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http://functions.wolfram.com/01.20.21.2381.01
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Integrate[(a + a Cosh[c z]^2)^\[Beta] Cosh[d z], z] ==
(1/(2 (d^2 - 4 c^2 \[Beta]^2))) (((6 + E^(-2 c z) + E^(2 c z))^\[Beta]
(E^(2 d z) (d + 2 c \[Beta]) AppellF1[d/(2 c) - \[Beta], -\[Beta],
-\[Beta], 1 + d/(2 c) - \[Beta], E^(2 c z)/(-3 - 2 Sqrt[2]),
E^(2 c z)/(-3 + 2 Sqrt[2])] - (d - 2 c \[Beta])
AppellF1[-((d + 2 c \[Beta])/(2 c)), -\[Beta], -\[Beta],
1 - d/(2 c) - \[Beta], E^(2 c z)/(-3 - 2 Sqrt[2]),
E^(2 c z)/(-3 + 2 Sqrt[2])]) (a (3 + Cosh[2 c z]))^\[Beta])/
(E^(d z) (4 + (4 E^(2 c z))/(3 - 2 Sqrt[2]))^\[Beta]
(1 + E^(2 c z)/(3 + 2 Sqrt[2]))^\[Beta] (3 + Cosh[2 c z])^\[Beta]))
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</mo> </mrow> <mo> + </mo> <mn> 3 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mi> β </mi> </msup> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <int /> <bvar> <ci> z </ci> </bvar> <apply> <times /> <apply> <power /> <apply> <plus /> <apply> <times /> <ci> a </ci> <apply> <power /> <apply> <cosh /> <apply> <times /> <ci> c </ci> <ci> z </ci> </apply> </apply> <cn type='integer'> 2 </cn> </apply> </apply> <ci> a </ci> </apply> <ci> β </ci> </apply> <apply> <cosh /> <apply> <times /> <ci> d </ci> <ci> z </ci> </apply> </apply> </apply> </apply> <apply> <times /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <plus /> <apply> <power /> <ci> d </ci> <cn type='integer'> 2 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 4 </cn> <apply> <power /> <ci> c </ci> <cn type='integer'> 2 </cn> 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