\(\int \frac {-4 x+4 x^2+(-2 x+2 x^2) \log (x)+(-6 x-4 x^2+2 x^3+4 x^4) \log (x) \log (\frac {1}{3} x \log ^2(x))+(-2 x+4 x^3) \log (x) \log (\frac {1}{3} x \log ^2(x)) \log (\log (\frac {1}{3} x \log ^2(x)))+((-3-2 x^2+2 x^3) \log (x) \log (\frac {1}{3} x \log ^2(x))+(-2 x+2 x^2) \log (x) \log (\frac {1}{3} x \log ^2(x)) \log (\log (\frac {1}{3} x \log ^2(x)))) \log (3+2 x^2-2 x^3+(2 x-2 x^2) \log (\log (\frac {1}{3} x \log ^2(x))))}{(-3-2 x^2+2 x^3) \log (x) \log (\frac {1}{3} x \log ^2(x))+(-2 x+2 x^2) \log (x) \log (\frac {1}{3} x \log ^2(x)) \log (\log (\frac {1}{3} x \log ^2(x)))} \, dx\) [7456]

Optimal result

Integrand size = 242, antiderivative size = 27 \[ \int \frac {-4 x+4 x^2+\left (-2 x+2 x^2\right ) \log (x)+\left (-6 x-4 x^2+2 x^3+4 x^4\right ) \log (x) \log \left (\frac {1}{3} x \log ^2(x)\right )+\left (-2 x+4 x^3\right ) \log (x) \log \left (\frac {1}{3} x \log ^2(x)\right ) \log \left (\log \left (\frac {1}{3} x \log ^2(x)\right )\right )+\left (\left (-3-2 x^2+2 x^3\right ) \log (x) \log \left (\frac {1}{3} x \log ^2(x)\right )+\left (-2 x+2 x^2\right ) \log (x) \log \left (\frac {1}{3} x \log ^2(x)\right ) \log \left (\log \left (\frac {1}{3} x \log ^2(x)\right )\right )\right ) \log \left (3+2 x^2-2 x^3+\left (2 x-2 x^2\right ) \log \left (\log \left (\frac {1}{3} x \log ^2(x)\right )\right )\right )}{\left (-3-2 x^2+2 x^3\right ) \log (x) \log \left (\frac {1}{3} x \log ^2(x)\right )+\left (-2 x+2 x^2\right ) \log (x) \log \left (\frac {1}{3} x \log ^2(x)\right ) \log \left (\log \left (\frac {1}{3} x \log ^2(x)\right )\right )} \, dx=x \left (x+\log \left (3+(2-2 x) x \left (x+\log \left (\log \left (\frac {1}{3} x \log ^2(x)\right )\right )\right )\right )\right ) \]

[Out]

x*(ln((ln(ln(1/3*x*ln(x)^2))+x)*x*(2-2*x)+3)+x)

Rubi [F]

\[ \int \frac {-4 x+4 x^2+\left (-2 x+2 x^2\right ) \log (x)+\left (-6 x-4 x^2+2 x^3+4 x^4\right ) \log (x) \log \left (\frac {1}{3} x \log ^2(x)\right )+\left (-2 x+4 x^3\right ) \log (x) \log \left (\frac {1}{3} x \log ^2(x)\right ) \log \left (\log \left (\frac {1}{3} x \log ^2(x)\right )\right )+\left (\left (-3-2 x^2+2 x^3\right ) \log (x) \log \left (\frac {1}{3} x \log ^2(x)\right )+\left (-2 x+2 x^2\right ) \log (x) \log \left (\frac {1}{3} x \log ^2(x)\right ) \log \left (\log \left (\frac {1}{3} x \log ^2(x)\right )\right )\right ) \log \left (3+2 x^2-2 x^3+\left (2 x-2 x^2\right ) \log \left (\log \left (\frac {1}{3} x \log ^2(x)\right )\right )\right )}{\left (-3-2 x^2+2 x^3\right ) \log (x) \log \left (\frac {1}{3} x \log ^2(x)\right )+\left (-2 x+2 x^2\right ) \log (x) \log \left (\frac {1}{3} x \log ^2(x)\right ) \log \left (\log \left (\frac {1}{3} x \log ^2(x)\right )\right )} \, dx=\int \frac {-4 x+4 x^2+\left (-2 x+2 x^2\right ) \log (x)+\left (-6 x-4 x^2+2 x^3+4 x^4\right ) \log (x) \log \left (\frac {1}{3} x \log ^2(x)\right )+\left (-2 x+4 x^3\right ) \log (x) \log \left (\frac {1}{3} x \log ^2(x)\right ) \log \left (\log \left (\frac {1}{3} x \log ^2(x)\right )\right )+\left (\left (-3-2 x^2+2 x^3\right ) \log (x) \log \left (\frac {1}{3} x \log ^2(x)\right )+\left (-2 x+2 x^2\right ) \log (x) \log \left (\frac {1}{3} x \log ^2(x)\right ) \log \left (\log \left (\frac {1}{3} x \log ^2(x)\right )\right )\right ) \log \left (3+2 x^2-2 x^3+\left (2 x-2 x^2\right ) \log \left (\log \left (\frac {1}{3} x \log ^2(x)\right )\right )\right )}{\left (-3-2 x^2+2 x^3\right ) \log (x) \log \left (\frac {1}{3} x \log ^2(x)\right )+\left (-2 x+2 x^2\right ) \log (x) \log \left (\frac {1}{3} x \log ^2(x)\right ) \log \left (\log \left (\frac {1}{3} x \log ^2(x)\right )\right )} \, dx \]

