\(\int \frac {-24-4 x-4 x \log (x^2)+(-12 x+4 x^2+(-12+4 x) \log (x^2)) \log (\frac {6-2 x}{x+\log (x^2)})+(12 x-4 x^2+(12-4 x) \log (x^2)) \log ^2(\frac {6-2 x}{x+\log (x^2)})}{(-3 x+x^2+(-3+x) \log (x^2)) \log ^2(\frac {6-2 x}{x+\log (x^2)})} \, dx\) [690]

Optimal result

Integrand size = 119, antiderivative size = 25 \[ \int \frac {-24-4 x-4 x \log \left (x^2\right )+\left (-12 x+4 x^2+(-12+4 x) \log \left (x^2\right )\right ) \log \left (\frac {6-2 x}{x+\log \left (x^2\right )}\right )+\left (12 x-4 x^2+(12-4 x) \log \left (x^2\right )\right ) \log ^2\left (\frac {6-2 x}{x+\log \left (x^2\right )}\right )}{\left (-3 x+x^2+(-3+x) \log \left (x^2\right )\right ) \log ^2\left (\frac {6-2 x}{x+\log \left (x^2\right )}\right )} \, dx=-4 x+\frac {4 x}{\log \left (\frac {2 (3-x)}{x+\log \left (x^2\right )}\right )} \]

[Out]

4*x/ln(2*(-x+3)/(ln(x^2)+x))-4*x

Rubi [F]

\[ \int \frac {-24-4 x-4 x \log \left (x^2\right )+\left (-12 x+4 x^2+(-12+4 x) \log \left (x^2\right )\right ) \log \left (\frac {6-2 x}{x+\log \left (x^2\right )}\right )+\left (12 x-4 x^2+(12-4 x) \log \left (x^2\right )\right ) \log ^2\left (\frac {6-2 x}{x+\log \left (x^2\right )}\right )}{\left (-3 x+x^2+(-3+x) \log \left (x^2\right )\right ) \log ^2\left (\frac {6-2 x}{x+\log \left (x^2\right )}\right )} \, dx=\int \frac {-24-4 x-4 x \log \left (x^2\right )+\left (-12 x+4 x^2+(-12+4 x) \log \left (x^2\right )\right ) \log \left (\frac {6-2 x}{x+\log \left (x^2\right )}\right )+\left (12 x-4 x^2+(12-4 x) \log \left (x^2\right )\right ) \log ^2\left (\frac {6-2 x}{x+\log \left (x^2\right )}\right )}{\left (-3 x+x^2+(-3+x) \log \left (x^2\right )\right ) \log ^2\left (\frac {6-2 x}{x+\log \left (x^2\right )}\right )} \, dx \]

[In]

Int[(-24 - 4*x - 4*x*Log[x^2] + (-12*x + 4*x^2 + (-12 + 4*x)*Log[x^2])*Log[(6 - 2*x)/(x + Log[x^2])] + (12*x -
4*x^2 + (12 - 4*x)*Log[x^2])*Log[(6 - 2*x)/(x + Log[x^2])]^2)/((-3*x + x^2 + (-3 + x)*Log[x^2])*Log[(6 - 2*x)
/(x + Log[x^2])]^2),x]

[Out]

-4*x - 4*Defer[Int][1/((x + Log[x^2])*Log[(6 - 2*x)/(x + Log[x^2])]^2), x] - 36*Defer[Int][1/((-3 + x)*(x + Lo
g[x^2])*Log[(6 - 2*x)/(x + Log[x^2])]^2), x] - 4*Defer[Int][Log[x^2]/((x + Log[x^2])*Log[(6 - 2*x)/(x + Log[x^
2])]^2), x] - 12*Defer[Int][Log[x^2]/((-3 + x)*(x + Log[x^2])*Log[(6 - 2*x)/(x + Log[x^2])]^2), x] + 4*Defer[I
nt][Log[(6 - 2*x)/(x + Log[x^2])]^(-1), x]

