\(\int \frac {-36 x+54 x^2-18 x^3+(-36 x+72 x^2-18 x^3) \log (x)+(36 x-54 x^2+18 x^3) \log (x) \log (\frac {(2 x-2 x^2) \log (x)}{-2+x})}{(2-3 x+x^2) \log (x) \log ^3(\frac {(2 x-2 x^2) \log (x)}{-2+x})} \, dx\) [8049]

Optimal result

Integrand size = 101, antiderivative size = 24 \[ \int \frac {-36 x+54 x^2-18 x^3+\left (-36 x+72 x^2-18 x^3\right ) \log (x)+\left (36 x-54 x^2+18 x^3\right ) \log (x) \log \left (\frac {\left (2 x-2 x^2\right ) \log (x)}{-2+x}\right )}{\left (2-3 x+x^2\right ) \log (x) \log ^3\left (\frac {\left (2 x-2 x^2\right ) \log (x)}{-2+x}\right )} \, dx=\frac {9 x^2}{\log ^2\left (\frac {2 \left (x-x^2\right ) \log (x)}{-2+x}\right )} \]

[Out]

9*x^2/ln(2*ln(x)/(-2+x)*(-x^2+x))^2

Rubi [F]

\[ \int \frac {-36 x+54 x^2-18 x^3+\left (-36 x+72 x^2-18 x^3\right ) \log (x)+\left (36 x-54 x^2+18 x^3\right ) \log (x) \log \left (\frac {\left (2 x-2 x^2\right ) \log (x)}{-2+x}\right )}{\left (2-3 x+x^2\right ) \log (x) \log ^3\left (\frac {\left (2 x-2 x^2\right ) \log (x)}{-2+x}\right )} \, dx=\int \frac {-36 x+54 x^2-18 x^3+\left (-36 x+72 x^2-18 x^3\right ) \log (x)+\left (36 x-54 x^2+18 x^3\right ) \log (x) \log \left (\frac {\left (2 x-2 x^2\right ) \log (x)}{-2+x}\right )}{\left (2-3 x+x^2\right ) \log (x) \log ^3\left (\frac {\left (2 x-2 x^2\right ) \log (x)}{-2+x}\right )} \, dx \]

[In]

Int[(-36*x + 54*x^2 - 18*x^3 + (-36*x + 72*x^2 - 18*x^3)*Log[x] + (36*x - 54*x^2 + 18*x^3)*Log[x]*Log[((2*x -
2*x^2)*Log[x])/(-2 + x)])/((2 - 3*x + x^2)*Log[x]*Log[((2*x - 2*x^2)*Log[x])/(-2 + x)]^3),x]

[Out]

18*Defer[Int][Log[(-2*(-1 + x)*x*Log[x])/(-2 + x)]^(-3), x] + 72*Defer[Int][1/((-2 + x)*Log[(-2*(-1 + x)*x*Log
[x])/(-2 + x)]^3), x] - 18*Defer[Int][1/((-1 + x)*Log[(-2*(-1 + x)*x*Log[x])/(-2 + x)]^3), x] - 18*Defer[Int][
x/Log[(-2*(-1 + x)*x*Log[x])/(-2 + x)]^3, x] - 18*Defer[Int][x/(Log[x]*Log[(-2*(-1 + x)*x*Log[x])/(-2 + x)]^3)
, x] + 18*Defer[Int][x/Log[(-2*(-1 + x)*x*Log[x])/(-2 + x)]^2, x]

