Integrand size = 141, antiderivative size = 27 \[ \int \frac {24-6 x+\left (12+24 x-6 x^2\right ) \log \left (\frac {2}{x^2}\right )+(12-3 x) \log \left (\frac {2}{x^2}\right ) \log \left (-\frac {x}{(-4+x) \log \left (\frac {2}{x^2}\right )}\right )}{\left (-4 x^4+x^5\right ) \log \left (\frac {2}{x^2}\right )+\left (-8 x^3+2 x^4\right ) \log \left (\frac {2}{x^2}\right ) \log \left (-\frac {x}{(-4+x) \log \left (\frac {2}{x^2}\right )}\right )+\left (-4 x^2+x^3\right ) \log \left (\frac {2}{x^2}\right ) \log ^2\left (-\frac {x}{(-4+x) \log \left (\frac {2}{x^2}\right )}\right )} \, dx=\frac {3}{x \left (x+\log \left (\frac {x}{(4-x) \log \left (\frac {2}{x^2}\right )}\right )\right )} \]
Time = 0.13 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.96 \[ \int \frac {24-6 x+\left (12+24 x-6 x^2\right ) \log \left (\frac {2}{x^2}\right )+(12-3 x) \log \left (\frac {2}{x^2}\right ) \log \left (-\frac {x}{(-4+x) \log \left (\frac {2}{x^2}\right )}\right )}{\left (-4 x^4+x^5\right ) \log \left (\frac {2}{x^2}\right )+\left (-8 x^3+2 x^4\right ) \log \left (\frac {2}{x^2}\right ) \log \left (-\frac {x}{(-4+x) \log \left (\frac {2}{x^2}\right )}\right )+\left (-4 x^2+x^3\right ) \log \left (\frac {2}{x^2}\right ) \log ^2\left (-\frac {x}{(-4+x) \log \left (\frac {2}{x^2}\right )}\right )} \, dx=\frac {3}{x \left (x+\log \left (-\frac {x}{(-4+x) \log \left (\frac {2}{x^2}\right )}\right )\right )} \]
Integrate[(24 - 6*x + (12 + 24*x - 6*x^2)*Log[2/x^2] + (12 - 3*x)*Log[2/x^ 2]*Log[-(x/((-4 + x)*Log[2/x^2]))])/((-4*x^4 + x^5)*Log[2/x^2] + (-8*x^3 + 2*x^4)*Log[2/x^2]*Log[-(x/((-4 + x)*Log[2/x^2]))] + (-4*x^2 + x^3)*Log[2/ x^2]*Log[-(x/((-4 + x)*Log[2/x^2]))]^2),x]
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {\left (-6 x^2+24 x+12\right ) \log \left (\frac {2}{x^2}\right )+(12-3 x) \log \left (\frac {2}{x^2}\right ) \log \left (-\frac {x}{(x-4) \log \left (\frac {2}{x^2}\right )}\right )-6 x+24}{\left (x^3-4 x^2\right ) \log \left (\frac {2}{x^2}\right ) \log ^2\left (-\frac {x}{(x-4) \log \left (\frac {2}{x^2}\right )}\right )+\left (x^5-4 x^4\right ) \log \left (\frac {2}{x^2}\right )+\left (2 x^4-8 x^3\right ) \log \left (\frac {2}{x^2}\right ) \log \left (-\frac {x}{(x-4) \log \left (\frac {2}{x^2}\right )}\right )} \, dx\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \int \frac {-\left (-6 x^2+24 x+12\right ) \log \left (\frac {2}{x^2}\right )-(12-3 x) \log \left (\frac {2}{x^2}\right ) \log \left (-\frac {x}{(x-4) \log \left (\frac {2}{x^2}\right )}\right )+6 x-24}{(4-x) x^2 \log \left (\frac {2}{x^2}\right ) \left (\log \left (-\frac {x}{(x-4) \log \left (\frac {2}{x^2}\right )}\right )+x\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (-\frac {3 \left (x^2 \log \left (\frac {2}{x^2}\right )-4 x \log \left (\frac {2}{x^2}\right )-4 \log \left (\frac {2}{x^2}\right )+2 x-8\right )}{(x-4) x^2 \log \left (\frac {2}{x^2}\right ) \left (\log \left (-\frac {x}{(x-4) \log \left (\frac {2}{x^2}\right )}\right )+x\right )^2}-\frac {3}{x^2 \left (\log \left (-\frac {x}{(x-4) \log \left (\frac {2}{x^2}\right )}\right )+x\right )}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {3}{4} \int \frac {1}{(x-4) \left (x+\log \left (-\frac {x}{(x-4) \log \left (\frac {2}{x^2}\right )}\right )\right )^2}dx-3 \int \frac {1}{x^2 \left (x+\log \left (-\frac {x}{(x-4) \log \left (\frac {2}{x^2}\right )}\right )\right )^2}dx-\frac {15}{4} \int \frac {1}{x \left (x+\log \left (-\frac {x}{(x-4) \log \left (\frac {2}{x^2}\right )}\right )\right )^2}dx-6 \int \frac {1}{x^2 \log \left (\frac {2}{x^2}\right ) \left (x+\log \left (-\frac {x}{(x-4) \log \left (\frac {2}{x^2}\right )}\right )\right )^2}dx-3 \int \frac {1}{x^2 \left (x+\log \left (-\frac {x}{(x-4) \log \left (\frac {2}{x^2}\right )}\right )\right )}dx\) |
Int[(24 - 6*x + (12 + 24*x - 6*x^2)*Log[2/x^2] + (12 - 3*x)*Log[2/x^2]*Log [-(x/((-4 + x)*Log[2/x^2]))])/((-4*x^4 + x^5)*Log[2/x^2] + (-8*x^3 + 2*x^4 )*Log[2/x^2]*Log[-(x/((-4 + x)*Log[2/x^2]))] + (-4*x^2 + x^3)*Log[2/x^2]*L og[-(x/((-4 + x)*Log[2/x^2]))]^2),x]
3.27.24.3.1 Defintions of rubi rules used
Time = 1.66 (sec) , antiderivative size = 54, normalized size of antiderivative = 2.00
| method | result | size |
| parallelrisch | \(\frac {48-3 x^{2}-3 \ln \left (-\frac {x}{\left (x -4\right ) \ln \left (\frac {2}{x^{2}}\right )}\right ) x}{16 x \left (\ln \left (-\frac {x}{\left (x -4\right ) \ln \left (\frac {2}{x^{2}}\right )}\right )+x \right )}\) | \(54\) |
int(((-3*x+12)*ln(2/x^2)*ln(-x/(x-4)/ln(2/x^2))+(-6*x^2+24*x+12)*ln(2/x^2) -6*x+24)/((x^3-4*x^2)*ln(2/x^2)*ln(-x/(x-4)/ln(2/x^2))^2+(2*x^4-8*x^3)*ln( 2/x^2)*ln(-x/(x-4)/ln(2/x^2))+(x^5-4*x^4)*ln(2/x^2)),x,method=_RETURNVERBO SE)
Time = 0.24 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.00 \[ \int \frac {24-6 x+\left (12+24 x-6 x^2\right ) \log \left (\frac {2}{x^2}\right )+(12-3 x) \log \left (\frac {2}{x^2}\right ) \log \left (-\frac {x}{(-4+x) \log \left (\frac {2}{x^2}\right )}\right )}{\left (-4 x^4+x^5\right ) \log \left (\frac {2}{x^2}\right )+\left (-8 x^3+2 x^4\right ) \log \left (\frac {2}{x^2}\right ) \log \left (-\frac {x}{(-4+x) \log \left (\frac {2}{x^2}\right )}\right )+\left (-4 x^2+x^3\right ) \log \left (\frac {2}{x^2}\right ) \log ^2\left (-\frac {x}{(-4+x) \log \left (\frac {2}{x^2}\right )}\right )} \, dx=\frac {3}{x^{2} + x \log \left (-\frac {x}{{\left (x - 4\right )} \log \left (\frac {2}{x^{2}}\right )}\right )} \]
integrate(((-3*x+12)*log(2/x^2)*log(-x/(x-4)/log(2/x^2))+(-6*x^2+24*x+12)* log(2/x^2)-6*x+24)/((x^3-4*x^2)*log(2/x^2)*log(-x/(x-4)/log(2/x^2))^2+(2*x ^4-8*x^3)*log(2/x^2)*log(-x/(x-4)/log(2/x^2))+(x^5-4*x^4)*log(2/x^2)),x, a lgorithm=\
