Integration is a linear operation, just like differentiation. Therefore, it "goes past" constants, and plus and minus signs, resulting in the right-hand side of the first equation. The right-hand side of the second equation results from finding antiderivatives of ${\sec }^2\left(x\right)$ and $\sin \left(x\right)$ by consulting, for example, Table 3.10.1  in Section 3.10. The rest is arithmetic.

Note that Maple does not tack on the additive constant. In other words, Maple returns an antiderivative, not the most general antiderivative. This is consistent with the space-saving requirements of printed tables of integrals, but not necessarily with practice in the typical calculus text.

Note, finally, that Maple's internal routines for finding the antiderivative of ${\sec }^2\left(x\right)$ causes it to express $\tan \left(x\right)$ as $\sin \left(x\right)\/\cos \left(x\right)$.