Rubi steps \begin{align*} \text {integral}& = \int \frac {18 x \left (-2+3 x-x^2+\log (x) \left (-2+4 x-x^2+\left (2-3 x+x^2\right ) \log \left (-\frac {2 (-1+x) x \log (x)}{-2+x}\right )\right )\right )}{\left (2-3 x+x^2\right ) \log (x) \log ^3\left (-\frac {2 (-1+x) x \log (x)}{-2+x}\right )} \, dx \\ & = 18 \int \frac {x \left (-2+3 x-x^2+\log (x) \left (-2+4 x-x^2+\left (2-3 x+x^2\right ) \log \left (-\frac {2 (-1+x) x \log (x)}{-2+x}\right )\right )\right )}{\left (2-3 x+x^2\right ) \log (x) \log ^3\left (-\frac {2 (-1+x) x \log (x)}{-2+x}\right )} \, dx \\ & = 18 \int \left (-\frac {x \left (2-3 x+x^2+2 \log (x)-4 x \log (x)+x^2 \log (x)\right )}{(-2+x) (-1+x) \log (x) \log ^3\left (-\frac {2 (-1+x) x \log (x)}{-2+x}\right )}+\frac {x}{\log ^2\left (-\frac {2 (-1+x) x \log (x)}{-2+x}\right )}\right ) \, dx \\ & = -\left (18 \int \frac {x \left (2-3 x+x^2+2 \log (x)-4 x \log (x)+x^2 \log (x)\right )}{(-2+x) (-1+x) \log (x) \log ^3\left (-\frac {2 (-1+x) x \log (x)}{-2+x}\right )} \, dx\right )+18 \int \frac {x}{\log ^2\left (-\frac {2 (-1+x) x \log (x)}{-2+x}\right )} \, dx \\ & = -\left (18 \int \left (\frac {-2+3 x-x^2-2 \log (x)+4 x \log (x)-x^2 \log (x)}{(-1+x) \log (x) \log ^3\left (-\frac {2 (-1+x) x \log (x)}{-2+x}\right )}+\frac {2 \left (2-3 x+x^2+2 \log (x)-4 x \log (x)+x^2 \log (x)\right )}{(-2+x) \log (x) \log ^3\left (-\frac {2 (-1+x) x \log (x)}{-2+x}\right )}\right ) \, dx\right )+18 \int \frac {x}{\log ^2\left (-\frac {2 (-1+x) x \log (x)}{-2+x}\right )} \, dx \\ & = -\left (18 \int \frac {-2+3 x-x^2-2 \log (x)+4 x \log (x)-x^2 \log (x)}{(-1+x) \log (x) \log ^3\left (-\frac {2 (-1+x) x \log (x)}{-2+x}\right )} \, dx\right )+18 \int \frac {x}{\log ^2\left (-\frac {2 (-1+x) x \log (x)}{-2+x}\right )} \, dx-36 \int \frac {2-3 x+x^2+2 \log (x)-4 x \log (x)+x^2 \log (x)}{(-2+x) \log (x) \log ^3\left (-\frac {2 (-1+x) x \log (x)}{-2+x}\right )} \, dx \\ & = -\left (18 \int \left (-\frac {2}{(-1+x) \log ^3\left (-\frac {2 (-1+x) x \log (x)}{-2+x}\right )}+\frac {4 x}{(-1+x) \log ^3\left (-\frac {2 (-1+x) x \log (x)}{-2+x}\right )}-\frac {x^2}{(-1+x) \log ^3\left (-\frac {2 (-1+x) x \log (x)}{-2+x}\right )}-\frac {2}{(-1+x) \log (x) \log ^3\left (-\frac {2 (-1+x) x \log (x)}{-2+x}\right )}+\frac {3 x}{(-1+x) \log (x) \log ^3\left (-\frac {2 (-1+x) x \log (x)}{-2+x}\right )}-\frac {x^2}{(-1+x) \log (x) \log ^3\left (-\frac {2 (-1+x) x \log (x)}{-2+x}\right )}\right ) \, dx\right )+18 \int \frac {x}{\log ^2\left (-\frac {2 (-1+x) x \log (x)}{-2+x}\right )} \, dx-36 \int \left (\frac {2}{(-2+x) \log ^3\left (-\frac {2 (-1+x) x \log (x)}{-2+x}\right )}-\frac {4 x}{(-2+x) \log ^3\left (-\frac {2 (-1+x) x \log (x)}{-2+x}\right )}+\frac {x^2}{(-2+x) \log ^3\left (-\frac {2 (-1+x) x \log (x)}{-2+x}\right )}+\frac {2}{(-2+x) \log (x) \log ^3\left (-\frac {2 (-1+x) x \log (x)}{-2+x}\right )}-\frac {3 x}{(-2+x) \log (x) \log ^3\left (-\frac {2 (-1+x) x \log (x)}{-2+x}\right )}+\frac {x^2}{(-2+x) \log (x) \log ^3\left (-\frac {2 (-1+x) x \log (x)}{-2+x}\right )}\right ) \, dx \\ & = 18 \int \frac {x^2}{(-1+x) \log ^3\left (-\frac {2 (-1+x) x \log (x)}{-2+x}\right )} \, dx+18 \int \frac {x^2}{(-1+x) \log (x) \log ^3\left (-\frac {2 (-1+x) x \log (x)}{-2+x}\right )} \, dx+18 \int \frac {x}{\log ^2\left (-\frac {2 (-1+x) x \log (x)}{-2+x}\right )} \, dx+36 \int \frac {1}{(-1+x) \log ^3\left (-\frac {2 (-1+x) x \log (x)}{-2+x}\right )} \, dx-36 \int \frac {x^2}{(-2+x) \log ^3\left (-\frac {2 (-1+x) x \log (x)}{-2+x}\right )} \, dx+36 \int \frac {1}{(-1+x) \log (x) \log ^3\left (-\frac {2 (-1+x) x \log (x)}{-2+x}\right )} \, dx-36 \int \frac {x^2}{(-2+x) \log (x) \log ^3\left (-\frac {2 (-1+x) x \log (x)}{-2+x}\right )} \, dx-54 \int \frac {x}{(-1+x) \log (x) \log ^3\left (-\frac {2 (-1+x) x \log (x)}{-2+x}\right )} \, dx-72 \int \frac {1}{(-2+x) \log ^3\left (-\frac {2 (-1+x) x \log (x)}{-2+x}\right )} \, dx-72 \int \frac {x}{(-1+x) \log ^3\left (-\frac {2 (-1+x) x \log (x)}{-2+x}\right )} \, dx-72 \int \frac {1}{(-2+x) \log (x) \log ^3\left (-\frac {2 (-1+x) x \log (x)}{-2+x}\right )} \, dx+108 \int \frac {x}{(-2+x) \log (x) \log ^3\left (-\frac {2 (-1+x) x \log (x)}{-2+x}\right )} \, dx+144 \int \frac {x}{(-2+x) \log ^3\left (-\frac {2 (-1+x) x \log (x)}{-2+x}\right )} \, dx \\ & = 18 \int \left (\frac {1}{\log ^3\left (-\frac {2 (-1+x) x \log (x)}{-2+x}\right )}+\frac {1}{(-1+x) \log ^3\left (-\frac {2 (-1+x) x \log (x)}{-2+x}\right )}+\frac {x}{\log ^3\left (-\frac {2 (-1+x) x \log (x)}{-2+x}\right )}\right ) \, dx+18 \int \left (\frac {1}{\log (x) \log ^3\left (-\frac {2 (-1+x) x \log (x)}{-2+x}\right )}+\frac {1}{(-1+x) \log (x) \log ^3\left (-\frac {2 (-1+x) x \log (x)}{-2+x}\right )}+\frac {x}{\log (x) \log ^3\left (-\frac {2 (-1+x) x \log (x)}{-2+x}\right )}\right ) \, dx+18 \int \frac {x}{\log ^2\left (-\frac {2 (-1+x) x \log (x)}{-2+x}\right )} \, dx-36 \int \left (\frac {2}{\log ^3\left (-\frac {2 (-1+x) x \log (x)}{-2+x}\right )}+\frac {4}{(-2+x) \log ^3\left (-\frac {2 (-1+x) x \log (x)}{-2+x}\right )}+\frac {x}{\log ^3\left (-\frac {2 (-1+x) x \log (x)}{-2+x}\right )}\right ) \, dx-36 \int \left (\frac {2}{\log (x) \log ^3\left (-\frac {2 (-1+x) x \log (x)}{-2+x}\right )}+\frac {4}{(-2+x) \log (x) \log ^3\left (-\frac {2 (-1+x) x \log (x)}{-2+x}\right )}+\frac {x}{\log (x) \log ^3\left (-\frac {2 (-1+x) x \log (x)}{-2+x}\right )}\right ) \, dx+36 \int \frac {1}{(-1+x) \log ^3\left (-\frac {2 (-1+x) x \log (x)}{-2+x}\right )} \, dx+36 \int \frac {1}{(-1+x) \log (x) \log ^3\left (-\frac {2 (-1+x) x \log (x)}{-2+x}\right )} \, dx-54 \int \left (\frac {1}{\log (x) \log ^3\left (-\frac {2 (-1+x) x \log (x)}{-2+x}\right )}+\frac {1}{(-1+x) \log (x) \log ^3\left (-\frac {2 (-1+x) x \log (x)}{-2+x}\right )}\right ) \, dx-72 \int \left (\frac {1}{\log ^3\left (-\frac {2 (-1+x) x \log (x)}{-2+x}\right )}+\frac {1}{(-1+x) \log ^3\left (-\frac {2 (-1+x) x \log (x)}{-2+x}\right )}\right ) \, dx-72 \int \frac {1}{(-2+x) \log ^3\left (-\frac {2 (-1+x) x \log (x)}{-2+x}\right )} \, dx-72 \int \frac {1}{(-2+x) \log (x) \log ^3\left (-\frac {2 (-1+x) x \log (x)}{-2+x}\right )} \, dx+108 \int \left (\frac {1}{\log (x) \log ^3\left (-\frac {2 (-1+x) x \log (x)}{-2+x}\right )}+\frac {2}{(-2+x) \log (x) \log ^3\left (-\frac {2 (-1+x) x \log (x)}{-2+x}\right )}\right ) \, dx+144 \int \left (\frac {1}{\log ^3\left (-\frac {2 (-1+x) x \log (x)}{-2+x}\right )}+\frac {2}{(-2+x) \log ^3\left (-\frac {2 (-1+x) x \log (x)}{-2+x}\right )}\right ) \, dx \\ & = 18 \int \frac {1}{\log ^3\left (-\frac {2 (-1+x) x \log (x)}{-2+x}\right )} \, dx+18 \int \frac {1}{(-1+x) \log ^3\left (-\frac {2 (-1+x) x \log (x)}{-2+x}\right )} \, dx+18 \int \frac {x}{\log ^3\left (-\frac {2 (-1+x) x \log (x)}{-2+x}\right )} \, dx+18 \int \frac {1}{\log (x) \log ^3\left (-\frac {2 (-1+x) x \log (x)}{-2+x}\right )} \, dx+18 \int \frac {1}{(-1+x) \log (x) \log ^3\left (-\frac {2 (-1+x) x \log (x)}{-2+x}\right )} \, dx+18 \int \frac {x}{\log (x) \log ^3\left (-\frac {2 (-1+x) x \log (x)}{-2+x}\right )} \, dx+18 \int \frac {x}{\log ^2\left (-\frac {2 (-1+x) x \log (x)}{-2+x}\right )} \, dx+36 \int \frac {1}{(-1+x) \log ^3\left (-\frac {2 (-1+x) x \log (x)}{-2+x}\right )} \, dx-36 \int \frac {x}{\log ^3\left (-\frac {2 (-1+x) x \log (x)}{-2+x}\right )} \, dx+36 \int \frac {1}{(-1+x) \log (x) \log ^3\left (-\frac {2 (-1+x) x \log (x)}{-2+x}\right )} \, dx-36 \int \frac {x}{\log (x) \log ^3\left (-\frac {2 (-1+x) x \log (x)}{-2+x}\right )} \, dx-54 \int \frac {1}{\log (x) \log ^3\left (-\frac {2 (-1+x) x \log (x)}{-2+x}\right )} \, dx-54 \int \frac {1}{(-1+x) \log (x) \log ^3\left (-\frac {2 (-1+x) x \log (x)}{-2+x}\right )} \, dx-2 \left (72 \int \frac {1}{\log ^3\left (-\frac {2 (-1+x) x \log (x)}{-2+x}\right )} \, dx\right )-72 \int \frac {1}{(-2+x) \log ^3\left (-\frac {2 (-1+x) x \log (x)}{-2+x}\right )} \, dx-72 \int \frac {1}{(-1+x) \log ^3\left (-\frac {2 (-1+x) x \log (x)}{-2+x}\right )} \, dx-72 \int \frac {1}{\log (x) \log ^3\left (-\frac {2 (-1+x) x \log (x)}{-2+x}\right )} \, dx-72 \int \frac {1}{(-2+x) \log (x) \log ^3\left (-\frac {2 (-1+x) x \log (x)}{-2+x}\right )} \, dx+108 \int \frac {1}{\log (x) \log ^3\left (-\frac {2 (-1+x) x \log (x)}{-2+x}\right )} \, dx+144 \int \frac {1}{\log ^3\left (-\frac {2 (-1+x) x \log (x)}{-2+x}\right )} \, dx-144 \int \frac {1}{(-2+x) \log ^3\left (-\frac {2 (-1+x) x \log (x)}{-2+x}\right )} \, dx-144 \int \frac {1}{(-2+x) \log (x) \log ^3\left (-\frac {2 (-1+x) x \log (x)}{-2+x}\right )} \, dx+216 \int \frac {1}{(-2+x) \log (x) \log ^3\left (-\frac {2 (-1+x) x \log (x)}{-2+x}\right )} \, dx+288 \int \frac {1}{(-2+x) \log ^3\left (-\frac {2 (-1+x) x \log (x)}{-2+x}\right )} \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 0.36 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.88 \[ \int \frac {-36 x+54 x^2-18 x^3+\left (-36 x+72 x^2-18 x^3\right ) \log (x)+\left (36 x-54 x^2+18 x^3\right ) \log (x) \log \left (\frac {\left (2 x-2 x^2\right ) \log (x)}{-2+x}\right )}{\left (2-3 x+x^2\right ) \log (x) \log ^3\left (\frac {\left (2 x-2 x^2\right ) \log (x)}{-2+x}\right )} \, dx=\frac {9 x^2}{\log ^2\left (-\frac {2 (-1+x) x \log (x)}{-2+x}\right )} \]