Time = 0.15 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.74 \[ \int \frac {24-6 x+\left (12+24 x-6 x^2\right ) \log \left (\frac {2}{x^2}\right )+(12-3 x) \log \left (\frac {2}{x^2}\right ) \log \left (-\frac {x}{(-4+x) \log \left (\frac {2}{x^2}\right )}\right )}{\left (-4 x^4+x^5\right ) \log \left (\frac {2}{x^2}\right )+\left (-8 x^3+2 x^4\right ) \log \left (\frac {2}{x^2}\right ) \log \left (-\frac {x}{(-4+x) \log \left (\frac {2}{x^2}\right )}\right )+\left (-4 x^2+x^3\right ) \log \left (\frac {2}{x^2}\right ) \log ^2\left (-\frac {x}{(-4+x) \log \left (\frac {2}{x^2}\right )}\right )} \, dx=\frac {3}{x^{2} + x \log {\left (- \frac {x}{\left (x - 4\right ) \log {\left (\frac {2}{x^{2}} \right )}} \right )}} \]
integrate(((-3*x+12)*ln(2/x**2)*ln(-x/(x-4)/ln(2/x**2))+(-6*x**2+24*x+12)* ln(2/x**2)-6*x+24)/((x**3-4*x**2)*ln(2/x**2)*ln(-x/(x-4)/ln(2/x**2))**2+(2 *x**4-8*x**3)*ln(2/x**2)*ln(-x/(x-4)/ln(2/x**2))+(x**5-4*x**4)*ln(2/x**2)) ,x)
Time = 0.35 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.19 \[ \int \frac {24-6 x+\left (12+24 x-6 x^2\right ) \log \left (\frac {2}{x^2}\right )+(12-3 x) \log \left (\frac {2}{x^2}\right ) \log \left (-\frac {x}{(-4+x) \log \left (\frac {2}{x^2}\right )}\right )}{\left (-4 x^4+x^5\right ) \log \left (\frac {2}{x^2}\right )+\left (-8 x^3+2 x^4\right ) \log \left (\frac {2}{x^2}\right ) \log \left (-\frac {x}{(-4+x) \log \left (\frac {2}{x^2}\right )}\right )+\left (-4 x^2+x^3\right ) \log \left (\frac {2}{x^2}\right ) \log ^2\left (-\frac {x}{(-4+x) \log \left (\frac {2}{x^2}\right )}\right )} \, dx=\frac {3}{x^{2} - x \log \left (x - 4\right ) + x \log \left (x\right ) - x \log \left (-\log \left (2\right ) + 2 \, \log \left (x\right )\right )} \]
integrate(((-3*x+12)*log(2/x^2)*log(-x/(x-4)/log(2/x^2))+(-6*x^2+24*x+12)* log(2/x^2)-6*x+24)/((x^3-4*x^2)*log(2/x^2)*log(-x/(x-4)/log(2/x^2))^2+(2*x ^4-8*x^3)*log(2/x^2)*log(-x/(x-4)/log(2/x^2))+(x^5-4*x^4)*log(2/x^2)),x, a lgorithm=\
Result contains complex when optimal does not.
Time = 2.45 (sec) , antiderivative size = 699, normalized size of antiderivative = 25.89 \[ \int \frac {24-6 x+\left (12+24 x-6 x^2\right ) \log \left (\frac {2}{x^2}\right )+(12-3 x) \log \left (\frac {2}{x^2}\right ) \log \left (-\frac {x}{(-4+x) \log \left (\frac {2}{x^2}\right )}\right )}{\left (-4 x^4+x^5\right ) \log \left (\frac {2}{x^2}\right )+\left (-8 x^3+2 x^4\right ) \log \left (\frac {2}{x^2}\right ) \log \left (-\frac {x}{(-4+x) \log \left (\frac {2}{x^2}\right )}\right )+\left (-4 x^2+x^3\right ) \log \left (\frac {2}{x^2}\right ) \log ^2\left (-\frac {x}{(-4+x) \log \left (\frac {2}{x^2}\right )}\right )} \, dx=\text {Too large to display} \]
integrate(((-3*x+12)*log(2/x^2)*log(-x/(x-4)/log(2/x^2))+(-6*x^2+24*x+12)* log(2/x^2)-6*x+24)/((x^3-4*x^2)*log(2/x^2)*log(-x/(x-4)/log(2/x^2))^2+(2*x ^4-8*x^3)*log(2/x^2)*log(-x/(x-4)/log(2/x^2))+(x^5-4*x^4)*log(2/x^2)),x, a lgorithm=\
