Integrand size = 179, antiderivative size = 35 \[ \int \frac {\left (2 x^2+8 x^3\right ) \log ^3(2) \log \left (\frac {2}{-2+x+2 x^2}\right )+\left (4 x-2 x^2-4 x^3\right ) \log ^3(2) \log ^2\left (\frac {2}{-2+x+2 x^2}\right )}{\left (2-x-2 x^2\right ) \log ^3(2)+\left (-6 x+3 x^2+6 x^3\right ) \log ^2(2) \log \left (\frac {2}{-2+x+2 x^2}\right )+\left (6 x^2-3 x^3-6 x^4\right ) \log (2) \log ^2\left (\frac {2}{-2+x+2 x^2}\right )+\left (-2 x^3+x^4+2 x^5\right ) \log ^3\left (\frac {2}{-2+x+2 x^2}\right )} \, dx=-3+\frac {x^2 \log ^2(2)}{\left (-x+\frac {\log (2)}{\log \left (\frac {2}{x+2 \left (-1+x^2\right )}\right )}\right )^2} \] Output:
ln(2)^2*x^2/(ln(2)/ln(2/(2*x^2+x-2))-x)^2-3
Time = 5.05 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.31 \[ \int \frac {\left (2 x^2+8 x^3\right ) \log ^3(2) \log \left (\frac {2}{-2+x+2 x^2}\right )+\left (4 x-2 x^2-4 x^3\right ) \log ^3(2) \log ^2\left (\frac {2}{-2+x+2 x^2}\right )}{\left (2-x-2 x^2\right ) \log ^3(2)+\left (-6 x+3 x^2+6 x^3\right ) \log ^2(2) \log \left (\frac {2}{-2+x+2 x^2}\right )+\left (6 x^2-3 x^3-6 x^4\right ) \log (2) \log ^2\left (\frac {2}{-2+x+2 x^2}\right )+\left (-2 x^3+x^4+2 x^5\right ) \log ^3\left (\frac {2}{-2+x+2 x^2}\right )} \, dx=-\frac {\log ^3(2) \left (\log (2)-2 x \log \left (\frac {2}{-2+x+2 x^2}\right )\right )}{\left (\log (2)-x \log \left (\frac {2}{-2+x+2 x^2}\right )\right )^2} \] Input:
Integrate[((2*x^2 + 8*x^3)*Log[2]^3*Log[2/(-2 + x + 2*x^2)] + (4*x - 2*x^2 - 4*x^3)*Log[2]^3*Log[2/(-2 + x + 2*x^2)]^2)/((2 - x - 2*x^2)*Log[2]^3 + (-6*x + 3*x^2 + 6*x^3)*Log[2]^2*Log[2/(-2 + x + 2*x^2)] + (6*x^2 - 3*x^3 - 6*x^4)*Log[2]*Log[2/(-2 + x + 2*x^2)]^2 + (-2*x^3 + x^4 + 2*x^5)*Log[2/(- 2 + x + 2*x^2)]^3),x]
Output:
-((Log[2]^3*(Log[2] - 2*x*Log[2/(-2 + x + 2*x^2)]))/(Log[2] - x*Log[2/(-2 + x + 2*x^2)])^2)
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.
