Integrand size = 242, antiderivative size = 27 \[ \int \frac {-4 x+4 x^2+\left (-2 x+2 x^2\right ) \log (x)+\left (-6 x-4 x^2+2 x^3+4 x^4\right ) \log (x) \log \left (\frac {1}{3} x \log ^2(x)\right )+\left (-2 x+4 x^3\right ) \log (x) \log \left (\frac {1}{3} x \log ^2(x)\right ) \log \left (\log \left (\frac {1}{3} x \log ^2(x)\right )\right )+\left (\left (-3-2 x^2+2 x^3\right ) \log (x) \log \left (\frac {1}{3} x \log ^2(x)\right )+\left (-2 x+2 x^2\right ) \log (x) \log \left (\frac {1}{3} x \log ^2(x)\right ) \log \left (\log \left (\frac {1}{3} x \log ^2(x)\right )\right )\right ) \log \left (3+2 x^2-2 x^3+\left (2 x-2 x^2\right ) \log \left (\log \left (\frac {1}{3} x \log ^2(x)\right )\right )\right )}{\left (-3-2 x^2+2 x^3\right ) \log (x) \log \left (\frac {1}{3} x \log ^2(x)\right )+\left (-2 x+2 x^2\right ) \log (x) \log \left (\frac {1}{3} x \log ^2(x)\right ) \log \left (\log \left (\frac {1}{3} x \log ^2(x)\right )\right )} \, dx=x \left (x+\log \left (3+(2-2 x) x \left (x+\log \left (\log \left (\frac {1}{3} x \log ^2(x)\right )\right )\right )\right )\right ) \]
Time = 0.20 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.33 \[ \int \frac {-4 x+4 x^2+\left (-2 x+2 x^2\right ) \log (x)+\left (-6 x-4 x^2+2 x^3+4 x^4\right ) \log (x) \log \left (\frac {1}{3} x \log ^2(x)\right )+\left (-2 x+4 x^3\right ) \log (x) \log \left (\frac {1}{3} x \log ^2(x)\right ) \log \left (\log \left (\frac {1}{3} x \log ^2(x)\right )\right )+\left (\left (-3-2 x^2+2 x^3\right ) \log (x) \log \left (\frac {1}{3} x \log ^2(x)\right )+\left (-2 x+2 x^2\right ) \log (x) \log \left (\frac {1}{3} x \log ^2(x)\right ) \log \left (\log \left (\frac {1}{3} x \log ^2(x)\right )\right )\right ) \log \left (3+2 x^2-2 x^3+\left (2 x-2 x^2\right ) \log \left (\log \left (\frac {1}{3} x \log ^2(x)\right )\right )\right )}{\left (-3-2 x^2+2 x^3\right ) \log (x) \log \left (\frac {1}{3} x \log ^2(x)\right )+\left (-2 x+2 x^2\right ) \log (x) \log \left (\frac {1}{3} x \log ^2(x)\right ) \log \left (\log \left (\frac {1}{3} x \log ^2(x)\right )\right )} \, dx=x^2+x \log \left (3+2 x^2-2 x^3-2 (-1+x) x \log \left (\log \left (\frac {1}{3} x \log ^2(x)\right )\right )\right ) \]
Integrate[(-4*x + 4*x^2 + (-2*x + 2*x^2)*Log[x] + (-6*x - 4*x^2 + 2*x^3 + 4*x^4)*Log[x]*Log[(x*Log[x]^2)/3] + (-2*x + 4*x^3)*Log[x]*Log[(x*Log[x]^2) /3]*Log[Log[(x*Log[x]^2)/3]] + ((-3 - 2*x^2 + 2*x^3)*Log[x]*Log[(x*Log[x]^ 2)/3] + (-2*x + 2*x^2)*Log[x]*Log[(x*Log[x]^2)/3]*Log[Log[(x*Log[x]^2)/3]] )*Log[3 + 2*x^2 - 2*x^3 + (2*x - 2*x^2)*Log[Log[(x*Log[x]^2)/3]]])/((-3 - 2*x^2 + 2*x^3)*Log[x]*Log[(x*Log[x]^2)/3] + (-2*x + 2*x^2)*Log[x]*Log[(x*L og[x]^2)/3]*Log[Log[(x*Log[x]^2)/3]]),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.