[In]

Integrate[(-36*x + 54*x^2 - 18*x^3 + (-36*x + 72*x^2 - 18*x^3)*Log[x] + (36*x - 54*x^2 + 18*x^3)*Log[x]*Log[((
2*x - 2*x^2)*Log[x])/(-2 + x)])/((2 - 3*x + x^2)*Log[x]*Log[((2*x - 2*x^2)*Log[x])/(-2 + x)]^3),x]

[Out]

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

Maple [A] (verified)

Time = 24.99 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.08

method	result	size
parallelrisch	\(\frac {9 x^{2}}{{\ln \left (\frac {\left (-2 x^{2}+2 x \right ) \ln \left (x \right )}{-2+x}\right )}^{2}}\)	\(26\)
risch	\(-\frac {36 x^{2}}{\left (\pi \,\operatorname {csgn}\left (i \ln \left (x \right )\right ) \operatorname {csgn}\left (\frac {i \left (-1+x \right )}{-2+x}\right ) \operatorname {csgn}\left (\frac {i \ln \left (x \right ) \left (-1+x \right )}{-2+x}\right )-\pi \,\operatorname {csgn}\left (i \ln \left (x \right )\right ) \operatorname {csgn}\left (\frac {i \ln \left (x \right ) \left (-1+x \right )}{-2+x}\right )^{2}+2 \pi \operatorname {csgn}\left (\frac {i \ln \left (x \right ) \left (-1+x \right ) x}{-2+x}\right )^{2}+\pi \,\operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (\frac {i \ln \left (x \right ) \left (-1+x \right )}{-2+x}\right ) \operatorname {csgn}\left (\frac {i \ln \left (x \right ) \left (-1+x \right ) x}{-2+x}\right )-\pi \,\operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (\frac {i \ln \left (x \right ) \left (-1+x \right ) x}{-2+x}\right )^{2}+\pi \operatorname {csgn}\left (\frac {i \left (-1+x \right )}{-2+x}\right )^{3}-\pi \,\operatorname {csgn}\left (\frac {i}{-2+x}\right ) \operatorname {csgn}\left (\frac {i \left (-1+x \right )}{-2+x}\right )^{2}-\pi \,\operatorname {csgn}\left (i \left (-1+x \right )\right ) \operatorname {csgn}\left (\frac {i \left (-1+x \right )}{-2+x}\right )^{2}-\pi \,\operatorname {csgn}\left (\frac {i \left (-1+x \right )}{-2+x}\right ) \operatorname {csgn}\left (\frac {i \ln \left (x \right ) \left (-1+x \right )}{-2+x}\right )^{2}+\pi \,\operatorname {csgn}\left (i \left (-1+x \right )\right ) \operatorname {csgn}\left (\frac {i}{-2+x}\right ) \operatorname {csgn}\left (\frac {i \left (-1+x \right )}{-2+x}\right )+\pi \operatorname {csgn}\left (\frac {i \ln \left (x \right ) \left (-1+x \right )}{-2+x}\right )^{3}-\pi \,\operatorname {csgn}\left (\frac {i \ln \left (x \right ) \left (-1+x \right )}{-2+x}\right ) \operatorname {csgn}\left (\frac {i \ln \left (x \right ) \left (-1+x \right ) x}{-2+x}\right )^{2}-\pi \operatorname {csgn}\left (\frac {i \ln \left (x \right ) \left (-1+x \right ) x}{-2+x}\right )^{3}-2 \pi +2 i \ln \left (\ln \left (x \right )\right )+2 i \ln \left (x \right )+2 i \ln \left (2\right )+2 i \ln \left (-1+x \right )-2 i \ln \left (-2+x \right )\right )^{2}}\)	\(381\)
default	\(\text {Expression too large to display}\)	\(13908\)