| \(\displaystyle \int \frac {\left (8 x^3+2 x^2\right ) \log ^3(2) \log \left (\frac {2}{2 x^2+x-2}\right )+\left (-4 x^3-2 x^2+4 x\right ) \log ^3(2) \log ^2\left (\frac {2}{2 x^2+x-2}\right )}{\left (-2 x^2-x+2\right ) \log ^3(2)+\left (6 x^3+3 x^2-6 x\right ) \log ^2(2) \log \left (\frac {2}{2 x^2+x-2}\right )+\left (-6 x^4-3 x^3+6 x^2\right ) \log (2) \log ^2\left (\frac {2}{2 x^2+x-2}\right )+\left (2 x^5+x^4-2 x^3\right ) \log ^3\left (\frac {2}{2 x^2+x-2}\right )} \, dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {2 x \log ^3(2) \log \left (\frac {2}{2 x^2+x-2}\right ) \left (x (4 x+1)-\left (2 x^2+x-2\right ) \log \left (\frac {2}{2 x^2+x-2}\right )\right )}{\left (-2 x^2-x+2\right ) \left (\log (2)-x \log \left (\frac {2}{2 x^2+x-2}\right )\right )^3}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 2 \log ^3(2) \int \frac {x \log \left (-\frac {2}{-2 x^2-x+2}\right ) \left (x (4 x+1)+\left (-2 x^2-x+2\right ) \log \left (-\frac {2}{-2 x^2-x+2}\right )\right )}{\left (-2 x^2-x+2\right ) \left (\log (2)-x \log \left (-\frac {2}{-2 x^2-x+2}\right )\right )^3}dx\) |
| \(\Big \downarrow \) 7279 |
| \(\displaystyle 2 \log ^3(2) \int \left (\frac {\log (2) \left (4 x^3+(1-\log (4)) x^2-\log (2) x+\log (4)\right )}{x \left (-2 x^2-x+2\right ) \left (\log (2)-x \log \left (\frac {2}{2 x^2+x-2}\right )\right )^3}-\frac {1}{x \left (x \log \left (\frac {2}{2 x^2+x-2}\right )-\log (2)\right )}+\frac {-4 x^3-(1-4 \log (2)) x^2+\log (4) x-\log (16)}{x \left (-2 x^2-x+2\right ) \left (\log (2)-x \log \left (\frac {2}{2 x^2+x-2}\right )\right )^2}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle 2 \log ^3(2) \left (-2 \log (2) \int \frac {1}{\left (\log (2)-x \log \left (\frac {2}{2 x^2+x-2}\right )\right )^3}dx+2 \int \frac {1}{\left (\log (2)-x \log \left (\frac {2}{2 x^2+x-2}\right )\right )^2}dx-\frac {16 \log (2) \int \frac {1}{\left (-4 x+\sqrt {17}-1\right ) \left (x \log \left (\frac {2}{2 x^2+x-2}\right )-\log (2)\right )^3}dx}{\sqrt {17}}-\frac {1}{2} \log (2) \log (4) \int \frac {1}{x \left (x \log \left (\frac {2}{2 x^2+x-2}\right )-\log (2)\right )^3}dx-\frac {1}{17} \left (17-\sqrt {17}\right ) \log (2) \int \frac {1}{\left (4 x-\sqrt {17}+1\right ) \left (x \log \left (\frac {2}{2 x^2+x-2}\right )-\log (2)\right )^3}dx-\frac {1}{17} \left (17+\sqrt {17}\right ) \log (2) \int \frac {1}{\left (4 x+\sqrt {17}+1\right ) \left (x \log \left (\frac {2}{2 x^2+x-2}\right )-\log (2)\right )^3}dx-\frac {16 \log (2) \int \frac {1}{\left (4 x+\sqrt {17}+1\right ) \left (x \log \left (\frac {2}{2 x^2+x-2}\right )-\log (2)\right )^3}dx}{\sqrt {17}}-\frac {16 \int \frac {1}{\left (-4 x+\sqrt {17}-1\right ) \left (x \log \left (\frac {2}{2 x^2+x-2}\right )-\log (2)\right )^2}dx}{\sqrt {17}}-\frac {1}{2} \log (16) \int \frac {1}{x \left (x \log \left (\frac {2}{2 x^2+x-2}\right )-\log (2)\right )^2}dx-\frac {1}{17} \left (17-\sqrt {17}\right ) \int \frac {1}{\left (4 x-\sqrt {17}+1\right ) \left (x \log \left (\frac {2}{2 x^2+x-2}\right )-\log (2)\right )^2}dx-\frac {1}{17} \left (17+\sqrt {17}\right ) \int \frac {1}{\left (4 x+\sqrt {17}+1\right ) \left (x \log \left (\frac {2}{2 x^2+x-2}\right )-\log (2)\right )^2}dx-\frac {16 \int \frac {1}{\left (4 x+\sqrt {17}+1\right ) \left (x \log \left (\frac {2}{2 x^2+x-2}\right )-\log (2)\right )^2}dx}{\sqrt {17}}-\int \frac {1}{x \left (x \log \left (\frac {2}{2 x^2+x-2}\right )-\log (2)\right )}dx\right )\) |