| \(\displaystyle \int \frac {\left (4 x^3-2 x\right ) \log (x) \log \left (\frac {1}{3} x \log ^2(x)\right ) \log \left (\log \left (\frac {1}{3} x \log ^2(x)\right )\right )+4 x^2+\left (2 x^2-2 x\right ) \log (x)+\left (\left (2 x^2-2 x\right ) \log (x) \log \left (\log \left (\frac {1}{3} x \log ^2(x)\right )\right ) \log \left (\frac {1}{3} x \log ^2(x)\right )+\left (2 x^3-2 x^2-3\right ) \log (x) \log \left (\frac {1}{3} x \log ^2(x)\right )\right ) \log \left (-2 x^3+2 x^2+\left (2 x-2 x^2\right ) \log \left (\log \left (\frac {1}{3} x \log ^2(x)\right )\right )+3\right )+\left (4 x^4+2 x^3-4 x^2-6 x\right ) \log (x) \log \left (\frac {1}{3} x \log ^2(x)\right )-4 x}{\left (2 x^2-2 x\right ) \log (x) \log \left (\log \left (\frac {1}{3} x \log ^2(x)\right )\right ) \log \left (\frac {1}{3} x \log ^2(x)\right )+\left (2 x^3-2 x^2-3\right ) \log (x) \log \left (\frac {1}{3} x \log ^2(x)\right )} \, dx\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \int \frac {-\left (4 x^3-2 x\right ) \log (x) \log \left (\frac {1}{3} x \log ^2(x)\right ) \log \left (\log \left (\frac {1}{3} x \log ^2(x)\right )\right )-4 x^2-\left (2 x^2-2 x\right ) \log (x)-\left (\left (2 x^2-2 x\right ) \log (x) \log \left (\log \left (\frac {1}{3} x \log ^2(x)\right )\right ) \log \left (\frac {1}{3} x \log ^2(x)\right )+\left (2 x^3-2 x^2-3\right ) \log (x) \log \left (\frac {1}{3} x \log ^2(x)\right )\right ) \log \left (-2 x^3+2 x^2+\left (2 x-2 x^2\right ) \log \left (\log \left (\frac {1}{3} x \log ^2(x)\right )\right )+3\right )-\left (4 x^4+2 x^3-4 x^2-6 x\right ) \log (x) \log \left (\frac {1}{3} x \log ^2(x)\right )+4 x}{\log (x) \log \left (\frac {1}{3} x \log ^2(x)\right ) \left (-2 x^3+2 x^2-2 x^2 \log \left (\log \left (\frac {1}{3} x \log ^2(x)\right )\right )+2 x \log \left (\log \left (\frac {1}{3} x \log ^2(x)\right )\right )+3\right )}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {4 x^2}{\log (x) \log \left (\frac {1}{3} x \log ^2(x)\right ) \left (2 x^3-2 x^2+2 x^2 \log \left (\log \left (\frac {1}{3} x \log ^2(x)\right )\right )-2 x \log \left (\log \left (\frac {1}{3} x \log ^2(x)\right )\right )-3\right )}+\frac {2 \left (2 x^3+x^2-2 x-3\right ) x}{2 x^3-2 x^2+2 x^2 \log \left (\log \left (\frac {1}{3} x \log ^2(x)\right )\right )-2 x \log \left (\log \left (\frac {1}{3} x \log ^2(x)\right )\right )-3}+\frac {2 \left (2 x^2-1\right ) x \log \left (\log \left (\frac {1}{3} x \log ^2(x)\right )\right )}{2 x^3-2 x^2+2 x^2 \log \left (\log \left (\frac {1}{3} x \log ^2(x)\right )\right )-2 x \log \left (\log \left (\frac {1}{3} x \log ^2(x)\right )\right )-3}+\frac {2 (x-1) x}{\log \left (\frac {1}{3} x \log ^2(x)\right ) \left (2 x^3-2 x^2+2 x^2 \log \left (\log \left (\frac {1}{3} x \log ^2(x)\right )\right )-2 x \log \left (\log \left (\frac {1}{3} x \log ^2(x)\right )\right )-3\right )}-\frac {4 x}{\log (x) \log \left (\frac {1}{3} x \log ^2(x)\right ) \left (2 x^3-2 x^2+2 x^2 \log \left (\log \left (\frac {1}{3} x \log ^2(x)\right )\right )-2 x \log \left (\log \left (\frac {1}{3} x \log ^2(x)\right )\right )-3\right )}+\log \left (-2 x^3+2 x^2-2 (x-1) x \log \left (\log \left (\frac {1}{3} x \log ^2(x)\right )\right )+3\right )\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle 6 \int \frac {1}{2 x^3+2 \log \left (\log \left (\frac {1}{3} x \log ^2(x)\right )\right ) x^2-2 x^2-2 \log \left (\log \left (\frac {1}{3} x \log ^2(x)\right )\right ) x-3}dx+3 \int \frac {1}{(x-1) \left (2 x^3+2 \log \left (\log \left (\frac {1}{3} x \log ^2(x)\right )\right ) x^2-2 x^2-2 \log \left (\log \left (\frac {1}{3} x \log ^2(x)\right )\right ) x-3\right )}dx-2 \int \frac {x^2}{2 x^3+2 \log \left (\log \left (\frac {1}{3} x \log ^2(x)\right )\right ) x^2-2 x^2-2 \log \left (\log \left (\frac {1}{3} x \log ^2(x)\right )\right ) x-3}dx+2 \int \frac {x^3}{2 x^3+2 \log \left (\log \left (\frac {1}{3} x \log ^2(x)\right )\right ) x^2-2 x^2-2 \log \left (\log \left (\frac {1}{3} x \log ^2(x)\right )\right ) x-3}dx-2 \int \frac {x}{\log \left (\frac {1}{3} x \log ^2(x)\right ) \left (2 x^3+2 \log \left (\log \left (\frac {1}{3} x \log ^2(x)\right )\right ) x^2-2 x^2-2 \log \left (\log \left (\frac {1}{3} x \log ^2(x)\right )\right ) x-3\right )}dx+2 \int \frac {x^2}{\log \left (\frac {1}{3} x \log ^2(x)\right ) \left (2 x^3+2 \log \left (\log \left (\frac {1}{3} x \log ^2(x)\right )\right ) x^2-2 x^2-2 \log \left (\log \left (\frac {1}{3} x \log ^2(x)\right )\right ) x-3\right )}dx-4 \int \frac {x}{\log (x) \log \left (\frac {1}{3} x \log ^2(x)\right ) \left (2 x^3+2 \log \left (\log \left (\frac {1}{3} x \log ^2(x)\right )\right ) x^2-2 x^2-2 \log \left (\log \left (\frac {1}{3} x \log ^2(x)\right )\right ) x-3\right )}dx+4 \int \frac {x^2}{\log (x) \log \left (\frac {1}{3} x \log ^2(x)\right ) \left (2 x^3+2 \log \left (\log \left (\frac {1}{3} x \log ^2(x)\right )\right ) x^2-2 x^2-2 \log \left (\log \left (\frac {1}{3} x \log ^2(x)\right )\right ) x-3\right )}dx+\int \log \left (-2 x^3+2 x^2-2 (x-1) \log \left (\log \left (\frac {1}{3} x \log ^2(x)\right )\right ) x+3\right )dx+x^2+2 x+\log (1-x)\) |
