1. **State the problem:** We need to evaluate the indefinite integral $$\int \sec^4(x) \tan(x) \, dx.$$\n\n2. **Recall relevant formulas and rules:** The derivative of $$\sec(x)$$ is $$\sec(x) \tan(x)$$ and the derivative of $$\tan(x)$$ is $$\sec^2(x)$$. This suggests substitution involving $$\sec(x)$$ or $$\tan(x)$$ might simplify the integral.\n\n3. **Choose substitution:** Let $$u = \sec(x)$$. Then, $$du = \sec(x) \tan(x) \, dx$$, which means $$\sec(x) \tan(x) \, dx = du$$.\n\n4. **Rewrite the integral:**\n$$\int \sec^4(x) \tan(x) \, dx = \int \sec^3(x) (\sec(x) \tan(x) \, dx) = \int u^3 \, du.$$\n\n5. **Integrate:**\n$$\int u^3 \, du = \frac{u^4}{4} + C = \frac{\sec^4(x)}{4} + C.$$\n\n**Final answer:** $$\boxed{\frac{1}{4} \sec^4(x) + C}.$$