\(\int \frac {8-4 x+(-20+4 x-x^2) \log (x^2) \log (\log (x^2))+4 \log (x^2) \log (\log (x^2)) \log (\log (\log (x^2)))}{(20-20 x+5 x^2) \log (x^2) \log (\log (x^2))} \, dx\) [3149]

Optimal result

Integrand size = 68, antiderivative size = 26 \[ \int \frac {8-4 x+\left (-20+4 x-x^2\right ) \log \left (x^2\right ) \log \left (\log \left (x^2\right )\right )+4 \log \left (x^2\right ) \log \left (\log \left (x^2\right )\right ) \log \left (\log \left (\log \left (x^2\right )\right )\right )}{\left (20-20 x+5 x^2\right ) \log \left (x^2\right ) \log \left (\log \left (x^2\right )\right )} \, dx=\frac {1}{5} x \left (-5+\frac {2 \left (-2 x+\log \left (\log \left (\log \left (x^2\right )\right )\right )\right )}{2-x}\right ) \]

[Out]

1/5*(2/(2-x)*(ln(ln(ln(x^2)))-2*x)-5)*x

Rubi [F]

\[ \int \frac {8-4 x+\left (-20+4 x-x^2\right ) \log \left (x^2\right ) \log \left (\log \left (x^2\right )\right )+4 \log \left (x^2\right ) \log \left (\log \left (x^2\right )\right ) \log \left (\log \left (\log \left (x^2\right )\right )\right )}{\left (20-20 x+5 x^2\right ) \log \left (x^2\right ) \log \left (\log \left (x^2\right )\right )} \, dx=\int \frac {8-4 x+\left (-20+4 x-x^2\right ) \log \left (x^2\right ) \log \left (\log \left (x^2\right )\right )+4 \log \left (x^2\right ) \log \left (\log \left (x^2\right )\right ) \log \left (\log \left (\log \left (x^2\right )\right )\right )}{\left (20-20 x+5 x^2\right ) \log \left (x^2\right ) \log \left (\log \left (x^2\right )\right )} \, dx \]

[In]

Int[(8 - 4*x + (-20 + 4*x - x^2)*Log[x^2]*Log[Log[x^2]] + 4*Log[x^2]*Log[Log[x^2]]*Log[Log[Log[x^2]]])/((20 -
20*x + 5*x^2)*Log[x^2]*Log[Log[x^2]]),x]

[Out]

-16/(5*(2 - x)) - x/5 - (4*Defer[Int][1/((-2 + x)*Log[x^2]*Log[Log[x^2]]), x])/5 + (4*Defer[Int][Log[Log[Log[x
^2]]]/(-2 + x)^2, x])/5