Int[(-4*x + 4*x^2 + (-2*x + 2*x^2)*Log[x] + (-6*x - 4*x^2 + 2*x^3 + 4*x^4) *Log[x]*Log[(x*Log[x]^2)/3] + (-2*x + 4*x^3)*Log[x]*Log[(x*Log[x]^2)/3]*Lo g[Log[(x*Log[x]^2)/3]] + ((-3 - 2*x^2 + 2*x^3)*Log[x]*Log[(x*Log[x]^2)/3] + (-2*x + 2*x^2)*Log[x]*Log[(x*Log[x]^2)/3]*Log[Log[(x*Log[x]^2)/3]])*Log[ 3 + 2*x^2 - 2*x^3 + (2*x - 2*x^2)*Log[Log[(x*Log[x]^2)/3]]])/((-3 - 2*x^2 + 2*x^3)*Log[x]*Log[(x*Log[x]^2)/3] + (-2*x + 2*x^2)*Log[x]*Log[(x*Log[x]^ 2)/3]*Log[Log[(x*Log[x]^2)/3]]),x]
3.15.56.3.1 Defintions of rubi rules used
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 5.10 (sec) , antiderivative size = 124, normalized size of antiderivative = 4.59
\[x^{2}+x \ln \left (\left (-2 x^{2}+2 x \right ) \ln \left (-\ln \left (3\right )+\ln \left (x \right )+2 \ln \left (\ln \left (x \right )\right )-\frac {i \pi \,\operatorname {csgn}\left (i \ln \left (x \right )^{2}\right ) {\left (-\operatorname {csgn}\left (i \ln \left (x \right )^{2}\right )+\operatorname {csgn}\left (i \ln \left (x \right )\right )\right )}^{2}}{2}-\frac {i \pi \,\operatorname {csgn}\left (i x \ln \left (x \right )^{2}\right ) \left (-\operatorname {csgn}\left (i x \ln \left (x \right )^{2}\right )+\operatorname {csgn}\left (i x \right )\right ) \left (-\operatorname {csgn}\left (i x \ln \left (x \right )^{2}\right )+\operatorname {csgn}\left (i \ln \left (x \right )^{2}\right )\right )}{2}\right )-2 x^{3}+2 x^{2}+3\right )\]
int((((2*x^2-2*x)*ln(x)*ln(1/3*x*ln(x)^2)*ln(ln(1/3*x*ln(x)^2))+(2*x^3-2*x ^2-3)*ln(x)*ln(1/3*x*ln(x)^2))*ln((-2*x^2+2*x)*ln(ln(1/3*x*ln(x)^2))-2*x^3 +2*x^2+3)+(4*x^3-2*x)*ln(x)*ln(1/3*x*ln(x)^2)*ln(ln(1/3*x*ln(x)^2))+(4*x^4 +2*x^3-4*x^2-6*x)*ln(x)*ln(1/3*x*ln(x)^2)+(2*x^2-2*x)*ln(x)+4*x^2-4*x)/((2 *x^2-2*x)*ln(x)*ln(1/3*x*ln(x)^2)*ln(ln(1/3*x*ln(x)^2))+(2*x^3-2*x^2-3)*ln (x)*ln(1/3*x*ln(x)^2)),x)
x^2+x*ln((-2*x^2+2*x)*ln(-ln(3)+ln(x)+2*ln(ln(x))-1/2*I*Pi*csgn(I*ln(x)^2) *(-csgn(I*ln(x)^2)+csgn(I*ln(x)))^2-1/2*I*Pi*csgn(I*x*ln(x)^2)*(-csgn(I*x* ln(x)^2)+csgn(I*x))*(-csgn(I*x*ln(x)^2)+csgn(I*ln(x)^2)))-2*x^3+2*x^2+3)
Time = 0.25 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.37 \[ \int \frac {-4 x+4 x^2+\left (-2 x+2 x^2\right ) \log (x)+\left (-6 x-4 x^2+2 x^3+4 x^4\right ) \log (x) \log \left (\frac {1}{3} x \log ^2(x)\right )+\left (-2 x+4 x^3\right ) \log (x) \log \left (\frac {1}{3} x \log ^2(x)\right ) \log \left (\log \left (\frac {1}{3} x \log ^2(x)\right )\right )+\left (\left (-3-2 x^2+2 x^3\right ) \log (x) \log \left (\frac {1}{3} x \log ^2(x)\right )+\left (-2 x+2 x^2\right ) \log (x) \log \left (\frac {1}{3} x \log ^2(x)\right ) \log \left (\log \left (\frac {1}{3} x \log ^2(x)\right )\right )\right ) \log \left (3+2 x^2-2 x^3+\left (2 x-2 x^2\right ) \log \left (\log \left (\frac {1}{3} x \log ^2(x)\right )\right )\right )}{\left (-3-2 x^2+2 x^3\right ) \log (x) \log \left (\frac {1}{3} x \log ^2(x)\right )+\left (-2 x+2 x^2\right ) \log (x) \log \left (\frac {1}{3} x \log ^2(x)\right ) \log \left (\log \left (\frac {1}{3} x \log ^2(x)\right )\right )} \, dx=x^{2} + x \log \left (-2 \, x^{3} + 2 \, x^{2} - 2 \, {\left (x^{2} - x\right )} \log \left (\log \left (\frac {1}{3} \, x \log \left (x\right )^{2}\right )\right ) + 3\right ) \]
integrate((((2*x^2-2*x)*log(x)*log(1/3*x*log(x)^2)*log(log(1/3*x*log(x)^2) )+(2*x^3-2*x^2-3)*log(x)*log(1/3*x*log(x)^2))*log((-2*x^2+2*x)*log(log(1/3 *x*log(x)^2))-2*x^3+2*x^2+3)+(4*x^3-2*x)*log(x)*log(1/3*x*log(x)^2)*log(lo g(1/3*x*log(x)^2))+(4*x^4+2*x^3-4*x^2-6*x)*log(x)*log(1/3*x*log(x)^2)+(2*x ^2-2*x)*log(x)+4*x^2-4*x)/((2*x^2-2*x)*log(x)*log(1/3*x*log(x)^2)*log(log( 1/3*x*log(x)^2))+(2*x^3-2*x^2-3)*log(x)*log(1/3*x*log(x)^2)),x, algorithm= \
Timed out. \[ \int \frac {-4 x+4 x^2+\left (-2 x+2 x^2\right ) \log (x)+\left (-6 x-4 x^2+2 x^3+4 x^4\right ) \log (x) \log \left (\frac {1}{3} x \log ^2(x)\right )+\left (-2 x+4 x^3\right ) \log (x) \log \left (\frac {1}{3} x \log ^2(x)\right ) \log \left (\log \left (\frac {1}{3} x \log ^2(x)\right )\right )+\left (\left (-3-2 x^2+2 x^3\right ) \log (x) \log \left (\frac {1}{3} x \log ^2(x)\right )+\left (-2 x+2 x^2\right ) \log (x) \log \left (\frac {1}{3} x \log ^2(x)\right ) \log \left (\log \left (\frac {1}{3} x \log ^2(x)\right )\right )\right ) \log \left (3+2 x^2-2 x^3+\left (2 x-2 x^2\right ) \log \left (\log \left (\frac {1}{3} x \log ^2(x)\right )\right )\right )}{\left (-3-2 x^2+2 x^3\right ) \log (x) \log \left (\frac {1}{3} x \log ^2(x)\right )+\left (-2 x+2 x^2\right ) \log (x) \log \left (\frac {1}{3} x \log ^2(x)\right ) \log \left (\log \left (\frac {1}{3} x \log ^2(x)\right )\right )} \, dx=\text {Timed out} \]
integrate((((2*x**2-2*x)*ln(x)*ln(1/3*x*ln(x)**2)*ln(ln(1/3*x*ln(x)**2))+( 2*x**3-2*x**2-3)*ln(x)*ln(1/3*x*ln(x)**2))*ln((-2*x**2+2*x)*ln(ln(1/3*x*ln (x)**2))-2*x**3+2*x**2+3)+(4*x**3-2*x)*ln(x)*ln(1/3*x*ln(x)**2)*ln(ln(1/3* x*ln(x)**2))+(4*x**4+2*x**3-4*x**2-6*x)*ln(x)*ln(1/3*x*ln(x)**2)+(2*x**2-2 *x)*ln(x)+4*x**2-4*x)/((2*x**2-2*x)*ln(x)*ln(1/3*x*ln(x)**2)*ln(ln(1/3*x*l n(x)**2))+(2*x**3-2*x**2-3)*ln(x)*ln(1/3*x*ln(x)**2)),x)
Leaf count of result is larger than twice the leaf count of optimal. 50 vs. \(2 (24) = 48\).
