$\int \frac{1}{(\cos x)^{2n+1}}dx$

Question

I am struggling to recreate the result from this article (part 3) see link (page 295):https://archive.org/details/sim_mathematics-magazine_1936-05_10_8/page/294/mode/2up

This is what I have done so far: My work so far

Answer 1

Perhaps this is of some use:

Integrate[1/Cos[x]^(2*m + 1), x]

(1/(2*m))*((Csc[x]*
    Hypergeometric2F1[1/2, -m, 
     1 - m, Cos[x]^2]*
    Sqrt[Sin[x]^2])/Cos[x]^(2*m))
    
FullSimplify[Evaluate[
     Integrate[1/Cos[x]^(2*m + 1), 
       x]], Element[m, Integers]]
       
(-(1/2))*Beta[Cos[x]^2, -m, 1/2]*
  Csc[x]*Sqrt[Sin[x]^2]
``