The substitution $x\=\frac{2}{3}\tan \left(\mathrm{\θ}\right)$ means $\mathrm{dx}\=\frac{2}{3}{\sec }^2\left(\mathrm{\θ}\right) d\mathrm{\θ}$, and turns $g\left(x\right)$ into $2 \sec \left(\mathrm{\θ}\right)$. From Figure 6.3.2, $\sec \left(\mathrm{\θ}\right)\=\frac{1}{2}\sqrt{4\+9 x^2}$. Hence, the evaluation of the given integral proceeds as follows.

In line 4, the absolute values inside the logarithm are dropped because the argument of the logarithm is positive for all real $x$.

Using the appropriate identity in Table 2.10.4, the alternate form of the solution, namely,

$\frac{x}{2}\sqrt{4\+9 x^2}\+\frac{2}{3}\ln \left(\frac{\sqrt{4\+9 x^2}}{2}\+\frac{3}{2}x\right)$

can be obtained from the Maple solution.

A stepwise solution that uses top-level commands except for one application of the Change command from the IntegrationTools package:

Even if the integral of $1\/{\cos }^3\left(\mathrm{\θ}\right)$ were written as the integral of ${\sec }^3\left(\mathrm{\θ}\right)$, Maple's antiderivative would still contain $\sin \left(\mathrm{\θ}\right)\/{\cos }^2\left(\mathrm{\θ}\right)$, instead of $\sec \left(\mathrm{\θ}\right)\tan \left(\mathrm{\θ}\right)$.

The stepwise solution provided by the  tutor when the Constant, Constant Multiple, and Sum rules are taken as Understood Rules begins with the substitution $u\=\sqrt{9 x^2\+4}-3 x$, and ends with

$-\frac{3x^2}{4}-\frac{1}{6}+\frac{x\sqrt{9x^2+4}}{4}-\frac{2\ln \left(\sqrt{9x^2+4}-3x\right)}{3}+\frac{2}{3{\left(\sqrt{9x^2+4}-3x\right)}^2}$

an expression equivalent to what has been obtained earlier, but which is difficult to transform to same. On the other hand, if the Change rule $x\=\frac{2}{3}\tan \left(\mathrm{\θ}\right)$ is invoked in the tutor, the immediate return is the expected, as shown in Table 6.3.9(a).