1. **State the problem:** We need to find the integral $$\int (\sec(x))^3 \, dx$$. 2. **Recall the formula and rules:** For integrals involving powers of secant, a useful approach is to split the power and use integration by parts or reduction formulas. One common formula is: $$\int \sec^n(x) \, dx = \frac{\sec^{n-2}(x) \tan(x)}{n-1} + \frac{n-2}{n-1} \int \sec^{n-2}(x) \, dx$$ for $n \neq 1$. 3. **Apply the formula for $n=3$:** $$\int \sec^3(x) \, dx = \frac{\sec(x) \tan(x)}{2} + \frac{1}{2} \int \sec(x) \, dx$$ 4. **Recall the integral of $\sec(x)$:** $$\int \sec(x) \, dx = \ln|\sec(x) + \tan(x)| + C$$ 5. **Substitute back:** $$\int \sec^3(x) \, dx = \frac{\sec(x) \tan(x)}{2} + \frac{1}{2} \ln|\sec(x) + \tan(x)| + C$$ 6. **Final answer:** $$\boxed{\int \sec^3(x) \, dx = \frac{\sec(x) \tan(x)}{2} + \frac{1}{2} \ln|\sec(x) + \tan(x)| + C}$$