Rubi steps \begin{align*} \text {integral}& = \int \frac {8-4 x+\left (-20+4 x-x^2\right ) \log \left (x^2\right ) \log \left (\log \left (x^2\right )\right )+4 \log \left (x^2\right ) \log \left (\log \left (x^2\right )\right ) \log \left (\log \left (\log \left (x^2\right )\right )\right )}{5 (-2+x)^2 \log \left (x^2\right ) \log \left (\log \left (x^2\right )\right )} \, dx \\ & = \frac {1}{5} \int \frac {8-4 x+\left (-20+4 x-x^2\right ) \log \left (x^2\right ) \log \left (\log \left (x^2\right )\right )+4 \log \left (x^2\right ) \log \left (\log \left (x^2\right )\right ) \log \left (\log \left (\log \left (x^2\right )\right )\right )}{(-2+x)^2 \log \left (x^2\right ) \log \left (\log \left (x^2\right )\right )} \, dx \\ & = \frac {1}{5} \int \left (\frac {8-4 x-20 \log \left (x^2\right ) \log \left (\log \left (x^2\right )\right )+4 x \log \left (x^2\right ) \log \left (\log \left (x^2\right )\right )-x^2 \log \left (x^2\right ) \log \left (\log \left (x^2\right )\right )}{(-2+x)^2 \log \left (x^2\right ) \log \left (\log \left (x^2\right )\right )}+\frac {4 \log \left (\log \left (\log \left (x^2\right )\right )\right )}{(-2+x)^2}\right ) \, dx \\ & = \frac {1}{5} \int \frac {8-4 x-20 \log \left (x^2\right ) \log \left (\log \left (x^2\right )\right )+4 x \log \left (x^2\right ) \log \left (\log \left (x^2\right )\right )-x^2 \log \left (x^2\right ) \log \left (\log \left (x^2\right )\right )}{(-2+x)^2 \log \left (x^2\right ) \log \left (\log \left (x^2\right )\right )} \, dx+\frac {4}{5} \int \frac {\log \left (\log \left (\log \left (x^2\right )\right )\right )}{(-2+x)^2} \, dx \\ & = \frac {1}{5} \int \frac {8-4 x-\left (20-4 x+x^2\right ) \log \left (x^2\right ) \log \left (\log \left (x^2\right )\right )}{(2-x)^2 \log \left (x^2\right ) \log \left (\log \left (x^2\right )\right )} \, dx+\frac {4}{5} \int \frac {\log \left (\log \left (\log \left (x^2\right )\right )\right )}{(-2+x)^2} \, dx \\ & = \frac {1}{5} \int \left (\frac {-20+4 x-x^2}{(-2+x)^2}-\frac {4}{(-2+x) \log \left (x^2\right ) \log \left (\log \left (x^2\right )\right )}\right ) \, dx+\frac {4}{5} \int \frac {\log \left (\log \left (\log \left (x^2\right )\right )\right )}{(-2+x)^2} \, dx \\ & = \frac {1}{5} \int \frac {-20+4 x-x^2}{(-2+x)^2} \, dx-\frac {4}{5} \int \frac {1}{(-2+x) \log \left (x^2\right ) \log \left (\log \left (x^2\right )\right )} \, dx+\frac {4}{5} \int \frac {\log \left (\log \left (\log \left (x^2\right )\right )\right )}{(-2+x)^2} \, dx \\ & = \frac {1}{5} \int \left (-1-\frac {16}{(-2+x)^2}\right ) \, dx-\frac {4}{5} \int \frac {1}{(-2+x) \log \left (x^2\right ) \log \left (\log \left (x^2\right )\right )} \, dx+\frac {4}{5} \int \frac {\log \left (\log \left (\log \left (x^2\right )\right )\right )}{(-2+x)^2} \, dx \\ & = -\frac {16}{5 (2-x)}-\frac {x}{5}-\frac {4}{5} \int \frac {1}{(-2+x) \log \left (x^2\right ) \log \left (\log \left (x^2\right )\right )} \, dx+\frac {4}{5} \int \frac {\log \left (\log \left (\log \left (x^2\right )\right )\right )}{(-2+x)^2} \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 0.15 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.00 \[ \int \frac {8-4 x+\left (-20+4 x-x^2\right ) \log \left (x^2\right ) \log \left (\log \left (x^2\right )\right )+4 \log \left (x^2\right ) \log \left (\log \left (x^2\right )\right ) \log \left (\log \left (\log \left (x^2\right )\right )\right )}{\left (20-20 x+5 x^2\right ) \log \left (x^2\right ) \log \left (\log \left (x^2\right )\right )} \, dx=-\frac {-16-2 x+x^2+2 x \log \left (\log \left (\log \left (x^2\right )\right )\right )}{5 (-2+x)} \]

[In]

Integrate[(8 - 4*x + (-20 + 4*x - x^2)*Log[x^2]*Log[Log[x^2]] + 4*Log[x^2]*Log[Log[x^2]]*Log[Log[Log[x^2]]])/(
(20 - 20*x + 5*x^2)*Log[x^2]*Log[Log[x^2]]),x]

[Out]

-1/5*(-16 - 2*x + x^2 + 2*x*Log[Log[Log[x^2]]])/(-2 + x)

Maple [A] (verified)

Time = 2.23 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.92

method	result	size
parallelrisch	\(\frac {40-4 \ln \left (\ln \left (\ln \left (x^{2}\right )\right )\right ) x -2 x^{2}}{10 x -20}\)	\(24\)

[In]

int((4*ln(x^2)*ln(ln(x^2))*ln(ln(ln(x^2)))+(-x^2+4*x-20)*ln(x^2)*ln(ln(x^2))-4*x+8)/(5*x^2-20*x+20)/ln(x^2)/ln
(ln(x^2)),x,method=_RETURNVERBOSE)

[Out]

1/10*(40-4*ln(ln(ln(x^2)))*x-2*x^2)/(-2+x)

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.92 \[ \int \frac {8-4 x+\left (-20+4 x-x^2\right ) \log \left (x^2\right ) \log \left (\log \left (x^2\right )\right )+4 \log \left (x^2\right ) \log \left (\log \left (x^2\right )\right ) \log \left (\log \left (\log \left (x^2\right )\right )\right )}{\left (20-20 x+5 x^2\right ) \log \left (x^2\right ) \log \left (\log \left (x^2\right )\right )} \, dx=-\frac {x^{2} + 2 \, x \log \left (\log \left (\log \left (x^{2}\right )\right )\right ) - 2 \, x - 16}{5 \, {\left (x - 2\right )}} \]

[In]

integrate((4*log(x^2)*log(log(x^2))*log(log(log(x^2)))+(-x^2+4*x-20)*log(x^2)*log(log(x^2))-4*x+8)/(5*x^2-20*x
+20)/log(x^2)/log(log(x^2)),x, algorithm="fricas")

[Out]

-1/5*(x^2 + 2*x*log(log(log(x^2))) - 2*x - 16)/(x - 2)

Sympy [A] (verification not implemented)