-3*(x^2*log(2)*log(2/x^2) - 2*x^2*log(x)*log(2/x^2) - 4*x*log(2)*log(2/x^2 ) + 8*x*log(x)*log(2/x^2) + 2*x*log(2/x^2) - 4*log(2)*log(2/x^2) + 8*log(x )*log(2/x^2) - 8*log(2/x^2))/(-I*pi*x^3*log(2)*log(2/x^2) - x^4*log(2)*log (2/x^2) + x^3*log(2)*log(x*log(2/x^2) - 4*log(2/x^2))*log(2/x^2) + 2*I*pi* x^3*log(x)*log(2/x^2) + 2*x^4*log(x)*log(2/x^2) - x^3*log(2)*log(x)*log(2/ x^2) - 2*x^3*log(x*log(2/x^2) - 4*log(2/x^2))*log(x)*log(2/x^2) + 2*x^3*lo g(x)^2*log(2/x^2) + 4*I*pi*x^2*log(2)*log(2/x^2) + 4*x^3*log(2)*log(2/x^2) - 4*x^2*log(2)*log(x*log(2/x^2) - 4*log(2/x^2))*log(2/x^2) - 8*I*pi*x^2*l og(x)*log(2/x^2) - 8*x^3*log(x)*log(2/x^2) + 4*x^2*log(2)*log(x)*log(2/x^2 ) + 8*x^2*log(x*log(2/x^2) - 4*log(2/x^2))*log(x)*log(2/x^2) - 8*x^2*log(x )^2*log(2/x^2) - 2*I*pi*x^2*log(2) - 2*x^3*log(2) + 2*x^2*log(2)*log(x*log (2/x^2) - 4*log(2/x^2)) + 4*I*pi*x^2*log(x) + 4*x^3*log(x) - 2*x^2*log(2)* log(x) - 4*x^2*log(x*log(2/x^2) - 4*log(2/x^2))*log(x) + 4*x^2*log(x)^2 + 4*I*pi*x*log(2)*log(2/x^2) + 4*x^2*log(2)*log(2/x^2) - 4*x*log(2)*log(x*lo g(2/x^2) - 4*log(2/x^2))*log(2/x^2) - 8*I*pi*x*log(x)*log(2/x^2) - 8*x^2*l og(x)*log(2/x^2) + 4*x*log(2)*log(x)*log(2/x^2) + 8*x*log(x*log(2/x^2) - 4 *log(2/x^2))*log(x)*log(2/x^2) - 8*x*log(x)^2*log(2/x^2) + 8*I*pi*x*log(2) + 8*x^2*log(2) - 8*x*log(2)*log(x*log(2/x^2) - 4*log(2/x^2)) - 16*I*pi*x* log(x) - 16*x^2*log(x) + 8*x*log(2)*log(x) + 16*x*log(x*log(2/x^2) - 4*log (2/x^2))*log(x) - 16*x*log(x)^2)
Timed out. \[ \int \frac {24-6 x+\left (12+24 x-6 x^2\right ) \log \left (\frac {2}{x^2}\right )+(12-3 x) \log \left (\frac {2}{x^2}\right ) \log \left (-\frac {x}{(-4+x) \log \left (\frac {2}{x^2}\right )}\right )}{\left (-4 x^4+x^5\right ) \log \left (\frac {2}{x^2}\right )+\left (-8 x^3+2 x^4\right ) \log \left (\frac {2}{x^2}\right ) \log \left (-\frac {x}{(-4+x) \log \left (\frac {2}{x^2}\right )}\right )+\left (-4 x^2+x^3\right ) \log \left (\frac {2}{x^2}\right ) \log ^2\left (-\frac {x}{(-4+x) \log \left (\frac {2}{x^2}\right )}\right )} \, dx=\int \frac {6\,x-\ln \left (\frac {2}{x^2}\right )\,\left (-6\,x^2+24\,x+12\right )+\ln \left (-\frac {x}{\ln \left (\frac {2}{x^2}\right )\,\left (x-4\right )}\right )\,\ln \left (\frac {2}{x^2}\right )\,\left (3\,x-12\right )-24}{\ln \left (\frac {2}{x^2}\right )\,\left (4\,x^2-x^3\right )\,{\ln \left (-\frac {x}{\ln \left (\frac {2}{x^2}\right )\,\left (x-4\right )}\right )}^2+\ln \left (\frac {2}{x^2}\right )\,\left (8\,x^3-2\,x^4\right )\,\ln \left (-\frac {x}{\ln \left (\frac {2}{x^2}\right )\,\left (x-4\right )}\right )+\ln \left (\frac {2}{x^2}\right )\,\left (4\,x^4-x^5\right )} \,d x \]
int((6*x - log(2/x^2)*(24*x - 6*x^2 + 12) + log(-x/(log(2/x^2)*(x - 4)))*l og(2/x^2)*(3*x - 12) - 24)/(log(2/x^2)*(4*x^4 - x^5) + log(-x/(log(2/x^2)* (x - 4)))*log(2/x^2)*(8*x^3 - 2*x^4) + log(-x/(log(2/x^2)*(x - 4)))^2*log( 2/x^2)*(4*x^2 - x^3)),x)