Input:
Int[((2*x^2 + 8*x^3)*Log[2]^3*Log[2/(-2 + x + 2*x^2)] + (4*x - 2*x^2 - 4*x ^3)*Log[2]^3*Log[2/(-2 + x + 2*x^2)]^2)/((2 - x - 2*x^2)*Log[2]^3 + (-6*x + 3*x^2 + 6*x^3)*Log[2]^2*Log[2/(-2 + x + 2*x^2)] + (6*x^2 - 3*x^3 - 6*x^4 )*Log[2]*Log[2/(-2 + x + 2*x^2)]^2 + (-2*x^3 + x^4 + 2*x^5)*Log[2/(-2 + x + 2*x^2)]^3),x]
Output:
$Aborted
Time = 0.84 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.34
| method | result | size |
| risch | \(-\frac {\left (-2 x \ln \left (\frac {2}{2 x^{2}+x -2}\right )+\ln \left (2\right )\right ) \ln \left (2\right )^{3}}{{\left (-x \ln \left (\frac {2}{2 x^{2}+x -2}\right )+\ln \left (2\right )\right )}^{2}}\) | \(47\) |
| norman | \(\frac {2 \ln \left (2\right )^{3} x \ln \left (\frac {2}{2 x^{2}+x -2}\right )-\ln \left (2\right )^{4}}{{\left (-x \ln \left (\frac {2}{2 x^{2}+x -2}\right )+\ln \left (2\right )\right )}^{2}}\) | \(50\) |
| parallelrisch | \(\frac {8 \ln \left (2\right )^{3} x \ln \left (\frac {2}{2 x^{2}+x -2}\right )-4 \ln \left (2\right )^{4}}{4 \ln \left (\frac {2}{2 x^{2}+x -2}\right )^{2} x^{2}-8 \ln \left (2\right ) \ln \left (\frac {2}{2 x^{2}+x -2}\right ) x +4 \ln \left (2\right )^{2}}\) | \(74\) |
Input:
int(((-4*x^3-2*x^2+4*x)*ln(2)^3*ln(2/(2*x^2+x-2))^2+(8*x^3+2*x^2)*ln(2)^3* ln(2/(2*x^2+x-2)))/((2*x^5+x^4-2*x^3)*ln(2/(2*x^2+x-2))^3+(-6*x^4-3*x^3+6* x^2)*ln(2)*ln(2/(2*x^2+x-2))^2+(6*x^3+3*x^2-6*x)*ln(2)^2*ln(2/(2*x^2+x-2)) +(-2*x^2-x+2)*ln(2)^3),x,method=_RETURNVERBOSE)
Output:
-(-2*x*ln(2/(2*x^2+x-2))+ln(2))*ln(2)^3/(-x*ln(2/(2*x^2+x-2))+ln(2))^2
Leaf count of result is larger than twice the leaf count of optimal. 72 vs. \(2 (33) = 66\).
Time = 0.11 (sec) , antiderivative size = 72, normalized size of antiderivative = 2.06 \[ \int \frac {\left (2 x^2+8 x^3\right ) \log ^3(2) \log \left (\frac {2}{-2+x+2 x^2}\right )+\left (4 x-2 x^2-4 x^3\right ) \log ^3(2) \log ^2\left (\frac {2}{-2+x+2 x^2}\right )}{\left (2-x-2 x^2\right ) \log ^3(2)+\left (-6 x+3 x^2+6 x^3\right ) \log ^2(2) \log \left (\frac {2}{-2+x+2 x^2}\right )+\left (6 x^2-3 x^3-6 x^4\right ) \log (2) \log ^2\left (\frac {2}{-2+x+2 x^2}\right )+\left (-2 x^3+x^4+2 x^5\right ) \log ^3\left (\frac {2}{-2+x+2 x^2}\right )} \, dx=\frac {2 \, x \log \left (2\right )^{3} \log \left (\frac {2}{2 \, x^{2} + x - 2}\right ) - \log \left (2\right )^{4}}{x^{2} \log \left (\frac {2}{2 \, x^{2} + x - 2}\right )^{2} - 2 \, x \log \left (2\right ) \log \left (\frac {2}{2 \, x^{2} + x - 2}\right ) + \log \left (2\right )^{2}} \] Input:
integrate(((-4*x^3-2*x^2+4*x)*log(2)^3*log(2/(2*x^2+x-2))^2+(8*x^3+2*x^2)* log(2)^3*log(2/(2*x^2+x-2)))/((2*x^5+x^4-2*x^3)*log(2/(2*x^2+x-2))^3+(-6*x ^4-3*x^3+6*x^2)*log(2)*log(2/(2*x^2+x-2))^2+(6*x^3+3*x^2-6*x)*log(2)^2*log (2/(2*x^2+x-2))+(-2*x^2-x+2)*log(2)^3),x, algorithm="fricas")
Output:
(2*x*log(2)^3*log(2/(2*x^2 + x - 2)) - log(2)^4)/(x^2*log(2/(2*x^2 + x - 2 ))^2 - 2*x*log(2)*log(2/(2*x^2 + x - 2)) + log(2)^2)
Leaf count of result is larger than twice the leaf count of optimal. 65 vs. \(2 (27) = 54\).
