\[ \int \frac {12 x-12 x^2-12 x^3+24 x^4-12 x^5+\left (-12+12 x+24 x^2-48 x^3+24 x^4\right ) \log (2)+\left (-12 x+24 x^2-12 x^3\right ) \log ^2(2)+\left (-16+44 x-40 x^2+44 x^3-40 x^4+12 x^5+\left (-16+44 x-88 x^2+80 x^3-24 x^4\right ) \log (2)+\left (-16+44 x-40 x^2+12 x^3\right ) \log ^2(2)\right ) \log \left (\frac {1}{2} (4-3 x)\right )}{\left (-4 x^2+11 x^3-10 x^4+3 x^5+\left (8 x-22 x^2+20 x^3-6 x^4\right ) \log (2)+\left (-4+11 x-10 x^2+3 x^3\right ) \log ^2(2)\right ) \log ^2\left (\frac {1}{2} (4-3 x)\right )} \, dx \]
Optimal antiderivative \[ \frac {4 x -\frac {4 x}{\left (\ln \left (2\right )-x \right ) \left (x^{2}-x \right )}}{\ln \! \left (2-\frac {3 x}{2}\right )} \]
command
integrate((((12*x^3-40*x^2+44*x-16)*log(2)^2+(-24*x^4+80*x^3-88*x^2+44*x-16)*log(2)+12*x^5-40*x^4+44*x^3-40*x^2+44*x-16)*log(2-3/2*x)+(-12*x^3+24*x^2-12*x)*log(2)^2+(24*x^4-48*x^3+24*x^2+12*x-12)*log(2)-12*x^5+24*x^4-12*x^3-12*x^2+12*x)/((3*x^3-10*x^2+11*x-4)*log(2)^2+(-6*x^4+20*x^3-22*x^2+8*x)*log(2)+3*x^5-10*x^4+11*x^3-4*x^2)/log(2-3/2*x)^2,x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \text {Timed out} \]
Giac 1.7.0 via sagemath 9.3 output
\[ -\frac {4 \, {\left (x^{3} - x^{2} \log \left (2\right ) - x^{2} + x \log \left (2\right ) + 1\right )}}{x^{2} \log \left (2\right ) - x \log \left (2\right )^{2} - x^{2} \log \left (-3 \, x + 4\right ) + x \log \left (2\right ) \log \left (-3 \, x + 4\right ) - x \log \left (2\right ) + \log \left (2\right )^{2} + x \log \left (-3 \, x + 4\right ) - \log \left (2\right ) \log \left (-3 \, x + 4\right )} \]