Time = 0.26 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.38 \[ \int \frac {8-4 x+\left (-20+4 x-x^2\right ) \log \left (x^2\right ) \log \left (\log \left (x^2\right )\right )+4 \log \left (x^2\right ) \log \left (\log \left (x^2\right )\right ) \log \left (\log \left (\log \left (x^2\right )\right )\right )}{\left (20-20 x+5 x^2\right ) \log \left (x^2\right ) \log \left (\log \left (x^2\right )\right )} \, dx=- \frac {x}{5} - \frac {2 \log {\left (\log {\left (\log {\left (x^{2} \right )} \right )} \right )}}{5} - \frac {4 \log {\left (\log {\left (\log {\left (x^{2} \right )} \right )} \right )}}{5 x - 10} + \frac {16}{5 x - 10} \]

[In]

integrate((4*ln(x**2)*ln(ln(x**2))*ln(ln(ln(x**2)))+(-x**2+4*x-20)*ln(x**2)*ln(ln(x**2))-4*x+8)/(5*x**2-20*x+2
0)/ln(x**2)/ln(ln(x**2)),x)

[Out]

-x/5 - 2*log(log(log(x**2)))/5 - 4*log(log(log(x**2)))/(5*x - 10) + 16/(5*x - 10)

Maxima [A] (verification not implemented)

none

Time = 0.32 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.96 \[ \int \frac {8-4 x+\left (-20+4 x-x^2\right ) \log \left (x^2\right ) \log \left (\log \left (x^2\right )\right )+4 \log \left (x^2\right ) \log \left (\log \left (x^2\right )\right ) \log \left (\log \left (\log \left (x^2\right )\right )\right )}{\left (20-20 x+5 x^2\right ) \log \left (x^2\right ) \log \left (\log \left (x^2\right )\right )} \, dx=-\frac {x^{2} + 2 \, x \log \left (\log \left (2\right ) + \log \left (\log \left (x\right )\right )\right ) - 2 \, x - 16}{5 \, {\left (x - 2\right )}} \]

[In]

integrate((4*log(x^2)*log(log(x^2))*log(log(log(x^2)))+(-x^2+4*x-20)*log(x^2)*log(log(x^2))-4*x+8)/(5*x^2-20*x
+20)/log(x^2)/log(log(x^2)),x, algorithm="maxima")

[Out]

-1/5*(x^2 + 2*x*log(log(2) + log(log(x))) - 2*x - 16)/(x - 2)

Giac [A] (verification not implemented)

none

Time = 0.36 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.23 \[ \int \frac {8-4 x+\left (-20+4 x-x^2\right ) \log \left (x^2\right ) \log \left (\log \left (x^2\right )\right )+4 \log \left (x^2\right ) \log \left (\log \left (x^2\right )\right ) \log \left (\log \left (\log \left (x^2\right )\right )\right )}{\left (20-20 x+5 x^2\right ) \log \left (x^2\right ) \log \left (\log \left (x^2\right )\right )} \, dx=-\frac {1}{5} \, x - \frac {4 \, \log \left (\log \left (\log \left (x^{2}\right )\right )\right )}{5 \, {\left (x - 2\right )}} + \frac {16}{5 \, {\left (x - 2\right )}} - \frac {2}{5} \, \log \left (\log \left (\log \left (x^{2}\right )\right )\right ) \]

[In]

integrate((4*log(x^2)*log(log(x^2))*log(log(log(x^2)))+(-x^2+4*x-20)*log(x^2)*log(log(x^2))-4*x+8)/(5*x^2-20*x
+20)/log(x^2)/log(log(x^2)),x, algorithm="giac")

[Out]

-1/5*x - 4/5*log(log(log(x^2)))/(x - 2) + 16/5/(x - 2) - 2/5*log(log(log(x^2)))

Mupad [B] (verification not implemented)

Time = 8.85 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.77 \[ \int \frac {8-4 x+\left (-20+4 x-x^2\right ) \log \left (x^2\right ) \log \left (\log \left (x^2\right )\right )+4 \log \left (x^2\right ) \log \left (\log \left (x^2\right )\right ) \log \left (\log \left (\log \left (x^2\right )\right )\right )}{\left (20-20 x+5 x^2\right ) \log \left (x^2\right ) \log \left (\log \left (x^2\right )\right )} \, dx=\frac {16}{5\,\left (x-2\right )}-\frac {2\,\ln \left (\ln \left (\ln \left (x^2\right )\right )\right )}{5}-\frac {x}{5}+\frac {\ln \left (\ln \left (\ln \left (x^2\right )\right )\right )\,\left (8\,x-4\,x^2\right )}{5\,x\,{\left (x-2\right )}^2} \]

[In]

int(-(4*x + log(x^2)*log(log(x^2))*(x^2 - 4*x + 20) - 4*log(log(log(x^2)))*log(x^2)*log(log(x^2)) - 8)/(log(x^
2)*log(log(x^2))*(5*x^2 - 20*x + 20)),x)

[Out]

16/(5*(x - 2)) - (2*log(log(log(x^2))))/5 - x/5 + (log(log(log(x^2)))*(8*x - 4*x^2))/(5*x*(x - 2)^2)