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.

Line 3 is obtained by applying the trig identity ${\tan }^2\left(\mathrm{\θ}\right)\={\sec }^2\left(\mathrm{\θ}\right)-1$. The integral of ${\sec }^3\left(\mathrm{\θ}\right)$ evaluated in line 5 is derived in Example 6.2.5. The integral of $\sec \left(\mathrm{\θ}\right)$ is derived in Table 6.2.10. The absolute values in line 5 are dropped in line 6 because the argument of the logarithm is positive for the $\mathrm{\θ}$-interval defined in Table 6.3.1.

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

$\frac{x}{18}\sqrt{4\+9 x^2}-\frac{2}{27}\ln \left(\frac{\sqrt{4\+9 x^2}}{2}\+\frac{3}{2}x\right)$

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:

From Figure 6.3.2, $\sin \left(\mathrm{\θ}\right)\=3 x\/\sqrt{9 x^2\+4}$, and $\cos \left(\mathrm{\θ}\right)\=2\/\sqrt{9 x^2\+4}$.

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 proceeds as shown in Table 6.3.15(a).

Note how the tutor requires the Rewrite rule where at top-level the expand command would suffice. Note further that the solution in Table 6.3.18 is not complete - the Revert rule has not been applied. If it were, the result would be the following, a result in dire need of a simplification that cannot be effected in the tutor.

$-\frac{1}{12}{x}^2-\frac{1}{54}\+\frac{1}{36}x\sqrt{9x^2\+4}\+\frac{2}{27}\ln \left(\sqrt{9x^2\+4}-3x\right)\+\frac{2}{27{\left(\sqrt{9x^2\+4}-3x\right)}^2}$

Table 6.3.15(b) shows the result when the Change rule $x\=\frac{2}{3}\tan \left(\mathrm{\θ}\right)$ is imposed on the tutor.

Note the use of the Rewrite rule to effect transformations that at top-level could be implemented more directly. Note also that the two integrals in the last line of Table 6.3.15(b) are not evaluated. Maple's stepwise evaluation of these two integrals reproduces the derivations in Example 6.2.5 and Table 6.2.10, respectively.