Time = 0.15 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.86 \[ \int \frac {\left (2 x^2+8 x^3\right ) \log ^3(2) \log \left (\frac {2}{-2+x+2 x^2}\right )+\left (4 x-2 x^2-4 x^3\right ) \log ^3(2) \log ^2\left (\frac {2}{-2+x+2 x^2}\right )}{\left (2-x-2 x^2\right ) \log ^3(2)+\left (-6 x+3 x^2+6 x^3\right ) \log ^2(2) \log \left (\frac {2}{-2+x+2 x^2}\right )+\left (6 x^2-3 x^3-6 x^4\right ) \log (2) \log ^2\left (\frac {2}{-2+x+2 x^2}\right )+\left (-2 x^3+x^4+2 x^5\right ) \log ^3\left (\frac {2}{-2+x+2 x^2}\right )} \, dx=\frac {2 x \log {\left (2 \right )}^{3} \log {\left (\frac {2}{2 x^{2} + x - 2} \right )} - \log {\left (2 \right )}^{4}}{x^{2} \log {\left (\frac {2}{2 x^{2} + x - 2} \right )}^{2} - 2 x \log {\left (2 \right )} \log {\left (\frac {2}{2 x^{2} + x - 2} \right )} + \log {\left (2 \right )}^{2}} \] Input:
integrate(((-4*x**3-2*x**2+4*x)*ln(2)**3*ln(2/(2*x**2+x-2))**2+(8*x**3+2*x **2)*ln(2)**3*ln(2/(2*x**2+x-2)))/((2*x**5+x**4-2*x**3)*ln(2/(2*x**2+x-2)) **3+(-6*x**4-3*x**3+6*x**2)*ln(2)*ln(2/(2*x**2+x-2))**2+(6*x**3+3*x**2-6*x )*ln(2)**2*ln(2/(2*x**2+x-2))+(-2*x**2-x+2)*ln(2)**3),x)
Output:
(2*x*log(2)**3*log(2/(2*x**2 + x - 2)) - log(2)**4)/(x**2*log(2/(2*x**2 + x - 2))**2 - 2*x*log(2)*log(2/(2*x**2 + x - 2)) + log(2)**2)
Leaf count of result is larger than twice the leaf count of optimal. 91 vs. \(2 (33) = 66\).
Time = 0.22 (sec) , antiderivative size = 91, normalized size of antiderivative = 2.60 \[ \int \frac {\left (2 x^2+8 x^3\right ) \log ^3(2) \log \left (\frac {2}{-2+x+2 x^2}\right )+\left (4 x-2 x^2-4 x^3\right ) \log ^3(2) \log ^2\left (\frac {2}{-2+x+2 x^2}\right )}{\left (2-x-2 x^2\right ) \log ^3(2)+\left (-6 x+3 x^2+6 x^3\right ) \log ^2(2) \log \left (\frac {2}{-2+x+2 x^2}\right )+\left (6 x^2-3 x^3-6 x^4\right ) \log (2) \log ^2\left (\frac {2}{-2+x+2 x^2}\right )+\left (-2 x^3+x^4+2 x^5\right ) \log ^3\left (\frac {2}{-2+x+2 x^2}\right )} \, dx=\frac {2 \, x \log \left (2\right )^{4} - 2 \, x \log \left (2\right )^{3} \log \left (2 \, x^{2} + x - 2\right ) - \log \left (2\right )^{4}}{x^{2} \log \left (2\right )^{2} + x^{2} \log \left (2 \, x^{2} + x - 2\right )^{2} - 2 \, x \log \left (2\right )^{2} + \log \left (2\right )^{2} - 2 \, {\left (x^{2} \log \left (2\right ) - x \log \left (2\right )\right )} \log \left (2 \, x^{2} + x - 2\right )} \] Input:
integrate(((-4*x^3-2*x^2+4*x)*log(2)^3*log(2/(2*x^2+x-2))^2+(8*x^3+2*x^2)* log(2)^3*log(2/(2*x^2+x-2)))/((2*x^5+x^4-2*x^3)*log(2/(2*x^2+x-2))^3+(-6*x ^4-3*x^3+6*x^2)*log(2)*log(2/(2*x^2+x-2))^2+(6*x^3+3*x^2-6*x)*log(2)^2*log (2/(2*x^2+x-2))+(-2*x^2-x+2)*log(2)^3),x, algorithm="maxima")
Output:
(2*x*log(2)^4 - 2*x*log(2)^3*log(2*x^2 + x - 2) - log(2)^4)/(x^2*log(2)^2 + x^2*log(2*x^2 + x - 2)^2 - 2*x*log(2)^2 + log(2)^2 - 2*(x^2*log(2) - x*l og(2))*log(2*x^2 + x - 2))
Result contains complex when optimal does not.
