The substitution $x\=\frac{2}{3}\sec \left(\mathrm{\θ}\right)$ means $\mathrm{dx}\=\frac{2}{3}\sec \left(\mathrm{\θ}\right)\tan \left(\mathrm{\θ}\right) d\mathrm{\θ}$, and turns $h\left(x\right)$ into $2 \tan \left(\mathrm{\θ}\right)$. From Figure 6.3.3, $\tan \left(\mathrm{\θ}\right)\=\frac{1}{2}\sqrt{9 x^2-4}$. Hence, the evaluation of the given integral proceeds as follows.

The integral of ${\sec }^3\left(\mathrm{\θ}\right)$ evaluated in line 3 is derived in Example 6.2.5. The absolute value in line 5 is retained in line 6 because the argument of the logarithm is negative for $\mathrm{\θ}\in \left[\mathrm{\π}\,3 \mathrm{\π}\/2\right)$.

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

From Figure 6.3.3, $\sin \left(\mathrm{\θ}\right)\=\frac{1}{3 x}\sqrt{9 x^2-4}$, $\cos \left(\mathrm{\θ}\right)\=\frac{2}{3 x}$, and $\tan \left(\mathrm{\θ}\right)\=\frac{1}{2}\sqrt{9 x^2-4}$. This solution differs from the previous one by an additive constant of integration.

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 6.3.23(a).

The change of variables selected by the tutor leads to a multiple of ${\left(u^2\+4\right)}^2\/u^3$, provided $u\ge 2$, and this corresponds to $x\le -2\/3$. Since the tutor does not have provision for routine simplifications, it takes several steps, including invocation of the Rewrite rule to massage the expression into a form where the power rule of integration applies.

Table 6.3.23(b) shows the result when the Change rule $x\=\frac{2}{3}\sec \left(\mathrm{\θ}\right)$ is imposed on the tutor.

It takes the Rewrite rule to remove the absolute value in the integrand, and without such, the tutor can go no further. The antiderivative of ${\sec }^3\left(\mathrm{\θ}\right)$ is derived in Example 6.2.5.