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$$\int \tan^n(x) \, dx = \int \tan^{n-2}(x) (sec^2(x) - 1) \, dx = \int \tan^{n-2}(x) \, d\tan(x)-\int \tan^{n-2}(x) \, dx$$
$$\int \tan^n(x) \, dx = \frac {\tan^{n-1}(x) }{n-1}-\int \tan^{n-2}(x) \, dx$$

Wolfram mathematica