$\begin{align}  & \int{{{\sec }^{3}}xdx}=\int{\sec x\cdot {{\sec }^{2}}xdx} \\ & \text{Using integration by part} \\ & \text{Let }u=\sec x,\text{ }du=\sec x\tan xdx \\ & \text{     }dv={{\sec }^{2}}xdx,\text{ }v=\tan x \\ & \int{udv}=uv-\int{vdu} \\ & \int{\sec x\cdot {{\sec }^{2}}xdx}=\sec x\tan x-\int{\tan x(\sec x\tan x)dx} \\ & \int{{{\sec }^{3}}xdx}=\sec x\tan x-\int{{{\tan }^{2}}x\sec xdx} \\ & \int{{{\sec }^{3}}xdx}=\sec x\tan x-\int{({{\sec }^{2}}x-1)\sec xdx} \\ & \int{{{\sec }^{3}}x}dx=\sec x\tan x-\int{({{\sec }^{3}}x-\sec x)dx} \\ & \int{{{\sec }^{3}}x}dx=\sec x\tan x-\int{{{\sec }^{3}}xdx}+\int{\sec xdx} \\ & \int{{{\sec }^{3}}xdx}+\int{{{\sec }^{3}}x}=\sec x\tan x+\int{\sec xdx} \\ & 2\int{{{\sec }^{3}}xdx}=\sec x\tan x+\ln (\sec x+\tan x) \\ & \int{{{\sec }^{3}}xdx}=\frac{1}{2}\left[ \sec x\tan x+(\ln \sec x+\tan x) \right]+C \\\end{align}$