[In]

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]*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*Log
[x]^2)/3]*Log[Log[(x*Log[x]^2)/3]]),x]

[Out]

2*x + x^2 + Log[1 - x] + 6*Defer[Int][(-3 - 2*x^2 + 2*x^3 + 2*(-1 + x)*x*Log[Log[(x*Log[x]^2)/3]])^(-1), x] -
2*Defer[Int][x^2/(-3 - 2*x^2 + 2*x^3 + 2*(-1 + x)*x*Log[Log[(x*Log[x]^2)/3]]), x] + 2*Defer[Int][x^3/(-3 - 2*x
^2 + 2*x^3 + 2*(-1 + x)*x*Log[Log[(x*Log[x]^2)/3]]), x] - 2*Defer[Int][x/(Log[(x*Log[x]^2)/3]*(-3 - 2*x^2 + 2*
x^3 + 2*(-1 + x)*x*Log[Log[(x*Log[x]^2)/3]])), x] + 2*Defer[Int][x^2/(Log[(x*Log[x]^2)/3]*(-3 - 2*x^2 + 2*x^3
+ 2*(-1 + x)*x*Log[Log[(x*Log[x]^2)/3]])), x] - 4*Defer[Int][x/(Log[x]*Log[(x*Log[x]^2)/3]*(-3 - 2*x^2 + 2*x^3
+ 2*(-1 + x)*x*Log[Log[(x*Log[x]^2)/3]])), x] + 4*Defer[Int][x^2/(Log[x]*Log[(x*Log[x]^2)/3]*(-3 - 2*x^2 + 2*
x^3 + 2*(-1 + x)*x*Log[Log[(x*Log[x]^2)/3]])), x] + 3*Defer[Int][1/((-1 + x)*(-3 - 2*x^2 + 2*x^3 - 2*x*Log[Log
[(x*Log[x]^2)/3]] + 2*x^2*Log[Log[(x*Log[x]^2)/3]])), x] + Defer[Int][Log[3 + 2*x^2 - 2*x^3 - 2*(-1 + x)*x*Log
[Log[(x*Log[x]^2)/3]]], x]

