The substitution $x\=\frac{2}{3}\tan \left(\mathrm{\θ}\right)$ means $\mathrm{dx}\=\frac{2}{3}{\sec }^2\left(\mathrm{\θ}\right) d\mathrm{\θ}$, $\mathrm{\θ}\=\arctan \left(\frac{3}{2}x\right)$, and turns $g\left(x\right)$ into $2 \sec \left(\mathrm{\θ}\right)$. Hence, the evaluation of the given integral proceeds as follows.

The identity $\frac{1}{u^2\left(1-u^2\right)}\=\frac{1\/2}{u\+1}-\frac{1\/2}{u-1}\+\frac{1}{u^2}$ is established by the algebraic device of partial fraction decomposition, that will be studied formally in Section 6.4. The absolute value in the logarithm of $\sin \left(\mathrm{\θ}\right)-1$ can't be dropped since this quantity can be negative, whereas $\sin \left(\mathrm{\θ}\right)\+1$ is nonnegative.

Figure 6.3.2 can be used to obtain $\sin \left(\mathrm{\θ}\right)\=3 x\/\sqrt{4\+9 x^2}$.

The last three lines of the solution given above are the result of algebraic manipulations. First, the properties of real logarithms are used to combine the two log terms to one. Then, rationalization of the denominator in the argument of this single log term is used to simplify the argument. The result is seen in the penultimate line. The simplification in the final line is again the result of applying a property of real logarithms.

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

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

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

The stepwise solution provided by the  tutor when the Constant, Constant Multiple, and Sum rules are taken as Understood Rules begins with the substitution $u^2\=9\+4\/x^2$ and proceeds as shown in Table 6.3.16(a).

Note that the solution in Table 6.3.16(a) is not complete - the antiderivatives have not been obtained and the Revert rule has not been applied. Indeed, the final result, a result in dire need of a simplification that cannot be effected in the tutor, is

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

Table 6.3.16(b) shows the result when the Change rule $x\=\frac{2}{3}\tan \left(\mathrm{\θ}\right)$ is imposed on the tutor.

The substitution $u\=\sin \left(\mathrm{\θ}\right)$ leads to a rational function in $u$, an expression that is resolved by the algebraic technique of the partial fraction decomposition, to be studied in detail in Section 6.4.