[In]

int(((18*x^3-54*x^2+36*x)*ln(x)*ln((-2*x^2+2*x)*ln(x)/(-2+x))+(-18*x^3+72*x^2-36*x)*ln(x)-18*x^3+54*x^2-36*x)/
(x^2-3*x+2)/ln(x)/ln((-2*x^2+2*x)*ln(x)/(-2+x))^3,x,method=_RETURNVERBOSE)

[Out]

9*x^2/ln((-2*x^2+2*x)*ln(x)/(-2+x))^2

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00 \[ \int \frac {-36 x+54 x^2-18 x^3+\left (-36 x+72 x^2-18 x^3\right ) \log (x)+\left (36 x-54 x^2+18 x^3\right ) \log (x) \log \left (\frac {\left (2 x-2 x^2\right ) \log (x)}{-2+x}\right )}{\left (2-3 x+x^2\right ) \log (x) \log ^3\left (\frac {\left (2 x-2 x^2\right ) \log (x)}{-2+x}\right )} \, dx=\frac {9 \, x^{2}}{\log \left (-\frac {2 \, {\left (x^{2} - x\right )} \log \left (x\right )}{x - 2}\right )^{2}} \]

[In]

integrate(((18*x^3-54*x^2+36*x)*log(x)*log((-2*x^2+2*x)*log(x)/(-2+x))+(-18*x^3+72*x^2-36*x)*log(x)-18*x^3+54*
x^2-36*x)/(x^2-3*x+2)/log(x)/log((-2*x^2+2*x)*log(x)/(-2+x))^3,x, algorithm="fricas")

[Out]

9*x^2/log(-2*(x^2 - x)*log(x)/(x - 2))^2

Sympy [A] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.92 \[ \int \frac {-36 x+54 x^2-18 x^3+\left (-36 x+72 x^2-18 x^3\right ) \log (x)+\left (36 x-54 x^2+18 x^3\right ) \log (x) \log \left (\frac {\left (2 x-2 x^2\right ) \log (x)}{-2+x}\right )}{\left (2-3 x+x^2\right ) \log (x) \log ^3\left (\frac {\left (2 x-2 x^2\right ) \log (x)}{-2+x}\right )} \, dx=\frac {9 x^{2}}{\log {\left (\frac {\left (- 2 x^{2} + 2 x\right ) \log {\left (x \right )}}{x - 2} \right )}^{2}} \]

[In]

integrate(((18*x**3-54*x**2+36*x)*ln(x)*ln((-2*x**2+2*x)*ln(x)/(-2+x))+(-18*x**3+72*x**2-36*x)*ln(x)-18*x**3+5
4*x**2-36*x)/(x**2-3*x+2)/ln(x)/ln((-2*x**2+2*x)*ln(x)/(-2+x))**3,x)

[Out]

9*x**2/log((-2*x**2 + 2*x)*log(x)/(x - 2))**2

Maxima [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.43 (sec) , antiderivative size = 124, normalized size of antiderivative = 5.17 \[ \int \frac {-36 x+54 x^2-18 x^3+\left (-36 x+72 x^2-18 x^3\right ) \log (x)+\left (36 x-54 x^2+18 x^3\right ) \log (x) \log \left (\frac {\left (2 x-2 x^2\right ) \log (x)}{-2+x}\right )}{\left (2-3 x+x^2\right ) \log (x) \log ^3\left (\frac {\left (2 x-2 x^2\right ) \log (x)}{-2+x}\right )} \, dx=-\frac {9 \, x^{2}}{\pi ^{2} - 2 i \, \pi \log \left (2\right ) - \log \left (2\right )^{2} + 2 \, {\left (-i \, \pi - \log \left (2\right ) + \log \left (x - 2\right ) - \log \left (x\right ) - \log \left (\log \left (x\right )\right )\right )} \log \left (x - 1\right ) - \log \left (x - 1\right )^{2} + 2 \, {\left (i \, \pi + \log \left (2\right ) + \log \left (x\right ) + \log \left (\log \left (x\right )\right )\right )} \log \left (x - 2\right ) - \log \left (x - 2\right )^{2} + 2 \, {\left (-i \, \pi - \log \left (2\right )\right )} \log \left (x\right ) - \log \left (x\right )^{2} + 2 \, {\left (-i \, \pi - \log \left (2\right ) - \log \left (x\right )\right )} \log \left (\log \left (x\right )\right ) - \log \left (\log \left (x\right )\right )^{2}} \]

[In]

integrate(((18*x^3-54*x^2+36*x)*log(x)*log((-2*x^2+2*x)*log(x)/(-2+x))+(-18*x^3+72*x^2-36*x)*log(x)-18*x^3+54*
x^2-36*x)/(x^2-3*x+2)/log(x)/log((-2*x^2+2*x)*log(x)/(-2+x))^3,x, algorithm="maxima")

[Out]

-9*x^2/(pi^2 - 2*I*pi*log(2) - log(2)^2 + 2*(-I*pi - log(2) + log(x - 2) - log(x) - log(log(x)))*log(x - 1) -
log(x - 1)^2 + 2*(I*pi + log(2) + log(x) + log(log(x)))*log(x - 2) - log(x - 2)^2 + 2*(-I*pi - log(2))*log(x)
- log(x)^2 + 2*(-I*pi - log(2) - log(x))*log(log(x)) - log(log(x))^2)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 525 vs. \(2 (24) = 48\).