Rubi steps \begin{align*} \text {integral}& = \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 )}{\log (x) \log \left (\frac {1}{3} x \log ^2(x)\right ) \left (3+2 x^2-2 x^3+2 x \log \left (\log \left (\frac {1}{3} x \log ^2(x)\right )\right )-2 x^2 \log \left (\log \left (\frac {1}{3} x \log ^2(x)\right )\right )\right )} \, dx \\ & = \int \left (\frac {2 x \left (-3-2 x+x^2+2 x^3\right )}{-3-2 x^2+2 x^3-2 x \log \left (\log \left (\frac {1}{3} x \log ^2(x)\right )\right )+2 x^2 \log \left (\log \left (\frac {1}{3} x \log ^2(x)\right )\right )}+\frac {2 (-1+x) x}{\log \left (\frac {1}{3} x \log ^2(x)\right ) \left (-3-2 x^2+2 x^3-2 x \log \left (\log \left (\frac {1}{3} x \log ^2(x)\right )\right )+2 x^2 \log \left (\log \left (\frac {1}{3} x \log ^2(x)\right )\right )\right )}-\frac {4 x}{\log (x) \log \left (\frac {1}{3} x \log ^2(x)\right ) \left (-3-2 x^2+2 x^3-2 x \log \left (\log \left (\frac {1}{3} x \log ^2(x)\right )\right )+2 x^2 \log \left (\log \left (\frac {1}{3} x \log ^2(x)\right )\right )\right )}+\frac {4 x^2}{\log (x) \log \left (\frac {1}{3} x \log ^2(x)\right ) \left (-3-2 x^2+2 x^3-2 x \log \left (\log \left (\frac {1}{3} x \log ^2(x)\right )\right )+2 x^2 \log \left (\log \left (\frac {1}{3} x \log ^2(x)\right )\right )\right )}+\frac {2 x \left (-1+2 x^2\right ) \log \left (\log \left (\frac {1}{3} x \log ^2(x)\right )\right )}{-3-2 x^2+2 x^3-2 x \log \left (\log \left (\frac {1}{3} x \log ^2(x)\right )\right )+2 x^2 \log \left (\log \left (\frac {1}{3} x \log ^2(x)\right )\right )}+\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 )\right ) \, dx \\ & = 2 \int \frac {x \left (-3-2 x+x^2+2 x^3\right )}{-3-2 x^2+2 x^3-2 x \log \left (\log \left (\frac {1}{3} x \log ^2(x)\right )\right )+2 x^2 \log \left (\log \left (\frac {1}{3} x \log ^2(x)\right )\right )} \, dx+2 \int \frac {(-1+x) x}{\log \left (\frac {1}{3} x \log ^2(x)\right ) \left (-3-2 x^2+2 x^3-2 x \log \left (\log \left (\frac {1}{3} x \log ^2(x)\right )\right )+2 x^2 \log \left (\log \left (\frac {1}{3} x \log ^2(x)\right )\right )\right )} \, dx+2 \int \frac {x \left (-1+2 x^2\right ) \log \left (\log \left (\frac {1}{3} x \log ^2(x)\right )\right )}{-3-2 x^2+2 x^3-2 x \log \left (\log \left (\frac {1}{3} x \log ^2(x)\right )\right )+2 x^2 \log \left (\log \left (\frac {1}{3} x \log ^2(x)\right )\right )} \, dx-4 \int \frac {x}{\log (x) \log \left (\frac {1}{3} x \log ^2(x)\right ) \left (-3-2 x^2+2 x^3-2 x \log \left (\log \left (\frac {1}{3} x \log ^2(x)\right )\right )+2 x^2 \log \left (\log \left (\frac {1}{3} x \log ^2(x)\right )\right )\right )} \, dx+4 \int \frac {x^2}{\log (x) \log \left (\frac {1}{3} x \log ^2(x)\right ) \left (-3-2 x^2+2 x^3-2 x \log \left (\log \left (\frac {1}{3} x \log ^2(x)\right )\right )+2 x^2 \log \left (\log \left (\frac {1}{3} x \log ^2(x)\right )\right )\right )} \, dx+\int \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 ) \, dx \\ & = 2 \int \left (-\frac {3 x}{-3-2 x^2+2 x^3-2 x \log \left (\log \left (\frac {1}{3} x \log ^2(x)\right )\right )+2 x^2 \log \left (\log \left (\frac {1}{3} x \log ^2(x)\right )\right )}-\frac {2 x^2}{-3-2 x^2+2 x^3-2 x \log \left (\log \left (\frac {1}{3} x \log ^2(x)\right )\right )+2 x^2 \log \left (\log \left (\frac {1}{3} x \log ^2(x)\right )\right )}+\frac {x^3}{-3-2 x^2+2 x^3-2 x \log \left (\log \left (\frac {1}{3} x \log ^2(x)\right )\right )+2 x^2 \log \left (\log \left (\frac {1}{3} x \log ^2(x)\right )\right )}+\frac {2 x^4}{-3-2 x^2+2 x^3-2 x \log \left (\log \left (\frac {1}{3} x \log ^2(x)\right )\right )+2 x^2 \log \left (\log \left (\frac {1}{3} x \log ^2(x)\right )\right )}\right ) \, dx+2 \int \left (\frac {-1+2 x^2}{2 (-1+x)}+\frac {-3+4 x^2+2 x^3+4 x^4-4 x^5}{2 (-1+x) \left (-3-2 x^2+2 x^3-2 x \log \left (\log \left (\frac {1}{3} x \log ^2(x)\right )\right )+2 x^2 \log \left (\log \left (\frac {1}{3} x \log ^2(x)\right )\right )\right )}\right ) \, dx+2 \int \left (-\frac {x}{\log \left (\frac {1}{3} x \log ^2(x)\right ) \left (-3-2 x^2+2 x^3-2 x \log \left (\log \left (\frac {1}{3} x \log ^2(x)\right )\right )+2 x^2 \log \left (\log \left (\frac {1}{3} x \log ^2(x)\right )\right )\right )}+\frac {x^2}{\log \left (\frac {1}{3} x \log ^2(x)\right ) \left (-3-2 x^2+2 x^3-2 x \log \left (\log \left (\frac {1}{3} x \log ^2(x)\right )\right )+2 x^2 \log \left (\log \left (\frac {1}{3} x \log ^2(x)\right )\right )\right )}\right ) \, dx-4 \int \frac {x}{\log (x) \log \left (\frac {1}{3} x \log ^2(x)\right ) \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 )} \, dx+4 \int \frac {x^2}{\log (x) \log \left (\frac {1}{3} x \log ^2(x)\right ) \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 )} \, dx+\int \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 ) \, dx \\ & = 2 \int \frac {x^3}{-3-2 x^2+2 x^3-2 x \log \left (\log \left (\frac {1}{3} x \log ^2(x)\right )\right )+2 x^2 \log \left (\log \left (\frac {1}{3} x \log ^2(x)\right )\right )} \, dx-2 \int \frac {x}{\log \left (\frac {1}{3} x \log ^2(x)\right ) \left (-3-2 x^2+2 x^3-2 x \log \left (\log \left (\frac {1}{3} x \log ^2(x)\right )\right )+2 x^2 \log \left (\log \left (\frac {1}{3} x \log ^2(x)\right )\right )\right )} \, dx+2 \int \frac {x^2}{\log \left (\frac {1}{3} x \log ^2(x)\right ) \left (-3-2 x^2+2 x^3-2 x \log \left (\log \left (\frac {1}{3} x \log ^2(x)\right )\right )+2 x^2 \log \left (\log \left (\frac {1}{3} x \log ^2(x)\right )\right )\right )} \, dx-4 \int \frac {x}{\log (x) \log \left (\frac {1}{3} x \log ^2(x)\right ) \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 )} \, dx+4 \int \frac {x^2}{\log (x) \log \left (\frac {1}{3} x \log ^2(x)\right ) \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 )} \, dx-4 \int \frac {x^2}{-3-2 x^2+2 x^3-2 x \log \left (\log \left (\frac {1}{3} x \log ^2(x)\right )\right )+2 x^2 \log \left (\log \left (\frac {1}{3} x \log ^2(x)\right )\right )} \, dx+4 \int \frac {x^4}{-3-2 x^2+2 x^3-2 x \log \left (\log \left (\frac {1}{3} x \log ^2(x)\right )\right )+2 x^2 \log \left (\log \left (\frac {1}{3} x \log ^2(x)\right )\right )} \, dx-6 \int \frac {x}{-3-2 x^2+2 x^3-2 x \log \left (\log \left (\frac {1}{3} x \log ^2(x)\right )\right )+2 x^2 \log \left (\log \left (\frac {1}{3} x \log ^2(x)\right )\right )} \, dx+\int \frac {-1+2 x^2}{-1+x} \, dx+\int \frac {-3+4 x^2+2 x^3+4 x^4-4 x^5}{(-1+x) \left (-3-2 x^2+2 x^3-2 x \log \left (\log \left (\frac {1}{3} x \log ^2(x)\right )\right )+2 x^2 \log \left (\log \left (\frac {1}{3} x \log ^2(x)\right )\right )\right )} \, dx+\int \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 ) \, dx \\ & = 2 \int \frac {x^3}{-3-2 x^2+2 x^3+2 (-1+x) x \log \left (\log \left (\frac {1}{3} x \log ^2(x)\right )\right )} \, dx-2 \int \frac {x}{\log \left (\frac {1}{3} x \log ^2(x)\right ) \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 )} \, dx+2 \int \frac {x^2}{\log \left (\frac {1}{3} x \log ^2(x)\right ) \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 )} \, dx-4 \int \frac {x^2}{-3-2 x^2+2 x^3+2 (-1+x) x \log \left (\log \left (\frac {1}{3} x \log ^2(x)\right )\right )} \, dx+4 \int \frac {x^4}{-3-2 x^2+2 x^3+2 (-1+x) x \log \left (\log \left (\frac {1}{3} x \log ^2(x)\right )\right )} \, dx-4 \int \frac {x}{\log (x) \log \left (\frac {1}{3} x \log ^2(x)\right ) \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 )} \, dx+4 \int \frac {x^2}{\log (x) \log \left (\frac {1}{3} x \log ^2(x)\right ) \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 )} \, dx-6 \int \frac {x}{-3-2 x^2+2 x^3+2 (-1+x) x \log \left (\log \left (\frac {1}{3} x \log ^2(x)\right )\right )} \, dx+\int \left (2+\frac {1}{-1+x}+2 x\right ) \, dx+\int \left (\frac {6}{-3-2 x^2+2 x^3-2 x \log \left (\log \left (\frac {1}{3} x \log ^2(x)\right )\right )+2 x^2 \log \left (\log \left (\frac {1}{3} x \log ^2(x)\right )\right )}+\frac {3}{(-1+x) \left (-3-2 x^2+2 x^3-2 x \log \left (\log \left (\frac {1}{3} x \log ^2(x)\right )\right )+2 x^2 \log \left (\log \left (\frac {1}{3} x \log ^2(x)\right )\right )\right )}+\frac {6 x}{-3-2 x^2+2 x^3-2 x \log \left (\log \left (\frac {1}{3} x \log ^2(x)\right )\right )+2 x^2 \log \left (\log \left (\frac {1}{3} x \log ^2(x)\right )\right )}+\frac {2 x^2}{-3-2 x^2+2 x^3-2 x \log \left (\log \left (\frac {1}{3} x \log ^2(x)\right )\right )+2 x^2 \log \left (\log \left (\frac {1}{3} x \log ^2(x)\right )\right )}-\frac {4 x^4}{-3-2 x^2+2 x^3-2 x \log \left (\log \left (\frac {1}{3} x \log ^2(x)\right )\right )+2 x^2 \log \left (\log \left (\frac {1}{3} x \log ^2(x)\right )\right )}\right ) \, dx+\int \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 ) \, dx \\ & = 2 x+x^2+\log (1-x)+2 \int \frac {x^3}{-3-2 x^2+2 x^3+2 (-1+x) x \log \left (\log \left (\frac {1}{3} x \log ^2(x)\right )\right )} \, dx-2 \int \frac {x}{\log \left (\frac {1}{3} x \log ^2(x)\right ) \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 )} \, dx+2 \int \frac {x^2}{\log \left (\frac {1}{3} x \log ^2(x)\right ) \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 )} \, dx+2 \int \frac {x^2}{-3-2 x^2+2 x^3-2 x \log \left (\log \left (\frac {1}{3} x \log ^2(x)\right )\right )+2 x^2 \log \left (\log \left (\frac {1}{3} x \log ^2(x)\right )\right )} \, dx+3 \int \frac {1}{(-1+x) \left (-3-2 x^2+2 x^3-2 x \log \left (\log \left (\frac {1}{3} x \log ^2(x)\right )\right )+2 x^2 \log \left (\log \left (\frac {1}{3} x \log ^2(x)\right )\right )\right )} \, dx-4 \int \frac {x^2}{-3-2 x^2+2 x^3+2 (-1+x) x \log \left (\log \left (\frac {1}{3} x \log ^2(x)\right )\right )} \, dx+4 \int \frac {x^4}{-3-2 x^2+2 x^3+2 (-1+x) x \log \left (\log \left (\frac {1}{3} x \log ^2(x)\right )\right )} \, dx-4 \int \frac {x}{\log (x) \log \left (\frac {1}{3} x \log ^2(x)\right ) \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 )} \, dx+4 \int \frac {x^2}{\log (x) \log \left (\frac {1}{3} x \log ^2(x)\right ) \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 )} \, dx-4 \int \frac {x^4}{-3-2 x^2+2 x^3-2 x \log \left (\log \left (\frac {1}{3} x \log ^2(x)\right )\right )+2 x^2 \log \left (\log \left (\frac {1}{3} x \log ^2(x)\right )\right )} \, dx-6 \int \frac {x}{-3-2 x^2+2 x^3+2 (-1+x) x \log \left (\log \left (\frac {1}{3} x \log ^2(x)\right )\right )} \, dx+6 \int \frac {1}{-3-2 x^2+2 x^3-2 x \log \left (\log \left (\frac {1}{3} x \log ^2(x)\right )\right )+2 x^2 \log \left (\log \left (\frac {1}{3} x \log ^2(x)\right )\right )} \, dx+6 \int \frac {x}{-3-2 x^2+2 x^3-2 x \log \left (\log \left (\frac {1}{3} x \log ^2(x)\right )\right )+2 x^2 \log \left (\log \left (\frac {1}{3} x \log ^2(x)\right )\right )} \, dx+\int \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 ) \, dx \\ & = 2 x+x^2+\log (1-x)+2 \int \frac {x^2}{-3-2 x^2+2 x^3+2 (-1+x) x \log \left (\log \left (\frac {1}{3} x \log ^2(x)\right )\right )} \, dx+2 \int \frac {x^3}{-3-2 x^2+2 x^3+2 (-1+x) x \log \left (\log \left (\frac {1}{3} x \log ^2(x)\right )\right )} \, dx-2 \int \frac {x}{\log \left (\frac {1}{3} x \log ^2(x)\right ) \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 )} \, dx+2 \int \frac {x^2}{\log \left (\frac {1}{3} x \log ^2(x)\right ) \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 )} \, dx+3 \int \frac {1}{(-1+x) \left (-3-2 x^2+2 x^3-2 x \log \left (\log \left (\frac {1}{3} x \log ^2(x)\right )\right )+2 x^2 \log \left (\log \left (\frac {1}{3} x \log ^2(x)\right )\right )\right )} \, dx-4 \int \frac {x^2}{-3-2 x^2+2 x^3+2 (-1+x) x \log \left (\log \left (\frac {1}{3} x \log ^2(x)\right )\right )} \, dx-4 \int \frac {x}{\log (x) \log \left (\frac {1}{3} x \log ^2(x)\right ) \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 )} \, dx+4 \int \frac {x^2}{\log (x) \log \left (\frac {1}{3} x \log ^2(x)\right ) \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 )} \, dx+6 \int \frac {1}{-3-2 x^2+2 x^3+2 (-1+x) x \log \left (\log \left (\frac {1}{3} x \log ^2(x)\right )\right )} \, dx+\int \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 ) \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 0.20 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.33 \[ \int \frac {-4 x+4 x^2+\left (-2 x+2 x^2\right ) \log (x)+\left (-6 x-4 x^2+2 x^3+4 x^4\right ) \log (x) \log \left (\frac {1}{3} x \log ^2(x)\right )+\left (-2 x+4 x^3\right ) \log (x) \log \left (\frac {1}{3} x \log ^2(x)\right ) \log \left (\log \left (\frac {1}{3} x \log ^2(x)\right )\right )+\left (\left (-3-2 x^2+2 x^3\right ) \log (x) \log \left (\frac {1}{3} x \log ^2(x)\right )+\left (-2 x+2 x^2\right ) \log (x) \log \left (\frac {1}{3} x \log ^2(x)\right ) \log \left (\log \left (\frac {1}{3} x \log ^2(x)\right )\right )\right ) \log \left (3+2 x^2-2 x^3+\left (2 x-2 x^2\right ) \log \left (\log \left (\frac {1}{3} x \log ^2(x)\right )\right )\right )}{\left (-3-2 x^2+2 x^3\right ) \log (x) \log \left (\frac {1}{3} x \log ^2(x)\right )+\left (-2 x+2 x^2\right ) \log (x) \log \left (\frac {1}{3} x \log ^2(x)\right ) \log \left (\log \left (\frac {1}{3} x \log ^2(x)\right )\right )} \, dx=x^2+x \log \left (3+2 x^2-2 x^3-2 (-1+x) x \log \left (\log \left (\frac {1}{3} x \log ^2(x)\right )\right )\right ) \]

