\int\tan(x)\,dx=\int\frac{\sin(x)}{\cos(x)}\,dx=\frac{-\cos(x)}{\cos(x)}-\int\frac{-\sin(x)\cos(x)}{\cos^2(x)}\,dx=-1+\int\frac{\sin(x)}{\cos(x)}=-1+\int\tan(x)\,dx