Products of secants and tangents are similar, but not identical, to products of sines and cosines. If you need to evaluate $\int \sec^n(x) \tan^m(x) dx$,

• If $n=2$, do the substitution $u=\tan(x)$, $du=\sec^2(x) dx$ to get $\int u^m du$.
• If $n$ is even, use the identity $\sec^2(x)=1+\tan^2(x)$ to convert all but two powers of secant into tangents. (If $n=0$, convert two powers of tangent into secants instead.)
• If $m=1$, do the substitution $u=\sec(x)$, $du=\sec(x)\tan(x) dx$ to get $\int u^{n-1} dx$.
• If $m$ is odd, 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 in Slide 12.