[In]

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*Log[x]^2)/3]*Log[Log[(x*Log[x]^2)/3]]),x]

[Out]

x^2 + x*Log[3 + 2*x^2 - 2*x^3 - 2*(-1 + x)*x*Log[Log[(x*Log[x]^2)/3]]]

Maple [C] (warning: unable to verify)

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

\[x^{2}+x \ln \left (\left (-2 x^{2}+2 x \right ) \ln \left (-\ln \left (3\right )+\ln \left (x \right )+2 \ln \left (\ln \left (x \right )\right )-\frac {i \pi \,\operatorname {csgn}\left (i \ln \left (x \right )^{2}\right ) {\left (-\operatorname {csgn}\left (i \ln \left (x \right )^{2}\right )+\operatorname {csgn}\left (i \ln \left (x \right )\right )\right )}^{2}}{2}-\frac {i \pi \,\operatorname {csgn}\left (i x \ln \left (x \right )^{2}\right ) \left (-\operatorname {csgn}\left (i x \ln \left (x \right )^{2}\right )+\operatorname {csgn}\left (i x \right )\right ) \left (-\operatorname {csgn}\left (i x \ln \left (x \right )^{2}\right )+\operatorname {csgn}\left (i \ln \left (x \right )^{2}\right )\right )}{2}\right )-2 x^{3}+2 x^{2}+3\right )\]

[In]

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)

[Out]

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)