Rubi steps \begin{align*} \text {integral}& = \int \frac {24+4 x+4 x \log \left (x^2\right )-\left (-12 x+4 x^2+(-12+4 x) \log \left (x^2\right )\right ) \log \left (\frac {6-2 x}{x+\log \left (x^2\right )}\right )-\left (12 x-4 x^2+(12-4 x) \log \left (x^2\right )\right ) \log ^2\left (\frac {6-2 x}{x+\log \left (x^2\right )}\right )}{(3-x) \left (x+\log \left (x^2\right )\right ) \log ^2\left (\frac {6-2 x}{x+\log \left (x^2\right )}\right )} \, dx \\ & = \int \frac {4 \left (6+x+x \log \left (x^2\right )-(-3+x) \left (x+\log \left (x^2\right )\right ) \log \left (\frac {6-2 x}{x+\log \left (x^2\right )}\right )+(-3+x) \left (x+\log \left (x^2\right )\right ) \log ^2\left (\frac {6-2 x}{x+\log \left (x^2\right )}\right )\right )}{(3-x) \left (x+\log \left (x^2\right )\right ) \log ^2\left (\frac {6-2 x}{x+\log \left (x^2\right )}\right )} \, dx \\ & = 4 \int \frac {6+x+x \log \left (x^2\right )-(-3+x) \left (x+\log \left (x^2\right )\right ) \log \left (\frac {6-2 x}{x+\log \left (x^2\right )}\right )+(-3+x) \left (x+\log \left (x^2\right )\right ) \log ^2\left (\frac {6-2 x}{x+\log \left (x^2\right )}\right )}{(3-x) \left (x+\log \left (x^2\right )\right ) \log ^2\left (\frac {6-2 x}{x+\log \left (x^2\right )}\right )} \, dx \\ & = 4 \int \left (-1+\frac {-6-x-x \log \left (x^2\right )}{(-3+x) \left (x+\log \left (x^2\right )\right ) \log ^2\left (\frac {6-2 x}{x+\log \left (x^2\right )}\right )}+\frac {1}{\log \left (\frac {6-2 x}{x+\log \left (x^2\right )}\right )}\right ) \, dx \\ & = -4 x+4 \int \frac {-6-x-x \log \left (x^2\right )}{(-3+x) \left (x+\log \left (x^2\right )\right ) \log ^2\left (\frac {6-2 x}{x+\log \left (x^2\right )}\right )} \, dx+4 \int \frac {1}{\log \left (\frac {6-2 x}{x+\log \left (x^2\right )}\right )} \, dx \\ & = -4 x+4 \int \left (-\frac {6}{(-3+x) \left (x+\log \left (x^2\right )\right ) \log ^2\left (\frac {6-2 x}{x+\log \left (x^2\right )}\right )}-\frac {x}{(-3+x) \left (x+\log \left (x^2\right )\right ) \log ^2\left (\frac {6-2 x}{x+\log \left (x^2\right )}\right )}-\frac {x \log \left (x^2\right )}{(-3+x) \left (x+\log \left (x^2\right )\right ) \log ^2\left (\frac {6-2 x}{x+\log \left (x^2\right )}\right )}\right ) \, dx+4 \int \frac {1}{\log \left (\frac {6-2 x}{x+\log \left (x^2\right )}\right )} \, dx \\ & = -4 x-4 \int \frac {x}{(-3+x) \left (x+\log \left (x^2\right )\right ) \log ^2\left (\frac {6-2 x}{x+\log \left (x^2\right )}\right )} \, dx-4 \int \frac {x \log \left (x^2\right )}{(-3+x) \left (x+\log \left (x^2\right )\right ) \log ^2\left (\frac {6-2 x}{x+\log \left (x^2\right )}\right )} \, dx+4 \int \frac {1}{\log \left (\frac {6-2 x}{x+\log \left (x^2\right )}\right )} \, dx-24 \int \frac {1}{(-3+x) \left (x+\log \left (x^2\right )\right ) \log ^2\left (\frac {6-2 x}{x+\log \left (x^2\right )}\right )} \, dx \\ & = -4 x-4 \int \left (\frac {1}{\left (x+\log \left (x^2\right )\right ) \log ^2\left (\frac {6-2 x}{x+\log \left (x^2\right )}\right )}+\frac {3}{(-3+x) \left (x+\log \left (x^2\right )\right ) \log ^2\left (\frac {6-2 x}{x+\log \left (x^2\right )}\right )}\right ) \, dx-4 \int \left (\frac {\log \left (x^2\right )}{\left (x+\log \left (x^2\right )\right ) \log ^2\left (\frac {6-2 x}{x+\log \left (x^2\right )}\right )}+\frac {3 \log \left (x^2\right )}{(-3+x) \left (x+\log \left (x^2\right )\right ) \log ^2\left (\frac {6-2 x}{x+\log \left (x^2\right )}\right )}\right ) \, dx+4 \int \frac {1}{\log \left (\frac {6-2 x}{x+\log \left (x^2\right )}\right )} \, dx-24 \int \frac {1}{(-3+x) \left (x+\log \left (x^2\right )\right ) \log ^2\left (\frac {6-2 x}{x+\log \left (x^2\right )}\right )} \, dx \\ & = -4 x-4 \int \frac {1}{\left (x+\log \left (x^2\right )\right ) \log ^2\left (\frac {6-2 x}{x+\log \left (x^2\right )}\right )} \, dx-4 \int \frac {\log \left (x^2\right )}{\left (x+\log \left (x^2\right )\right ) \log ^2\left (\frac {6-2 x}{x+\log \left (x^2\right )}\right )} \, dx+4 \int \frac {1}{\log \left (\frac {6-2 x}{x+\log \left (x^2\right )}\right )} \, dx-12 \int \frac {1}{(-3+x) \left (x+\log \left (x^2\right )\right ) \log ^2\left (\frac {6-2 x}{x+\log \left (x^2\right )}\right )} \, dx-12 \int \frac {\log \left (x^2\right )}{(-3+x) \left (x+\log \left (x^2\right )\right ) \log ^2\left (\frac {6-2 x}{x+\log \left (x^2\right )}\right )} \, dx-24 \int \frac {1}{(-3+x) \left (x+\log \left (x^2\right )\right ) \log ^2\left (\frac {6-2 x}{x+\log \left (x^2\right )}\right )} \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 0.10 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.96 \[ \int \frac {-24-4 x-4 x \log \left (x^2\right )+\left (-12 x+4 x^2+(-12+4 x) \log \left (x^2\right )\right ) \log \left (\frac {6-2 x}{x+\log \left (x^2\right )}\right )+\left (12 x-4 x^2+(12-4 x) \log \left (x^2\right )\right ) \log ^2\left (\frac {6-2 x}{x+\log \left (x^2\right )}\right )}{\left (-3 x+x^2+(-3+x) \log \left (x^2\right )\right ) \log ^2\left (\frac {6-2 x}{x+\log \left (x^2\right )}\right )} \, dx=-4 \left (x-\frac {x}{\log \left (\frac {6-2 x}{x+\log \left (x^2\right )}\right )}\right ) \]