Time = 0.25 (sec) , antiderivative size = 145, normalized size of antiderivative = 4.14 \[ \int \frac {\left (2 x^2+8 x^3\right ) \log ^3(2) \log \left (\frac {2}{-2+x+2 x^2}\right )+\left (4 x-2 x^2-4 x^3\right ) \log ^3(2) \log ^2\left (\frac {2}{-2+x+2 x^2}\right )}{\left (2-x-2 x^2\right ) \log ^3(2)+\left (-6 x+3 x^2+6 x^3\right ) \log ^2(2) \log \left (\frac {2}{-2+x+2 x^2}\right )+\left (6 x^2-3 x^3-6 x^4\right ) \log (2) \log ^2\left (\frac {2}{-2+x+2 x^2}\right )+\left (-2 x^3+x^4+2 x^5\right ) \log ^3\left (\frac {2}{-2+x+2 x^2}\right )} \, dx=\frac {-4 i \, \pi x \log \left (2\right )^{3} - 2 \, x \log \left (2\right )^{4} + 2 \, x \log \left (2\right )^{3} \log \left (2 \, x^{2} + x - 2\right ) + \log \left (2\right )^{4}}{4 \, \pi ^{2} x^{2} - 4 i \, \pi x^{2} \log \left (2\right ) - x^{2} \log \left (2\right )^{2} + 4 i \, \pi x^{2} \log \left (2 \, x^{2} + x - 2\right ) + 2 \, x^{2} \log \left (2\right ) \log \left (2 \, x^{2} + x - 2\right ) - x^{2} \log \left (2 \, x^{2} + x - 2\right )^{2} + 4 i \, \pi x \log \left (2\right ) + 2 \, x \log \left (2\right )^{2} - 2 \, x \log \left (2\right ) \log \left (2 \, x^{2} + x - 2\right ) - \log \left (2\right )^{2}} \] Input:
integrate(((-4*x^3-2*x^2+4*x)*log(2)^3*log(2/(2*x^2+x-2))^2+(8*x^3+2*x^2)* log(2)^3*log(2/(2*x^2+x-2)))/((2*x^5+x^4-2*x^3)*log(2/(2*x^2+x-2))^3+(-6*x ^4-3*x^3+6*x^2)*log(2)*log(2/(2*x^2+x-2))^2+(6*x^3+3*x^2-6*x)*log(2)^2*log (2/(2*x^2+x-2))+(-2*x^2-x+2)*log(2)^3),x, algorithm="giac")
Output:
(-4*I*pi*x*log(2)^3 - 2*x*log(2)^4 + 2*x*log(2)^3*log(2*x^2 + x - 2) + log (2)^4)/(4*pi^2*x^2 - 4*I*pi*x^2*log(2) - x^2*log(2)^2 + 4*I*pi*x^2*log(2*x ^2 + x - 2) + 2*x^2*log(2)*log(2*x^2 + x - 2) - x^2*log(2*x^2 + x - 2)^2 + 4*I*pi*x*log(2) + 2*x*log(2)^2 - 2*x*log(2)*log(2*x^2 + x - 2) - log(2)^2 )
Timed out. \[ \int \frac {\left (2 x^2+8 x^3\right ) \log ^3(2) \log \left (\frac {2}{-2+x+2 x^2}\right )+\left (4 x-2 x^2-4 x^3\right ) \log ^3(2) \log ^2\left (\frac {2}{-2+x+2 x^2}\right )}{\left (2-x-2 x^2\right ) \log ^3(2)+\left (-6 x+3 x^2+6 x^3\right ) \log ^2(2) \log \left (\frac {2}{-2+x+2 x^2}\right )+\left (6 x^2-3 x^3-6 x^4\right ) \log (2) \log ^2\left (\frac {2}{-2+x+2 x^2}\right )+\left (-2 x^3+x^4+2 x^5\right ) \log ^3\left (\frac {2}{-2+x+2 x^2}\right )} \, dx=-\int \frac {\ln \left (\frac {2}{2\,x^2+x-2}\right )\,{\ln \left (2\right )}^3\,\left (8\,x^3+2\,x^2\right )-{\ln \left (\frac {2}{2\,x^2+x-2}\right )}^2\,{\ln \left (2\right )}^3\,\left (4\,x^3+2\,x^2-4\,x\right )}{\left (-2\,x^5-x^4+2\,x^3\right )\,{\ln \left (\frac {2}{2\,x^2+x-2}\right )}^3+\ln \left (2\right )\,\left (6\,x^4+3\,x^3-6\,x^2\right )\,{\ln \left (\frac {2}{2\,x^2+x-2}\right )}^2-{\ln \left (2\right )}^2\,\left (6\,x^3+3\,x^2-6\,x\right )\,\ln \left (\frac {2}{2\,x^2+x-2}\right )+{\ln \left (2\right )}^3\,\left (2\,x^2+x-2\right )} \,d x \] Input:
int(-(log(2/(x + 2*x^2 - 2))*log(2)^3*(2*x^2 + 8*x^3) - log(2/(x + 2*x^2 - 2))^2*log(2)^3*(2*x^2 - 4*x + 4*x^3))/(log(2)^3*(x + 2*x^2 - 2) - log(2/( x + 2*x^2 - 2))^3*(x^4 - 2*x^3 + 2*x^5) - log(2/(x + 2*x^2 - 2))*log(2)^2* (3*x^2 - 6*x + 6*x^3) + log(2/(x + 2*x^2 - 2))^2*log(2)*(3*x^3 - 6*x^2 + 6 *x^4)),x)
Output:
-int((log(2/(x + 2*x^2 - 2))*log(2)^3*(2*x^2 + 8*x^3) - log(2/(x + 2*x^2 - 2))^2*log(2)^3*(2*x^2 - 4*x + 4*x^3))/(log(2)^3*(x + 2*x^2 - 2) - log(2/( x + 2*x^2 - 2))^3*(x^4 - 2*x^3 + 2*x^5) - log(2/(x + 2*x^2 - 2))*log(2)^2* (3*x^2 - 6*x + 6*x^3) + log(2/(x + 2*x^2 - 2))^2*log(2)*(3*x^3 - 6*x^2 + 6 *x^4)), x)
Time = 0.17 (sec) , antiderivative size = 67, normalized size of antiderivative = 1.91 \[ \int \frac {\left (2 x^2+8 x^3\right ) \log ^3(2) \log \left (\frac {2}{-2+x+2 x^2}\right )+\left (4 x-2 x^2-4 x^3\right ) \log ^3(2) \log ^2\left (\frac {2}{-2+x+2 x^2}\right )}{\left (2-x-2 x^2\right ) \log ^3(2)+\left (-6 x+3 x^2+6 x^3\right ) \log ^2(2) \log \left (\frac {2}{-2+x+2 x^2}\right )+\left (6 x^2-3 x^3-6 x^4\right ) \log (2) \log ^2\left (\frac {2}{-2+x+2 x^2}\right )+\left (-2 x^3+x^4+2 x^5\right ) \log ^3\left (\frac {2}{-2+x+2 x^2}\right )} \, dx=\frac {\mathrm {log}\left (\frac {2}{2 x^{2}+x -2}\right )^{2} \mathrm {log}\left (2\right )^{2} x^{2}}{\mathrm {log}\left (\frac {2}{2 x^{2}+x -2}\right )^{2} x^{2}-2 \,\mathrm {log}\left (\frac {2}{2 x^{2}+x -2}\right ) \mathrm {log}\left (2\right ) x +\mathrm {log}\left (2\right )^{2}} \] Input:
int(((-4*x^3-2*x^2+4*x)*log(2)^3*log(2/(2*x^2+x-2))^2+(8*x^3+2*x^2)*log(2) ^3*log(2/(2*x^2+x-2)))/((2*x^5+x^4-2*x^3)*log(2/(2*x^2+x-2))^3+(-6*x^4-3*x ^3+6*x^2)*log(2)*log(2/(2*x^2+x-2))^2+(6*x^3+3*x^2-6*x)*log(2)^2*log(2/(2* x^2+x-2))+(-2*x^2-x+2)*log(2)^3),x)
Output:
(log(2/(2*x**2 + x - 2))**2*log(2)**2*x**2)/(log(2/(2*x**2 + x - 2))**2*x* *2 - 2*log(2/(2*x**2 + x - 2))*log(2)*x + log(2)**2)