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.37 \[ \int \frac {-4 x+4 x^2+\left (-2 x+2 x^2\right ) \log (x)+\left (-6 x-4 x^2+2 x^3+4 x^4\right ) \log (x) \log \left (\frac {1}{3} x \log ^2(x)\right )+\left (-2 x+4 x^3\right ) \log (x) \log \left (\frac {1}{3} x \log ^2(x)\right ) \log \left (\log \left (\frac {1}{3} x \log ^2(x)\right )\right )+\left (\left (-3-2 x^2+2 x^3\right ) \log (x) \log \left (\frac {1}{3} x \log ^2(x)\right )+\left (-2 x+2 x^2\right ) \log (x) \log \left (\frac {1}{3} x \log ^2(x)\right ) \log \left (\log \left (\frac {1}{3} x \log ^2(x)\right )\right )\right ) \log \left (3+2 x^2-2 x^3+\left (2 x-2 x^2\right ) \log \left (\log \left (\frac {1}{3} x \log ^2(x)\right )\right )\right )}{\left (-3-2 x^2+2 x^3\right ) \log (x) \log \left (\frac {1}{3} x \log ^2(x)\right )+\left (-2 x+2 x^2\right ) \log (x) \log \left (\frac {1}{3} x \log ^2(x)\right ) \log \left (\log \left (\frac {1}{3} x \log ^2(x)\right )\right )} \, dx=x^{2} + x \log \left (-2 \, x^{3} + 2 \, x^{2} - 2 \, {\left (x^{2} - x\right )} \log \left (\log \left (\frac {1}{3} \, x \log \left (x\right )^{2}\right )\right ) + 3\right ) \]

[In]

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*l
og(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(
log(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, algori
thm="fricas")

[Out]

x^2 + x*log(-2*x^3 + 2*x^2 - 2*(x^2 - x)*log(log(1/3*x*log(x)^2)) + 3)

Sympy [F(-1)]

Timed out. \[ \int \frac {-4 x+4 x^2+\left (-2 x+2 x^2\right ) \log (x)+\left (-6 x-4 x^2+2 x^3+4 x^4\right ) \log (x) \log \left (\frac {1}{3} x \log ^2(x)\right )+\left (-2 x+4 x^3\right ) \log (x) \log \left (\frac {1}{3} x \log ^2(x)\right ) \log \left (\log \left (\frac {1}{3} x \log ^2(x)\right )\right )+\left (\left (-3-2 x^2+2 x^3\right ) \log (x) \log \left (\frac {1}{3} x \log ^2(x)\right )+\left (-2 x+2 x^2\right ) \log (x) \log \left (\frac {1}{3} x \log ^2(x)\right ) \log \left (\log \left (\frac {1}{3} x \log ^2(x)\right )\right )\right ) \log \left (3+2 x^2-2 x^3+\left (2 x-2 x^2\right ) \log \left (\log \left (\frac {1}{3} x \log ^2(x)\right )\right )\right )}{\left (-3-2 x^2+2 x^3\right ) \log (x) \log \left (\frac {1}{3} x \log ^2(x)\right )+\left (-2 x+2 x^2\right ) \log (x) \log \left (\frac {1}{3} x \log ^2(x)\right ) \log \left (\log \left (\frac {1}{3} x \log ^2(x)\right )\right )} \, dx=\text {Timed out} \]

[In]

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*ln(x)**2))+(2*x**3-2*x**2-3)*ln(x)*ln(1/3*x*ln(x)**2)),x)

[Out]

Timed out

Maxima [B] (verification not implemented)

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 \[ \int \frac {-4 x+4 x^2+\left (-2 x+2 x^2\right ) \log (x)+\left (-6 x-4 x^2+2 x^3+4 x^4\right ) \log (x) \log \left (\frac {1}{3} x \log ^2(x)\right )+\left (-2 x+4 x^3\right ) \log (x) \log \left (\frac {1}{3} x \log ^2(x)\right ) \log \left (\log \left (\frac {1}{3} x \log ^2(x)\right )\right )+\left (\left (-3-2 x^2+2 x^3\right ) \log (x) \log \left (\frac {1}{3} x \log ^2(x)\right )+\left (-2 x+2 x^2\right ) \log (x) \log \left (\frac {1}{3} x \log ^2(x)\right ) \log \left (\log \left (\frac {1}{3} x \log ^2(x)\right )\right )\right ) \log \left (3+2 x^2-2 x^3+\left (2 x-2 x^2\right ) \log \left (\log \left (\frac {1}{3} x \log ^2(x)\right )\right )\right )}{\left (-3-2 x^2+2 x^3\right ) \log (x) \log \left (\frac {1}{3} x \log ^2(x)\right )+\left (-2 x+2 x^2\right ) \log (x) \log \left (\frac {1}{3} x \log ^2(x)\right ) \log \left (\log \left (\frac {1}{3} x \log ^2(x)\right )\right )} \, dx=x^{2} + x \log \left (-2 \, x^{3} - 2 \, x^{2} {\left (\log \left (-\log \left (3\right ) + \log \left (x\right ) + 2 \, \log \left (\log \left (x\right )\right )\right ) - 1\right )} + 2 \, x \log \left (-\log \left (3\right ) + \log \left (x\right ) + 2 \, \log \left (\log \left (x\right )\right )\right ) + 3\right ) \]

