Note that there is a discontinuity in the integrand at $x\=0$, which is in $\left[-\frac{2}{3}\,\frac{2}{3}\right]$, the interval for which the integrand remains real.

The substitution $x\=\frac{2}{3}\sin \left(\mathrm{\θ}\right)$ means $\mathrm{dx}\=\frac{2}{3}\cos \left(\mathrm{\θ}\right) d\mathrm{\θ}$, and turns $f\left(x\right)$ into $2 \cos \left(\mathrm{\θ}\right)$. From Figure 6.3.1, $\cos \left(\mathrm{\θ}\right)\=\frac{1}{2}\sqrt{4-9 x^2}$, $\cot \left(\mathrm{\θ}\right)\=\frac{1}{3 x}\sqrt{4-9 x^2}$, and $\csc \left(\mathrm{\θ}\right)\=\frac{2}{3 x}$. Hence, the evaluation of the given integral proceeds as follows.

The antiderivative of $\csc \left(\mathrm{\θ}\right)$ is obtained in Table 6.2.10.

Recall that the given integrand is defined over the reals only when $x\in \left(0\,2\/3\right]\bigcup \left[-2\/3\,0\right)$. The solution given by Maple is not real for this domain because on this domain $2\/\sqrt{4-9 x^2}\ge 1$. (The inverse hyperbolic tangent function is complex for such arguments.)

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

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

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:

Without a restriction on $\mathrm{\θ}$, the logarithm in the penultimate step could be complex. Its argument is nonnegative only for $\mathrm{\θ}\in \left[0\,\mathrm{\π}\/2\right]$. It would be better to apply the logarithm to the absolute value of the  arguments in this and the last step.

Table 6.3.5(a) displays the annotated stepwise solution provided by the  tutor when the Constant, Constant Multiple, and Sum rules are taken as Understood Rules.

Maple's stepwise solution will next re-derive the antiderivative of $\csc \left(u\right)$ as per Table 6.2.10.