[In]

Integrate[(-24 - 4*x - 4*x*Log[x^2] + (-12*x + 4*x^2 + (-12 + 4*x)*Log[x^2])*Log[(6 - 2*x)/(x + Log[x^2])] + (
12*x - 4*x^2 + (12 - 4*x)*Log[x^2])*Log[(6 - 2*x)/(x + Log[x^2])]^2)/((-3*x + x^2 + (-3 + x)*Log[x^2])*Log[(6
- 2*x)/(x + Log[x^2])]^2),x]

[Out]

-4*(x - x/Log[(6 - 2*x)/(x + Log[x^2])])

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(55\) vs. \(2(25)=50\).

Time = 0.35 (sec) , antiderivative size = 56, normalized size of antiderivative = 2.24

method	result	size
parallelrisch	\(\frac {-8 \ln \left (-\frac {2 \left (-3+x \right )}{\ln \left (x^{2}\right )+x}\right ) x +8 x -48 \ln \left (-\frac {2 \left (-3+x \right )}{\ln \left (x^{2}\right )+x}\right )}{2 \ln \left (-\frac {2 \left (-3+x \right )}{\ln \left (x^{2}\right )+x}\right )}\)	\(56\)

[In]

int((((-4*x+12)*ln(x^2)-4*x^2+12*x)*ln((6-2*x)/(ln(x^2)+x))^2+((4*x-12)*ln(x^2)+4*x^2-12*x)*ln((6-2*x)/(ln(x^2
)+x))-4*x*ln(x^2)-4*x-24)/((-3+x)*ln(x^2)+x^2-3*x)/ln((6-2*x)/(ln(x^2)+x))^2,x,method=_RETURNVERBOSE)

[Out]

1/2*(-8*ln(-2*(-3+x)/(ln(x^2)+x))*x+8*x-48*ln(-2*(-3+x)/(ln(x^2)+x)))/ln(-2*(-3+x)/(ln(x^2)+x))

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.52 \[ \int \frac {-24-4 x-4 x \log \left (x^2\right )+\left (-12 x+4 x^2+(-12+4 x) \log \left (x^2\right )\right ) \log \left (\frac {6-2 x}{x+\log \left (x^2\right )}\right )+\left (12 x-4 x^2+(12-4 x) \log \left (x^2\right )\right ) \log ^2\left (\frac {6-2 x}{x+\log \left (x^2\right )}\right )}{\left (-3 x+x^2+(-3+x) \log \left (x^2\right )\right ) \log ^2\left (\frac {6-2 x}{x+\log \left (x^2\right )}\right )} \, dx=-\frac {4 \, {\left (x \log \left (-\frac {2 \, {\left (x - 3\right )}}{x + \log \left (x^{2}\right )}\right ) - x\right )}}{\log \left (-\frac {2 \, {\left (x - 3\right )}}{x + \log \left (x^{2}\right )}\right )} \]

[In]

integrate((((-4*x+12)*log(x^2)-4*x^2+12*x)*log((6-2*x)/(log(x^2)+x))^2+((4*x-12)*log(x^2)+4*x^2-12*x)*log((6-2
*x)/(log(x^2)+x))-4*x*log(x^2)-4*x-24)/((-3+x)*log(x^2)+x^2-3*x)/log((6-2*x)/(log(x^2)+x))^2,x, algorithm="fri
cas")

[Out]

-4*(x*log(-2*(x - 3)/(x + log(x^2))) - x)/log(-2*(x - 3)/(x + log(x^2)))

Sympy [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.76 \[ \int \frac {-24-4 x-4 x \log \left (x^2\right )+\left (-12 x+4 x^2+(-12+4 x) \log \left (x^2\right )\right ) \log \left (\frac {6-2 x}{x+\log \left (x^2\right )}\right )+\left (12 x-4 x^2+(12-4 x) \log \left (x^2\right )\right ) \log ^2\left (\frac {6-2 x}{x+\log \left (x^2\right )}\right )}{\left (-3 x+x^2+(-3+x) \log \left (x^2\right )\right ) \log ^2\left (\frac {6-2 x}{x+\log \left (x^2\right )}\right )} \, dx=- 4 x + \frac {4 x}{\log {\left (\frac {6 - 2 x}{x + \log {\left (x^{2} \right )}} \right )}} \]

[In]

integrate((((-4*x+12)*ln(x**2)-4*x**2+12*x)*ln((6-2*x)/(ln(x**2)+x))**2+((4*x-12)*ln(x**2)+4*x**2-12*x)*ln((6-
2*x)/(ln(x**2)+x))-4*x*ln(x**2)-4*x-24)/((-3+x)*ln(x**2)+x**2-3*x)/ln((6-2*x)/(ln(x**2)+x))**2,x)

[Out]

-4*x + 4*x/log((6 - 2*x)/(x + log(x**2)))