Time = 0.38 (sec) , antiderivative size = 525, normalized size of antiderivative = 21.88 \[ \int \frac {-36 x+54 x^2-18 x^3+\left (-36 x+72 x^2-18 x^3\right ) \log (x)+\left (36 x-54 x^2+18 x^3\right ) \log (x) \log \left (\frac {\left (2 x-2 x^2\right ) \log (x)}{-2+x}\right )}{\left (2-3 x+x^2\right ) \log (x) \log ^3\left (\frac {\left (2 x-2 x^2\right ) \log (x)}{-2+x}\right )} \, dx=\frac {9 \, {\left (x^{4} \log \left (x\right ) + x^{4} - 4 \, x^{3} \log \left (x\right ) - 3 \, x^{3} + 2 \, x^{2} \log \left (x\right ) + 2 \, x^{2}\right )}}{x^{2} \log \left (-2 \, x \log \left (x\right ) + 2 \, \log \left (x\right )\right )^{2} \log \left (x\right ) - 2 \, x^{2} \log \left (-2 \, x \log \left (x\right ) + 2 \, \log \left (x\right )\right ) \log \left (x - 2\right ) \log \left (x\right ) + x^{2} \log \left (x - 2\right )^{2} \log \left (x\right ) + 2 \, x^{2} \log \left (-2 \, x \log \left (x\right ) + 2 \, \log \left (x\right )\right ) \log \left (x\right )^{2} - 2 \, x^{2} \log \left (x - 2\right ) \log \left (x\right )^{2} + x^{2} \log \left (x\right )^{3} + x^{2} \log \left (-2 \, x \log \left (x\right ) + 2 \, \log \left (x\right )\right )^{2} - 2 \, x^{2} \log \left (-2 \, x \log \left (x\right ) + 2 \, \log \left (x\right )\right ) \log \left (x - 2\right ) + x^{2} \log \left (x - 2\right )^{2} + 2 \, x^{2} \log \left (-2 \, x \log \left (x\right ) + 2 \, \log \left (x\right )\right ) \log \left (x\right ) - 4 \, x \log \left (-2 \, x \log \left (x\right ) + 2 \, \log \left (x\right )\right )^{2} \log \left (x\right ) - 2 \, x^{2} \log \left (x - 2\right ) \log \left (x\right ) + 8 \, x \log \left (-2 \, x \log \left (x\right ) + 2 \, \log \left (x\right )\right ) \log \left (x - 2\right ) \log \left (x\right ) - 4 \, x \log \left (x - 2\right )^{2} \log \left (x\right ) + x^{2} \log \left (x\right )^{2} - 8 \, x \log \left (-2 \, x \log \left (x\right ) + 2 \, \log \left (x\right )\right ) \log \left (x\right )^{2} + 8 \, x \log \left (x - 2\right ) \log \left (x\right )^{2} - 4 \, x \log \left (x\right )^{3} - 3 \, x \log \left (-2 \, x \log \left (x\right ) + 2 \, \log \left (x\right )\right )^{2} + 6 \, x \log \left (-2 \, x \log \left (x\right ) + 2 \, \log \left (x\right )\right ) \log \left (x - 2\right ) - 3 \, x \log \left (x - 2\right )^{2} - 6 \, x \log \left (-2 \, x \log \left (x\right ) + 2 \, \log \left (x\right )\right ) \log \left (x\right ) + 2 \, \log \left (-2 \, x \log \left (x\right ) + 2 \, \log \left (x\right )\right )^{2} \log \left (x\right ) + 6 \, x \log \left (x - 2\right ) \log \left (x\right ) - 4 \, \log \left (-2 \, x \log \left (x\right ) + 2 \, \log \left (x\right )\right ) \log \left (x - 2\right ) \log \left (x\right ) + 2 \, \log \left (x - 2\right )^{2} \log \left (x\right ) - 3 \, x \log \left (x\right )^{2} + 4 \, \log \left (-2 \, x \log \left (x\right ) + 2 \, \log \left (x\right )\right ) \log \left (x\right )^{2} - 4 \, \log \left (x - 2\right ) \log \left (x\right )^{2} + 2 \, \log \left (x\right )^{3} + 2 \, \log \left (-2 \, x \log \left (x\right ) + 2 \, \log \left (x\right )\right )^{2} - 4 \, \log \left (-2 \, x \log \left (x\right ) + 2 \, \log \left (x\right )\right ) \log \left (x - 2\right ) + 2 \, \log \left (x - 2\right )^{2} + 4 \, \log \left (-2 \, x \log \left (x\right ) + 2 \, \log \left (x\right )\right ) \log \left (x\right ) - 4 \, \log \left (x - 2\right ) \log \left (x\right ) + 2 \, \log \left (x\right )^{2}} \]

[In]

integrate(((18*x^3-54*x^2+36*x)*log(x)*log((-2*x^2+2*x)*log(x)/(-2+x))+(-18*x^3+72*x^2-36*x)*log(x)-18*x^3+54*
x^2-36*x)/(x^2-3*x+2)/log(x)/log((-2*x^2+2*x)*log(x)/(-2+x))^3,x, algorithm="giac")

[Out]