Time = 0.35 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.85 \[ \int \frac {-4 x+4 x^2+\left (-2 x+2 x^2\right ) \log (x)+\left (-6 x-4 x^2+2 x^3+4 x^4\right ) \log (x) \log \left (\frac {1}{3} x \log ^2(x)\right )+\left (-2 x+4 x^3\right ) \log (x) \log \left (\frac {1}{3} x \log ^2(x)\right ) \log \left (\log \left (\frac {1}{3} x \log ^2(x)\right )\right )+\left (\left (-3-2 x^2+2 x^3\right ) \log (x) \log \left (\frac {1}{3} x \log ^2(x)\right )+\left (-2 x+2 x^2\right ) \log (x) \log \left (\frac {1}{3} x \log ^2(x)\right ) \log \left (\log \left (\frac {1}{3} x \log ^2(x)\right )\right )\right ) \log \left (3+2 x^2-2 x^3+\left (2 x-2 x^2\right ) \log \left (\log \left (\frac {1}{3} x \log ^2(x)\right )\right )\right )}{\left (-3-2 x^2+2 x^3\right ) \log (x) \log \left (\frac {1}{3} x \log ^2(x)\right )+\left (-2 x+2 x^2\right ) \log (x) \log \left (\frac {1}{3} x \log ^2(x)\right ) \log \left (\log \left (\frac {1}{3} x \log ^2(x)\right )\right )} \, dx=x^{2} + x \log \left (-2 \, x^{3} - 2 \, x^{2} {\left (\log \left (-\log \left (3\right ) + \log \left (x\right ) + 2 \, \log \left (\log \left (x\right )\right )\right ) - 1\right )} + 2 \, x \log \left (-\log \left (3\right ) + \log \left (x\right ) + 2 \, \log \left (\log \left (x\right )\right )\right ) + 3\right ) \]
integrate((((2*x^2-2*x)*log(x)*log(1/3*x*log(x)^2)*log(log(1/3*x*log(x)^2) )+(2*x^3-2*x^2-3)*log(x)*log(1/3*x*log(x)^2))*log((-2*x^2+2*x)*log(log(1/3 *x*log(x)^2))-2*x^3+2*x^2+3)+(4*x^3-2*x)*log(x)*log(1/3*x*log(x)^2)*log(lo g(1/3*x*log(x)^2))+(4*x^4+2*x^3-4*x^2-6*x)*log(x)*log(1/3*x*log(x)^2)+(2*x ^2-2*x)*log(x)+4*x^2-4*x)/((2*x^2-2*x)*log(x)*log(1/3*x*log(x)^2)*log(log( 1/3*x*log(x)^2))+(2*x^3-2*x^2-3)*log(x)*log(1/3*x*log(x)^2)),x, algorithm= \
x^2 + x*log(-2*x^3 - 2*x^2*(log(-log(3) + log(x) + 2*log(log(x))) - 1) + 2 *x*log(-log(3) + log(x) + 2*log(log(x))) + 3)
Leaf count of result is larger than twice the leaf count of optimal. 53 vs. \(2 (24) = 48\).
Time = 4.00 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.96 \[ \int \frac {-4 x+4 x^2+\left (-2 x+2 x^2\right ) \log (x)+\left (-6 x-4 x^2+2 x^3+4 x^4\right ) \log (x) \log \left (\frac {1}{3} x \log ^2(x)\right )+\left (-2 x+4 x^3\right ) \log (x) \log \left (\frac {1}{3} x \log ^2(x)\right ) \log \left (\log \left (\frac {1}{3} x \log ^2(x)\right )\right )+\left (\left (-3-2 x^2+2 x^3\right ) \log (x) \log \left (\frac {1}{3} x \log ^2(x)\right )+\left (-2 x+2 x^2\right ) \log (x) \log \left (\frac {1}{3} x \log ^2(x)\right ) \log \left (\log \left (\frac {1}{3} x \log ^2(x)\right )\right )\right ) \log \left (3+2 x^2-2 x^3+\left (2 x-2 x^2\right ) \log \left (\log \left (\frac {1}{3} x \log ^2(x)\right )\right )\right )}{\left (-3-2 x^2+2 