[In]

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*l
og(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(
log(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, algori
thm="maxima")

[Out]

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)

Giac [B] (verification not implemented)

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 \[ \int \frac {-4 x+4 x^2+\left (-2 x+2 x^2\right ) \log (x)+\left (-6 x-4 x^2+2 x^3+4 x^4\right ) \log (x) \log \left (\frac {1}{3} x \log ^2(x)\right )+\left (-2 x+4 x^3\right ) \log (x) \log \left (\frac {1}{3} x \log ^2(x)\right ) \log \left (\log \left (\frac {1}{3} x \log ^2(x)\right )\right )+\left (\left (-3-2 x^2+2 x^3\right ) \log (x) \log \left (\frac {1}{3} x \log ^2(x)\right )+\left (-2 x+2 x^2\right ) \log (x) \log \left (\frac {1}{3} x \log ^2(x)\right ) \log \left (\log \left (\frac {1}{3} x \log ^2(x)\right )\right )\right ) \log \left (3+2 x^2-2 x^3+\left (2 x-2 x^2\right ) \log \left (\log \left (\frac {1}{3} x \log ^2(x)\right )\right )\right )}{\left (-3-2 x^2+2 x^3\right ) \log (x) \log \left (\frac {1}{3} x \log ^2(x)\right )+\left (-2 x+2 x^2\right ) \log (x) \log \left (\frac {1}{3} x \log ^2(x)\right ) \log \left (\log \left (\frac {1}{3} x \log ^2(x)\right )\right )} \, dx=x^{2} + x \log \left (-2 \, x^{3} - 2 \, x^{2} \log \left (-\log \left (3\right ) + \log \left (\log \left (x\right )^{2}\right ) + \log \left (x\right )\right ) + 2 \, x^{2} + 2 \, x \log \left (-\log \left (3\right ) + \log \left (\log \left (x\right )^{2}\right ) + \log \left (x\right )\right ) + 3\right ) \]

[In]

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*l
og(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(
log(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, algori
thm="giac")

[Out]

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) + l
og(x)) + 3)

Mupad [B] (verification not implemented)

Time = 13.79 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.33 \[ \int \frac {-4 x+4 x^2+\left (-2 x+2 x^2\right ) \log (x)+\left (-6 x-4 x^2+2 x^3+4 x^4\right ) \log (x) \log \left (\frac {1}{3} x \log ^2(x)\right )+\left (-2 x+4 x^3\right ) \log (x) \log \left (\frac {1}{3} x \log ^2(x)\right ) \log \left (\log \left (\frac {1}{3} x \log ^2(x)\right )\right )+\left (\left (-3-2 x^2+2 x^3\right ) \log (x) \log \left (\frac {1}{3} x \log ^2(x)\right )+\left (-2 x+2 x^2\right ) \log (x) \log \left (\frac {1}{3} x \log ^2(x)\right ) \log \left (\log \left (\frac {1}{3} x \log ^2(x)\right )\right )\right ) \log \left (3+2 x^2-2 x^3+\left (2 x-2 x^2\right ) \log \left (\log \left (\frac {1}{3} x \log ^2(x)\right )\right )\right )}{\left (-3-2 x^2+2 x^3\right ) \log (x) \log \left (\frac {1}{3} x \log ^2(x)\right )+\left (-2 x+2 x^2\right ) \log (x) \log \left (\frac {1}{3} x \log ^2(x)\right ) \log \left (\log \left (\frac {1}{3} x \log ^2(x)\right )\right )} \, dx=x\,\left (x+\ln \left (\ln \left (\ln \left (\frac {x\,{\ln \left (x\right )}^2}{3}\right )\right )\,\left (2\,x-2\,x^2\right )+2\,x^2-2\,x^3+3\right )\right ) \]

[In]

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*l
og(x)^2)/3))*(2*x - 2*x^2)),x)

[Out]

x*(x + log(log(log((x*log(x)^2)/3))*(2*x - 2*x^2) + 2*x^2 - 2*x^3 + 3))