I=\int{\frac{\frac{a}{\cos^2{t}}dt}{(a^2tg^2{t}+a^2)^{\frac{3}{2}}}}=\int{\frac{dt}{a^2\cos^2{t}(tg^2{t}+1)^{\frac{3}{2}}}}=\frac{1}{a^2}\int{\frac{dt}{\cos^2{t}\Bigg (\frac{sin^2{t}}{cos^2{}t}+1\Bigg)^{\frac{3}{2}}}}=\frac{1}{a^2}\int{\frac{dt}{\cos^2{t}\Bigg (\frac{1}{cos^2{t}}\Bigg)^{\frac{3}{2}}}}=\frac{1}{a^2}\int{\cos{t}}=\frac{1}{a^2}\sin{t}+C