\(\int \frac {120 x^2-120 x \log (\frac {x}{4})+120 x \log ^2(\frac {x}{4})}{4 x^2-4 x^3+x^4+(4 x-2 x^2) \log ^2(\frac {x}{4})+\log ^4(\frac {x}{4})} \, dx\) [7882]

Optimal result

Integrand size = 69, antiderivative size = 22 \[ \int \frac {120 x^2-120 x \log \left (\frac {x}{4}\right )+120 x \log ^2\left (\frac {x}{4}\right )}{4 x^2-4 x^3+x^4+\left (4 x-2 x^2\right ) \log ^2\left (\frac {x}{4}\right )+\log ^4\left (\frac {x}{4}\right )} \, dx=\frac {60 x}{2-x+\frac {\log ^2\left (\frac {x}{4}\right )}{x}} \]

[Out]

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

Rubi [F]

\[ \int \frac {120 x^2-120 x \log \left (\frac {x}{4}\right )+120 x \log ^2\left (\frac {x}{4}\right )}{4 x^2-4 x^3+x^4+\left (4 x-2 x^2\right ) \log ^2\left (\frac {x}{4}\right )+\log ^4\left (\frac {x}{4}\right )} \, dx=\int \frac {120 x^2-120 x \log \left (\frac {x}{4}\right )+120 x \log ^2\left (\frac {x}{4}\right )}{4 x^2-4 x^3+x^4+\left (4 x-2 x^2\right ) \log ^2\left (\frac {x}{4}\right )+\log ^4\left (\frac {x}{4}\right )} \, dx \]

[In]

Int[(120*x^2 - 120*x*Log[x/4] + 120*x*Log[x/4]^2)/(4*x^2 - 4*x^3 + x^4 + (4*x - 2*x^2)*Log[x/4]^2 + Log[x/4]^4
),x]

[Out]

120*Log[4]*Defer[Int][x/(-2*x + x^2 - Log[x/4]^2)^2, x] - 120*Defer[Int][x^2/(-2*x + x^2 - Log[x/4]^2)^2, x] +
120*Defer[Int][x^3/(-2*x + x^2 - Log[x/4]^2)^2, x] - 120*Defer[Int][x/(-2*x + x^2 - Log[x/4]^2), x] - 120*Def
er[Int][(x*Log[x])/(-2*x + x^2 - Log[x/4]^2)^2, x]