9*(x^4*log(x) + x^4 - 4*x^3*log(x) - 3*x^3 + 2*x^2*log(x) + 2*x^2)/(x^2*log(-2*x*log(x) + 2*log(x))^2*log(x) -
2*x^2*log(-2*x*log(x) + 2*log(x))*log(x - 2)*log(x) + x^2*log(x - 2)^2*log(x) + 2*x^2*log(-2*x*log(x) + 2*log
(x))*log(x)^2 - 2*x^2*log(x - 2)*log(x)^2 + x^2*log(x)^3 + x^2*log(-2*x*log(x) + 2*log(x))^2 - 2*x^2*log(-2*x*
log(x) + 2*log(x))*log(x - 2) + x^2*log(x - 2)^2 + 2*x^2*log(-2*x*log(x) + 2*log(x))*log(x) - 4*x*log(-2*x*log
(x) + 2*log(x))^2*log(x) - 2*x^2*log(x - 2)*log(x) + 8*x*log(-2*x*log(x) + 2*log(x))*log(x - 2)*log(x) - 4*x*l
og(x - 2)^2*log(x) + x^2*log(x)^2 - 8*x*log(-2*x*log(x) + 2*log(x))*log(x)^2 + 8*x*log(x - 2)*log(x)^2 - 4*x*l
og(x)^3 - 3*x*log(-2*x*log(x) + 2*log(x))^2 + 6*x*log(-2*x*log(x) + 2*log(x))*log(x - 2) - 3*x*log(x - 2)^2 -
6*x*log(-2*x*log(x) + 2*log(x))*log(x) + 2*log(-2*x*log(x) + 2*log(x))^2*log(x) + 6*x*log(x - 2)*log(x) - 4*lo
g(-2*x*log(x) + 2*log(x))*log(x - 2)*log(x) + 2*log(x - 2)^2*log(x) - 3*x*log(x)^2 + 4*log(-2*x*log(x) + 2*log
(x))*log(x)^2 - 4*log(x - 2)*log(x)^2 + 2*log(x)^3 + 2*log(-2*x*log(x) + 2*log(x))^2 - 4*log(-2*x*log(x) + 2*l
og(x))*log(x - 2) + 2*log(x - 2)^2 + 4*log(-2*x*log(x) + 2*log(x))*log(x) - 4*log(x - 2)*log(x) + 2*log(x)^2)

Mupad [B] (verification not implemented)

Time = 14.01 (sec) , antiderivative size = 843, normalized size of antiderivative = 35.12 \[ \int \frac {-36 x+54 x^2-18 x^3+\left (-36 x+72 x^2-18 x^3\right ) \log (x)+\left (36 x-54 x^2+18 x^3\right ) \log (x) \log \left (\frac {\left (2 x-2 x^2\right ) \log (x)}{-2+x}\right )}{\left (2-3 x+x^2\right ) \log (x) \log ^3\left (\frac {\left (2 x-2 x^2\right ) \log (x)}{-2+x}\right )} \, dx=27\,x+\frac {\frac {9\,x^2\,\ln \left (x\right )\,\left (x^2-3\,x+2\right )}{2\,\ln \left (x\right )-3\,x+x^2\,\ln \left (x\right )-4\,x\,\ln \left (x\right )+x^2+2}-\frac {9\,x\,\ln \left (\frac {\ln \left (x\right )\,\left (2\,x-2\,x^2\right )}{x-2}\right )\,\ln \left (x\right )\,\left (x^2-3\,x+2\right )\,\left (2\,x^5\,{\ln \left (x\right )}^2+2\,x^5\,\ln \left (x\right )+x^5-15\,x^4\,{\ln \left (x\right )}^2-12\,x^4\,\ln \left (x\right )-6\,x^4+32\,x^3\,{\ln \left (x\right )}^2+26\,x^3\,\ln \left (x\right )+13\,x^3-26\,x^2\,{\ln \left (x\right )}^2-24\,x^2\,\ln \left (x\right )-12\,x^2+8\,x\,{\ln \left (x\right )}^2+8\,x\,\ln \left (x\right )+4\,x\right )}{{\left (2\,\ln \left (x\right )-3\,x+x^2\,\ln \left (x\right )-4\,x\,\ln \left (x\right )+x^2+2\right )}^3}}{\ln \left (\frac {\ln \left (x\right )\,\left (2\,x-2\,x^2\right )}{x-2}\right )}+\frac {9\,x^2-\frac {9\,x^2\,\ln \left (\frac {\ln \left (x\right )\,\left (2\,x-2\,x^2\right )}{x-2}\right )\,\ln \left (x\right )\,\left (x^2-3\,x+2\right )}{2\,\ln \left (x\right )-3\,x+x^2\,\ln \left (x\right )-4\,x\,\ln \left (x\right )+x^2+2}}{{\ln \left (\frac {\ln \left (x\right )\,\left (2\,x-2\,x^2\right )}{x-2}\right )}^2}+\frac {162\,x^5-1944\,x^4+6336\,x^3-7452\,x^2+3672\,x-648}{x^6-12\,x^5+54\,x^4-112\,x^3+108\,x^2-48\,x+8}+18\,x^2+\frac {9\,\left (3\,x^{17}-79\,x^{16}+904\,x^{15}-5986\,x^{14}+25783\,x^{13}-76855\,x^{12}+164354\,x^{11}-257452\,x^{10}+298436\,x^9-256268\,x^8+161528\,x^7-72992\,x^6+22528\,x^5-4288\,x^4+384\,x^3\right )}{{\left (x^2-4\,x+2\right )}^3\,\left ({\ln \left (x\right )}^2\,{\left (x^2-4\,x+2\right )}^2+{\left (x^2-3\,x+2\right )}^2+2\,\ln \left (x\right )\,\left (x^2-3\,x+2\right )\,\left (x^2-4\,x+2\right )\right )\,\left (x^5-9\,x^4+20\,x^3-14\,x^2+4\,x\right )}-\frac {9\,\left (x^{19}-30\,x^{18}+399\,x^{17}-3136\,x^{16}+16379\,x^{15}-60526\,x^{14}+164269\,x^{13}-335036\,x^{12}+520484\,x^{11}-619816\,x^{10}+565812\,x^9-393296\,x^8+204976\,x^7-77824\,x^6+20416\,x^5-3328\,x^4+256\,x^3\right )}{{\left (x^2-4\,x+2\right )}^3\,\left ({\ln \left (x\right )}^3\,{\left (x^2-4\,x+2\right )}^3+{\left (x^2-3\,x+2\right )}^3+3\,{\ln \left (x\right )}^2\,\left (x^2-3\,x+2\right )\,{\left (x^2-4\,x+2\right )}^2+3\,\ln \left (x\right )\,{\left (x^2-3\,x+2\right )}^2\,\left (x^2-4\,x+2\right )\right )\,\left (x^5-9\,x^4+20\,x^3-14\,x^2+4\,x\right )}-\frac {9\,\left (4\,x^{15}-91\,x^{14}+877\,x^{13}-4759\,x^{12}+16327\,x^{11}-37610\,x^{10}+60124\,x^9-67772\,x^8+53948\,x^7-29864\,x^6+11024\,x^5-2464\,x^4+256\,x^3\right )}{{\left (x^2-4\,x+2\right )}^3\,\left (\ln \left (x\right )\,\left (x^2-4\,x+2\right )-3\,x+x^2+2\right )\,\left (x^5-9\,x^4+20\,x^3-14\,x^2+4\,x\right )} \]

