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 the final step, the absolute values in 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,

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

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:

Maple resists changing the integral of $1\/\cos \left(\mathrm{\θ}\right)$ to the integral of $\sec \left(\mathrm{\θ}\right)$.

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

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

an antiderivative that differs from the earlier solution by $\ln \left(2\right)$, an additive constant of integration. Also, the argument of the logarithm has to be rationalized to make one expression look like the other.

On the other hand, Table 6.3.10(a) shows the result when the Change rule $x\=\frac{2}{3}\tan \left(\mathrm{\θ}\right)$, the sec rule, and the Revert rule are applied in the tutor.