Maxima [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.31 (sec) , antiderivative size = 53, normalized size of antiderivative = 2.12 \[ \int \frac {-24-4 x-4 x \log \left (x^2\right )+\left (-12 x+4 x^2+(-12+4 x) \log \left (x^2\right )\right ) \log \left (\frac {6-2 x}{x+\log \left (x^2\right )}\right )+\left (12 x-4 x^2+(12-4 x) \log \left (x^2\right )\right ) \log ^2\left (\frac {6-2 x}{x+\log \left (x^2\right )}\right )}{\left (-3 x+x^2+(-3+x) \log \left (x^2\right )\right ) \log ^2\left (\frac {6-2 x}{x+\log \left (x^2\right )}\right )} \, dx=-\frac {4 \, {\left ({\left (-i \, \pi - \log \left (2\right ) + 1\right )} x + x \log \left (x + 2 \, \log \left (x\right )\right ) - x \log \left (x - 3\right )\right )}}{-i \, \pi - \log \left (2\right ) + \log \left (x + 2 \, \log \left (x\right )\right ) - \log \left (x - 3\right )} \]

[In]

integrate((((-4*x+12)*log(x^2)-4*x^2+12*x)*log((6-2*x)/(log(x^2)+x))^2+((4*x-12)*log(x^2)+4*x^2-12*x)*log((6-2
*x)/(log(x^2)+x))-4*x*log(x^2)-4*x-24)/((-3+x)*log(x^2)+x^2-3*x)/log((6-2*x)/(log(x^2)+x))^2,x, algorithm="max
ima")

[Out]

-4*((-I*pi - log(2) + 1)*x + x*log(x + 2*log(x)) - x*log(x - 3))/(-I*pi - log(2) + log(x + 2*log(x)) - log(x -
3))

Giac [A] (verification not implemented)

none

Time = 0.82 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00 \[ \int \frac {-24-4 x-4 x \log \left (x^2\right )+\left (-12 x+4 x^2+(-12+4 x) \log \left (x^2\right )\right ) \log \left (\frac {6-2 x}{x+\log \left (x^2\right )}\right )+\left (12 x-4 x^2+(12-4 x) \log \left (x^2\right )\right ) \log ^2\left (\frac {6-2 x}{x+\log \left (x^2\right )}\right )}{\left (-3 x+x^2+(-3+x) \log \left (x^2\right )\right ) \log ^2\left (\frac {6-2 x}{x+\log \left (x^2\right )}\right )} \, dx=-4 \, x - \frac {4 \, x}{\log \left (x + \log \left (x^{2}\right )\right ) - \log \left (-2 \, x + 6\right )} \]

[In]

integrate((((-4*x+12)*log(x^2)-4*x^2+12*x)*log((6-2*x)/(log(x^2)+x))^2+((4*x-12)*log(x^2)+4*x^2-12*x)*log((6-2
*x)/(log(x^2)+x))-4*x*log(x^2)-4*x-24)/((-3+x)*log(x^2)+x^2-3*x)/log((6-2*x)/(log(x^2)+x))^2,x, algorithm="gia
c")

[Out]

-4*x - 4*x/(log(x + log(x^2)) - log(-2*x + 6))

Mupad [B] (verification not implemented)

Time = 9.10 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00 \[ \int \frac {-24-4 x-4 x \log \left (x^2\right )+\left (-12 x+4 x^2+(-12+4 x) \log \left (x^2\right )\right ) \log \left (\frac {6-2 x}{x+\log \left (x^2\right )}\right )+\left (12 x-4 x^2+(12-4 x) \log \left (x^2\right )\right ) \log ^2\left (\frac {6-2 x}{x+\log \left (x^2\right )}\right )}{\left (-3 x+x^2+(-3+x) \log \left (x^2\right )\right ) \log ^2\left (\frac {6-2 x}{x+\log \left (x^2\right )}\right )} \, dx=\frac {4\,x}{\ln \left (-\frac {2\,x-6}{x+\ln \left (x^2\right )}\right )}-4\,x \]

[In]

int(-(4*x + 4*x*log(x^2) - log(-(2*x - 6)/(x + log(x^2)))*(4*x^2 - 12*x + log(x^2)*(4*x - 12)) + log(-(2*x - 6
)/(x + log(x^2)))^2*(4*x^2 - 12*x + log(x^2)*(4*x - 12)) + 24)/(log(-(2*x - 6)/(x + log(x^2)))^2*(log(x^2)*(x
- 3) - 3*x + x^2)),x)

[Out]

(4*x)/log(-(2*x - 6)/(x + log(x^2))) - 4*x