[In]

int(-(36*x - 54*x^2 + 18*x^3 + log(x)*(36*x - 72*x^2 + 18*x^3) - log((log(x)*(2*x - 2*x^2))/(x - 2))*log(x)*(3
6*x - 54*x^2 + 18*x^3))/(log((log(x)*(2*x - 2*x^2))/(x - 2))^3*log(x)*(x^2 - 3*x + 2)),x)

[Out]

27*x + ((9*x^2*log(x)*(x^2 - 3*x + 2))/(2*log(x) - 3*x + x^2*log(x) - 4*x*log(x) + x^2 + 2) - (9*x*log((log(x)
*(2*x - 2*x^2))/(x - 2))*log(x)*(x^2 - 3*x + 2)*(4*x + 8*x*log(x)^2 - 24*x^2*log(x) + 26*x^3*log(x) - 12*x^4*l
og(x) + 2*x^5*log(x) - 26*x^2*log(x)^2 + 32*x^3*log(x)^2 - 15*x^4*log(x)^2 + 2*x^5*log(x)^2 + 8*x*log(x) - 12*
x^2 + 13*x^3 - 6*x^4 + x^5))/(2*log(x) - 3*x + x^2*log(x) - 4*x*log(x) + x^2 + 2)^3)/log((log(x)*(2*x - 2*x^2)
)/(x - 2)) + (9*x^2 - (9*x^2*log((log(x)*(2*x - 2*x^2))/(x - 2))*log(x)*(x^2 - 3*x + 2))/(2*log(x) - 3*x + x^2
*log(x) - 4*x*log(x) + x^2 + 2))/log((log(x)*(2*x - 2*x^2))/(x - 2))^2 + (3672*x - 7452*x^2 + 6336*x^3 - 1944*
x^4 + 162*x^5 - 648)/(108*x^2 - 48*x - 112*x^3 + 54*x^4 - 12*x^5 + x^6 + 8) + 18*x^2 + (9*(384*x^3 - 4288*x^4
+ 22528*x^5 - 72992*x^6 + 161528*x^7 - 256268*x^8 + 298436*x^9 - 257452*x^10 + 164354*x^11 - 76855*x^12 + 2578
3*x^13 - 5986*x^14 + 904*x^15 - 79*x^16 + 3*x^17))/((x^2 - 4*x + 2)^3*(log(x)^2*(x^2 - 4*x + 2)^2 + (x^2 - 3*x
+ 2)^2 + 2*log(x)*(x^2 - 3*x + 2)*(x^2 - 4*x + 2))*(4*x - 14*x^2 + 20*x^3 - 9*x^4 + x^5)) - (9*(256*x^3 - 332
8*x^4 + 20416*x^5 - 77824*x^6 + 204976*x^7 - 393296*x^8 + 565812*x^9 - 619816*x^10 + 520484*x^11 - 335036*x^12
+ 164269*x^13 - 60526*x^14 + 16379*x^15 - 3136*x^16 + 399*x^17 - 30*x^18 + x^19))/((x^2 - 4*x + 2)^3*(log(x)^
3*(x^2 - 4*x + 2)^3 + (x^2 - 3*x + 2)^3 + 3*log(x)^2*(x^2 - 3*x + 2)*(x^2 - 4*x + 2)^2 + 3*log(x)*(x^2 - 3*x +
2)^2*(x^2 - 4*x + 2))*(4*x - 14*x^2 + 20*x^3 - 9*x^4 + x^5)) - (9*(256*x^3 - 2464*x^4 + 11024*x^5 - 29864*x^6
+ 53948*x^7 - 67772*x^8 + 60124*x^9 - 37610*x^10 + 16327*x^11 - 4759*x^12 + 877*x^13 - 91*x^14 + 4*x^15))/((x
^2 - 4*x + 2)^3*(log(x)*(x^2 - 4*x + 2) - 3*x + x^2 + 2)*(4*x - 14*x^2 + 20*x^3 - 9*x^4 + x^5))