Products of secants and tangents are similar, but not identical, to
products of sines and cosines. To evaluate $$\int \sec^n(x)
\tan^m(x)\, dx$$
If $n=2$, we can use the substitution $u=\tan(x)$, $du=\sec^2(x)\,
dx$ to get $\int u^m\, du$.
If $n$ is even, we can use the identity $\sec^2(x)=1+\tan^2(x)$
to convert all but two powers of secant into tangents. (If $n=0$, we
can convert two powers of tangent into secants instead.)
If $m=1$, we can substitute $u=\sec(x)$, $du=\sec(x)\tan(x)\, dx$ to get $\int u^{n-1}\, dx$.
If $m$ is odd, we can use the identity $\tan^2(x) = \sec^2(x)-1$ to convert all but one power of secant into tangents.
If $n$ is odd and $m$ is even, then the problem is much harder. This case is described here.