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.

Line 3 is obtained by applying the trig identity ${\tan }^2\left(\mathrm{\θ}\right)\={\sec }^2\left(\mathrm{\θ}\right)-1$. Line 5 is obtained by applying the reduction formula derived in Example 6.2.9. The integral of ${\sec }^3\left(\mathrm{\θ}\right)$ evaluated in line 7 is derived in Example 6.2.5. The absolute value in line 7 is retained in line 8 because the argument of the logarithm is negative for $\mathrm{\θ}\in \left[\mathrm{\π}\,3 \mathrm{\π}\/2\right)$. (The astute reader will recognize that the trig integral in this example is exactly the same as the one in Example 6.3.14.)$$

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 $\tan \left(\mathrm{\θ}\right)\=\frac{1}{2}\sqrt{9 x^2-4}$.

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 6.3.22(a).

The change of variables selected by the tutor leads to $\left(-u^3\+32\/u-256\/u^5\right)\/432$, 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 6.3.22(b) shows the result when the Change rule $x\=\frac{2}{3}\sec \left(\mathrm{\θ}\right)$ is imposed on the tutor.

After making the initial change of variables, the integrand is actually a multiple of ${\sec }^3\left(\mathrm{\θ}\right){\tan }^2\left(\mathrm{\θ}\right)$, which occurred in Example 6.3.14, and is completely detailed there. The tutor makes the further change of variables $u\=\tan \left(\mathrm{\θ}\right)$ and after some six more steps (including another change of variables and a rewrite) arrives at the form ${\sec }^3\left(\mathrm{\θ}\right){\tan }^2\left(\mathrm{\θ}\right)$. Mercifully, these details have been suppressed in Table 6.3.22(b).