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 $\sec \left(\mathrm{\&theta;}\right)$ is derived in Table 6.2.10. The absolute value in line 3 is retained in line 4 because the argument of the logarithm is negative for $\mathrm{\&theta;}\in \left[\mathrm{\&pi;}\&comma;3 \mathrm{\&pi;}\&sol;2\right)$, which corresponds to $x<-2\&sol;3$

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, $\tan \left(\mathrm{\&theta;}\right)\&equals;\frac{1}{2}\sqrt{9 x^2-4}$. Note how the simplification isolates the additive constant of integration, $-\ln \left(2\right)\&sol;3$.

The stepwise solution provided by the  tutor when the Constant, Constant Multiple, and Sum rules are taken as Understood Rules begins with the substitution $u=\sqrt{9x^2-4}-3x$ and proceeds as shown in Table 6.3.18(a).

Table 6.3.18(b) shows the result when the Change rule $x\&equals;\frac{2}{3}\sec \left(\mathrm{\&theta;}\right)$ is imposed on the tutor.

The integrand has been simplified without any restriction on $\mathrm{\&theta;}$; hence, the absolute value of $\sec \left(\mathrm{\&theta;}\right)$ appears. The only way to proceed in the tutor is via the Rewrite rule whereby $\left|\sec \left(\mathrm{\&theta;}\right)\right|$ is simply replaced with $\sec \left(\mathrm{\&theta;}\right)$, at which point the stepwise code will re-derive the antiderivative of $\sec \left(\mathrm{\&theta;}\right)$ as per Table 6.2.10.