Integral of ln(x^2-3) dx

1. Use integration by parts:

   $$\int \operatorname{u} \operatorname{dv} = \operatorname{u}\operatorname{v} - \int \operatorname{v} \operatorname{du}$$

   Let $u{\left(x \right)} = \log{\left(x^{2} - 3 \right)}$ and let $\operatorname{dv}{\left(x \right)} = 1$.

   Then $\operatorname{du}{\left(x \right)} = \frac{2 x}{x^{2} - 3}$.

   To find $v{\left(x \right)}$:

   1. The integral of a constant is the constant times the variable of integration:

      $$\int 1\, dx = x$$

   Now evaluate the sub-integral.

2. The integral of a constant times a function is the constant times the integral of the function:

   $$\int \frac{2 x^{2}}{x^{2} - 3}\, dx = 2 \int \frac{x^{2}}{x^{2} - 3}\, dx$$

   1. Rewrite the integrand:

      $$\frac{x^{2}}{x^{2} - 3} = 1 + \frac{3}{x^{2} - 3}$$

   2. Integrate term-by-term:

      1. The integral of a constant is the constant times the variable of integration:

         $$\int 1\, dx = x$$

      1. The integral of a constant times a function is the constant times the integral of the function:

         $$\int \frac{3}{x^{2} - 3}\, dx = 3 \int \frac{1}{x^{2} - 3}\, dx$$

         PiecewiseRule(subfunctions=[(ArctanRule(a=1, b=1, c=-3, context=1/(x**2 - 3), symbol=x), False), (ArccothRule(a=1, b=1, c=-3, context=1/(x**2 - 3), symbol=x), x**2 > 3), (ArctanhRule(a=1, b=1, c=-3, context=1/(x**2 - 3), symbol=x), x**2 < 3)], context=1/(x**2 - 3), symbol=x)

         So, the result is: $3 \left(\begin{cases} - \frac{\sqrt{3} \operatorname{acoth}{\left(\frac{\sqrt{3} x}{3} \right)}}{3} & \text{for}\: x^{2} > 3 \\- \frac{\sqrt{3} \operatorname{atanh}{\left(\frac{\sqrt{3} x}{3} \right)}}{3} & \text{for}\: x^{2} < 3 \end{cases}\right)$

      The result is: $x + 3 \left(\begin{cases} - \frac{\sqrt{3} \operatorname{acoth}{\left(\frac{\sqrt{3} x}{3} \right)}}{3} & \text{for}\: x^{2} > 3 \\- \frac{\sqrt{3} \operatorname{atanh}{\left(\frac{\sqrt{3} x}{3} \right)}}{3} & \text{for}\: x^{2} < 3 \end{cases}\right)$

   So, the result is: $2 x + 6 \left(\begin{cases} - \frac{\sqrt{3} \operatorname{acoth}{\left(\frac{\sqrt{3} x}{3} \right)}}{3} & \text{for}\: x^{2} > 3 \\- \frac{\sqrt{3} \operatorname{atanh}{\left(\frac{\sqrt{3} x}{3} \right)}}{3} & \text{for}\: x^{2} < 3 \end{cases}\right)$

3. Now simplify:

   $$\begin{cases} x \log{\left(x^{2} - 3 \right)} - 2 x + 2 \sqrt{3} \operatorname{acoth}{\left(\frac{\sqrt{3} x}{3} \right)} & \text{for}\: x^{2} > 3 \\x \log{\left(x^{2} - 3 \right)} - 2 x + 2 \sqrt{3} \operatorname{atanh}{\left(\frac{\sqrt{3} x}{3} \right)} & \text{for}\: x^{2} < 3 \end{cases}$$

4. Add the constant of integration:

   $$\begin{cases} x \log{\left(x^{2} - 3 \right)} - 2 x + 2 \sqrt{3} \operatorname{acoth}{\left(\frac{\sqrt{3} x}{3} \right)} & \text{for}\: x^{2} > 3 \\x \log{\left(x^{2} - 3 \right)} - 2 x + 2 \sqrt{3} \operatorname{atanh}{\left(\frac{\sqrt{3} x}{3} \right)} & \text{for}\: x^{2} < 3 \end{cases}+ \mathrm{constant}$$

The answer is:

$$\begin{cases} x \log{\left(x^{2} - 3 \right)} - 2 x + 2 \sqrt{3} \operatorname{acoth}{\left(\frac{\sqrt{3} x}{3} \right)} & \text{for}\: x^{2} > 3 \\x \log{\left(x^{2} - 3 \right)} - 2 x + 2 \sqrt{3} \operatorname{atanh}{\left(\frac{\sqrt{3} x}{3} \right)} & \text{for}\: x^{2} < 3 \end{cases}+ \mathrm{constant}$$