Rubi steps \begin{align*} \text {integral}& = \int \frac {120 x \left (x+\log (4)+\log ^2\left (\frac {x}{4}\right )-\log (x)\right )}{\left ((-2+x) x-\log ^2\left (\frac {x}{4}\right )\right )^2} \, dx \\ & = 120 \int \frac {x \left (x+\log (4)+\log ^2\left (\frac {x}{4}\right )-\log (x)\right )}{\left ((-2+x) x-\log ^2\left (\frac {x}{4}\right )\right )^2} \, dx \\ & = 120 \int \left (\frac {x \left (x+\log (4)+\log ^2\left (\frac {x}{4}\right )\right )}{\left (-2 x+x^2-\log ^2\left (\frac {x}{4}\right )\right )^2}-\frac {x \log (x)}{\left (-2 x+x^2-\log ^2\left (\frac {x}{4}\right )\right )^2}\right ) \, dx \\ & = 120 \int \frac {x \left (x+\log (4)+\log ^2\left (\frac {x}{4}\right )\right )}{\left (-2 x+x^2-\log ^2\left (\frac {x}{4}\right )\right )^2} \, dx-120 \int \frac {x \log (x)}{\left (-2 x+x^2-\log ^2\left (\frac {x}{4}\right )\right )^2} \, dx \\ & = 120 \int \left (\frac {x \left (-x+x^2+\log (4)\right )}{\left (-2 x+x^2-\log ^2\left (\frac {x}{4}\right )\right )^2}-\frac {x}{-2 x+x^2-\log ^2\left (\frac {x}{4}\right )}\right ) \, dx-120 \int \frac {x \log (x)}{\left (-2 x+x^2-\log ^2\left (\frac {x}{4}\right )\right )^2} \, dx \\ & = 120 \int \frac {x \left (-x+x^2+\log (4)\right )}{\left (-2 x+x^2-\log ^2\left (\frac {x}{4}\right )\right )^2} \, dx-120 \int \frac {x}{-2 x+x^2-\log ^2\left (\frac {x}{4}\right )} \, dx-120 \int \frac {x \log (x)}{\left (-2 x+x^2-\log ^2\left (\frac {x}{4}\right )\right )^2} \, dx \\ & = -\left (120 \int \frac {x}{-2 x+x^2-\log ^2\left (\frac {x}{4}\right )} \, dx\right )+120 \int \left (-\frac {x^2}{\left (-2 x+x^2-\log ^2\left (\frac {x}{4}\right )\right )^2}+\frac {x^3}{\left (-2 x+x^2-\log ^2\left (\frac {x}{4}\right )\right )^2}+\frac {x \log (4)}{\left (-2 x+x^2-\log ^2\left (\frac {x}{4}\right )\right )^2}\right ) \, dx-120 \int \frac {x \log (x)}{\left (-2 x+x^2-\log ^2\left (\frac {x}{4}\right )\right )^2} \, dx \\ & = -\left (120 \int \frac {x^2}{\left (-2 x+x^2-\log ^2\left (\frac {x}{4}\right )\right )^2} \, dx\right )+120 \int \frac {x^3}{\left (-2 x+x^2-\log ^2\left (\frac {x}{4}\right )\right )^2} \, dx-120 \int \frac {x}{-2 x+x^2-\log ^2\left (\frac {x}{4}\right )} \, dx-120 \int \frac {x \log (x)}{\left (-2 x+x^2-\log ^2\left (\frac {x}{4}\right )\right )^2} \, dx+(120 \log (4)) \int \frac {x}{\left (-2 x+x^2-\log ^2\left (\frac {x}{4}\right )\right )^2} \, dx \\ \end{align*}

Mathematica [F(-1)]

Timed out. \[ \int \frac {120 x^2-120 x \log \left (\frac {x}{4}\right )+120 x \log ^2\left (\frac {x}{4}\right )}{4 x^2-4 x^3+x^4+\left (4 x-2 x^2\right ) \log ^2\left (\frac {x}{4}\right )+\log ^4\left (\frac {x}{4}\right )} \, dx=\text {\$Aborted} \]

[In]

Integrate[(120*x^2 - 120*x*Log[x/4] + 120*x*Log[x/4]^2)/(4*x^2 - 4*x^3 + x^4 + (4*x - 2*x^2)*Log[x/4]^2 + Log[
x/4]^4),x]

[Out]

$Aborted

Maple [A] (verified)

Time = 0.29 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.05

method	result	size
derivativedivides	\(\frac {60 x^{2}}{\ln \left (\frac {x}{4}\right )^{2}-x^{2}+2 x}\)	\(23\)
default	\(\frac {60 x^{2}}{\ln \left (\frac {x}{4}\right )^{2}-x^{2}+2 x}\)	\(23\)
risch	\(-\frac {60 x^{2}}{x^{2}-\ln \left (\frac {x}{4}\right )^{2}-2 x}\)	\(23\)
parallelrisch	\(-\frac {60 x^{2}}{x^{2}-\ln \left (\frac {x}{4}\right )^{2}-2 x}\)	\(23\)
norman	\(\frac {-120 x -60 \ln \left (\frac {x}{4}\right )^{2}}{x^{2}-\ln \left (\frac {x}{4}\right )^{2}-2 x}\)	\(31\)

[In]

int((120*x*ln(1/4*x)^2-120*x*ln(1/4*x)+120*x^2)/(ln(1/4*x)^4+(-2*x^2+4*x)*ln(1/4*x)^2+x^4-4*x^3+4*x^2),x,metho
d=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00 \[ \int \frac {120 x^2-120 x \log \left (\frac {x}{4}\right )+120 x \log ^2\left (\frac {x}{4}\right )}{4 x^2-4 x^3+x^4+\left (4 x-2 x^2\right ) \log ^2\left (\frac {x}{4}\right )+\log ^4\left (\frac {x}{4}\right )} \, dx=-\frac {60 \, x^{2}}{x^{2} - \log \left (\frac {1}{4} \, x\right )^{2} - 2 \, x} \]

