d + c z))])/
(2 a s - a v - c \[Mu])), {s, 0, Floor[(1/2) (-1 + v)]}]))/
(2^v (1 - E^(2 (d + c z)))^\[Mu])) /; Element[v, Integers] && v > 0
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<mrow> <mo> ⌊ </mo> <mfrac> <mrow> <mi> v </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mn> 2 </mn> </mfrac> <mo> ⌋ </mo> </mrow> </munderover> <mrow> <msup> <mi> ⅇ </mi> <mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> b </mi> <mo> ⁢ </mo> <mi> s </mi> </mrow> <mo> - </mo> <mrow> <mi> b </mi> <mo> ⁢ </mo> <mi> v </mi> </mrow> </mrow> </msup> <mo> ⁢ </mo> <semantics> <mrow> <mo> ( </mo> <mtable> <mtr> <mtd> <mi> v </mi> </mtd> </mtr> <mtr> <mtd> <mi> s </mi> </mtd> </mtr> </mtable> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["(", GridBox[List[List[TagBox["v", Identity]], List[TagBox["s", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] </annotation> </semantics> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mfrac> <mrow> <msup> <mi> ⅇ </mi> <mrow> <mi> a </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> s </mi> </mrow> <mo> - </mo> <mi> v </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mi> z </mi> </mrow> </msup> <mtext> </mtext> </mrow> <mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> a </mi> <mo> ⁢ </mo> <mi> s </mi> </mrow> <mo> - </mo> <mrow> <mi> a </mi> <mo> ⁢ </mo> <mi> v </mi> </mrow> <mo> - </mo> <mrow> <mi> c </mi> <mo> ⁢ </mo> <mi> μ </mi> </mrow> </mrow> </mfrac> <mo> ⁢ </mo> <semantics> <mrow> <mrow> <msub> <mo>   </mo> <mn> 2 </mn> </msub> <msub> <mi> F </mi> <mn> 1 </mn> </msub> </mrow> <mo> ⁡ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mo> - </mo> <mfrac> <mrow> <mrow> <mrow> <mo> - </mo> <mn> 2 </mn> </mrow> <mo> ⁢ </mo> <mi> a </mi> <mo> ⁢ </mo> <mi> s </mi> </mrow> <mo> + </mo> <mrow> <mi> a </mi> <mo> ⁢ </mo> <mi> v </mi> </mrow> <mo> + </mo> <mrow> <mi> c </mi> <mo> ⁢ </mo> <mi> μ </mi> </mrow> </mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> c </mi> </mrow> </mfrac> </mrow> <mo> , </mo> <mrow> <mo> - </mo> <mi> μ </mi> </mrow> </mrow> <mo> ; </mo> <mrow> <mo> - </mo> <mfrac> <mrow> <mrow> <mi> a </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> v </mi> <mo> - </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> s </mi> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mi> c </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> μ </mi> <mo> - </mo> <mn> 2 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> c </mi> </mrow> </mfrac> </mrow> <mo> ; </mo> <msup> <mi> ⅇ </mi> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> d </mi> <mo> + </mo> <mrow> <mi> c </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </msup> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation encoding='Mathematica'> TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["2", TraditionalForm]], SubscriptBox["F", FormBox["1", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox[RowBox[List["-", FractionBox[RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "a", " ", "s"]], "+", RowBox[List["a", " ", 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</mo> <mrow> <mi> v </mi> <mo> - </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> s </mi> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> b </mi> </mrow> <mo> + </mo> <mrow> <mi> a </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </msup> <mtext> </mtext> </mrow> <mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> a </mi> <mo> ⁢ </mo> <mi> s </mi> </mrow> <mo> - </mo> <mrow> <mi> a </mi> <mo> ⁢ </mo> <mi> v </mi> </mrow> <mo> + </mo> <mrow> <mi> c </mi> <mo> ⁢ </mo> <mi> μ </mi> </mrow> </mrow> </mfrac> <mo> ⁢ </mo> <semantics> <mrow> <mrow> <msub> <mo>   </mo> <mn> 2 </mn> </msub> <msub> <mi> F </mi> <mn> 1 </mn> </msub> </mrow> <mo> ⁡ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mo> - </mo> <mfrac> <mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> a </mi> <mo> ⁢ </mo> <mi> s </mi> </mrow> <mo> - </mo> <mrow> <mi> a </mi> <mo> ⁢ </mo> <mi> v </mi> </mrow> <mo> + </mo> <mrow> <mi> c </mi> <mo> ⁢ </mo> <mi> μ </mi> </mrow> </mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> c </mi> </mrow> </mfrac> </mrow> <mo> , </mo> <mrow> <mo> - </mo> <mi> μ </mi> </mrow> </mrow> <mo> ; </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mfrac> <mrow> <mi> a </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> v </mi> <mo> - </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> s </mi> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mi> c </mi> </mfrac> <mo> - </mo> <mi> μ </mi> <mo> + </mo> <mn> 2 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ; </mo> <msup> <mi> ⅇ </mi> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> d </mi> <mo> + </mo> <mrow> <mi> c </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </msup> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation encoding='Mathematica'> TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["2", TraditionalForm]], SubscriptBox["F", FormBox["1", 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