x^3\right ) \log (x) \log \left (\frac {1}{3} x \log ^2(x)\right )+\left (-2 x+2 x^2\right ) \log (x) \log \left (\frac {1}{3} x \log ^2(x)\right ) \log \left (\log \left (\frac {1}{3} x \log ^2(x)\right )\right )} \, dx=x^{2} + x \log \left (-2 \, x^{3} - 2 \, x^{2} \log \left (-\log \left (3\right ) + \log \left (\log \left (x\right )^{2}\right ) + \log \left (x\right )\right ) + 2 \, x^{2} + 2 \, x \log \left (-\log \left (3\right ) + \log \left (\log \left (x\right )^{2}\right ) + \log \left (x\right )\right ) + 3\right ) \]
integrate((((2*x^2-2*x)*log(x)*log(1/3*x*log(x)^2)*log(log(1/3*x*log(x)^2) )+(2*x^3-2*x^2-3)*log(x)*log(1/3*x*log(x)^2))*log((-2*x^2+2*x)*log(log(1/3 *x*log(x)^2))-2*x^3+2*x^2+3)+(4*x^3-2*x)*log(x)*log(1/3*x*log(x)^2)*log(lo g(1/3*x*log(x)^2))+(4*x^4+2*x^3-4*x^2-6*x)*log(x)*log(1/3*x*log(x)^2)+(2*x ^2-2*x)*log(x)+4*x^2-4*x)/((2*x^2-2*x)*log(x)*log(1/3*x*log(x)^2)*log(log( 1/3*x*log(x)^2))+(2*x^3-2*x^2-3)*log(x)*log(1/3*x*log(x)^2)),x, algorithm= \
x^2 + x*log(-2*x^3 - 2*x^2*log(-log(3) + log(log(x)^2) + log(x)) + 2*x^2 + 2*x*log(-log(3) + log(log(x)^2) + log(x)) + 3)
Time = 13.79 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.33 \[ \int \frac {-4 x+4 x^2+\left (-2 x+2 x^2\right ) \log (x)+\left (-6 x-4 x^2+2 x^3+4 x^4\right ) \log (x) \log \left (\frac {1}{3} x \log ^2(x)\right )+\left (-2 x+4 x^3\right ) \log (x) \log \left (\frac {1}{3} x \log ^2(x)\right ) \log \left (\log \left (\frac {1}{3} x \log ^2(x)\right )\right )+\left (\left (-3-2 x^2+2 x^3\right ) \log (x) \log \left (\frac {1}{3} x \log ^2(x)\right )+\left (-2 x+2 x^2\right ) \log (x) \log \left (\frac {1}{3} x \log ^2(x)\right ) \log \left (\log \left (\frac {1}{3} x \log ^2(x)\right )\right )\right ) \log \left (3+2 x^2-2 x^3+\left (2 x-2 x^2\right ) \log \left (\log \left (\frac {1}{3} x \log ^2(x)\right )\right )\right )}{\left (-3-2 x^2+2 x^3\right ) \log (x) \log \left (\frac {1}{3} x \log ^2(x)\right )+\left (-2 x+2 x^2\right ) \log (x) \log \left (\frac {1}{3} x \log ^2(x)\right ) \log \left (\log \left (\frac {1}{3} x \log ^2(x)\right )\right )} \, dx=x\,\left (x+\ln \left (\ln \left (\ln \left (\frac {x\,{\ln \left (x\right )}^2}{3}\right )\right )\,\left (2\,x-2\,x^2\right )+2\,x^2-2\,x^3+3\right )\right ) \]
int((4*x + log(log(log((x*log(x)^2)/3))*(2*x - 2*x^2) + 2*x^2 - 2*x^3 + 3) *(log((x*log(x)^2)/3)*log(x)*(2*x^2 - 2*x^3 + 3) + log((x*log(x)^2)/3)*log (x)*log(log((x*log(x)^2)/3))*(2*x - 2*x^2)) + log(x)*(2*x - 2*x^2) - 4*x^2 + log((x*log(x)^2)/3)*log(x)*(6*x + 4*x^2 - 2*x^3 - 4*x^4) + log((x*log(x )^2)/3)*log(x)*log(log((x*log(x)^2)/3))*(2*x - 4*x^3))/(log((x*log(x)^2)/3 )*log(x)*(2*x^2 - 2*x^3 + 3) + log((x*log(x)^2)/3)*log(x)*log(log((x*log(x )^2)/3))*(2*x - 2*x^2)),x)