(∫_a^0) (1/(Sqrt[Cos[t] - Cos[a]])) \[DifferentialD]t

Integrate[1/Sqrt[Sin[t/2]^2 - Sin[a/2]^2], {t, 0, a}]

4 (Sqrt[(L/g)]) 
      (∫_0^((π/2)))(1/(Sqrt[1 - (K^2) Sin[p]])) 
          \[DifferentialD]p

T[a_] = 
      8 (Sqrt[(1/(1 - (K^2)))]) (Sqrt[(L/g)]) 
          EllipticF[(π/4), -((2 (K^2))/(1 - (K^2)))] /. 
        K -> Sin[a/2]

\(\(Plot[T[a]/\((2  Pi\ \@\(L\/g\))\), {a, 0, Pi}, 
      AxesLabel -> {"α", "ratio"}, 
      PlotLabel -> 
        StringForm[\*""T[α]/(2Pi \!\(\@\(L\/g\)\))""]]; \)\)