There are several methods for integrating this type of integral without using tables. One method is to use integration by parts, where you choose $u=\arctan(x)$ and $dv=x\,dx$. Another method is to use a substitution, where you let $u=\arctan(x)$ and $dx=\frac{du}{1+x^2}$. Both of these methods will eventually lead to an answer without the use of tables.
Yes, you can use trigonometric identities to integrate this type of integral. One example is to use the identity $\arctan(x)=\frac{1}{2}i\ln\left(\frac{1+ix}{1-ix}\right)$, which can help simplify the integral and make it easier to solve.
Unfortunately, there is no shortcut for integrating this type of integral. It requires some algebraic manipulation and knowledge of integration techniques in order to arrive at the answer. However, with practice and familiarity, the process can become easier and less time-consuming.
Yes, you can use a computer program or calculator to integrate this type of integral. However, it is still important to understand the steps and techniques involved in the integration process in order to verify the accuracy of the answer and be able to solve similar integrals in the future.
One helpful tip for integrating this type of integral is to pay attention to the symmetry of the integrand. Since $\arctan(x)$ is an odd function and $x$ is an even function, the product $x\arctan(x)$ is an odd function. This means that the integral will be symmetric about the origin, which can help simplify the integration process.