[In]

integrate((120*x*log(1/4*x)^2-120*x*log(1/4*x)+120*x^2)/(log(1/4*x)^4+(-2*x^2+4*x)*log(1/4*x)^2+x^4-4*x^3+4*x^
2),x, algorithm="fricas")

[Out]

-60*x^2/(x^2 - log(1/4*x)^2 - 2*x)

Sympy [A] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.77 \[ \int \frac {120 x^2-120 x \log \left (\frac {x}{4}\right )+120 x \log ^2\left (\frac {x}{4}\right )}{4 x^2-4 x^3+x^4+\left (4 x-2 x^2\right ) \log ^2\left (\frac {x}{4}\right )+\log ^4\left (\frac {x}{4}\right )} \, dx=\frac {60 x^{2}}{- x^{2} + 2 x + \log {\left (\frac {x}{4} \right )}^{2}} \]

[In]

integrate((120*x*ln(1/4*x)**2-120*x*ln(1/4*x)+120*x**2)/(ln(1/4*x)**4+(-2*x**2+4*x)*ln(1/4*x)**2+x**4-4*x**3+4
*x**2),x)

[Out]

60*x**2/(-x**2 + 2*x + log(x/4)**2)

Maxima [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.45 \[ \int \frac {120 x^2-120 x \log \left (\frac {x}{4}\right )+120 x \log ^2\left (\frac {x}{4}\right )}{4 x^2-4 x^3+x^4+\left (4 x-2 x^2\right ) \log ^2\left (\frac {x}{4}\right )+\log ^4\left (\frac {x}{4}\right )} \, dx=-\frac {60 \, x^{2}}{x^{2} - 4 \, \log \left (2\right )^{2} + 4 \, \log \left (2\right ) \log \left (x\right ) - \log \left (x\right )^{2} - 2 \, x} \]

[In]

integrate((120*x*log(1/4*x)^2-120*x*log(1/4*x)+120*x^2)/(log(1/4*x)^4+(-2*x^2+4*x)*log(1/4*x)^2+x^4-4*x^3+4*x^
2),x, algorithm="maxima")

[Out]

-60*x^2/(x^2 - 4*log(2)^2 + 4*log(2)*log(x) - log(x)^2 - 2*x)

Giac [A] (verification not implemented)

none

Time = 0.33 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00 \[ \int \frac {120 x^2-120 x \log \left (\frac {x}{4}\right )+120 x \log ^2\left (\frac {x}{4}\right )}{4 x^2-4 x^3+x^4+\left (4 x-2 x^2\right ) \log ^2\left (\frac {x}{4}\right )+\log ^4\left (\frac {x}{4}\right )} \, dx=-\frac {60 \, x^{2}}{x^{2} - \log \left (\frac {1}{4} \, x\right )^{2} - 2 \, x} \]

[In]

integrate((120*x*log(1/4*x)^2-120*x*log(1/4*x)+120*x^2)/(log(1/4*x)^4+(-2*x^2+4*x)*log(1/4*x)^2+x^4-4*x^3+4*x^
2),x, algorithm="giac")

[Out]

-60*x^2/(x^2 - log(1/4*x)^2 - 2*x)

Mupad [B] (verification not implemented)

Time = 11.93 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00 \[ \int \frac {120 x^2-120 x \log \left (\frac {x}{4}\right )+120 x \log ^2\left (\frac {x}{4}\right )}{4 x^2-4 x^3+x^4+\left (4 x-2 x^2\right ) \log ^2\left (\frac {x}{4}\right )+\log ^4\left (\frac {x}{4}\right )} \, dx=\frac {60\,x^2}{-x^2+2\,x+{\ln \left (\frac {x}{4}\right )}^2} \]

[In]

int((120*x*log(x/4)^2 - 120*x*log(x/4) + 120*x^2)/(log(x/4)^2*(4*x - 2*x^2) + log(x/4)^4 + 4*x^2 - 4*x^3 + x^4
),x)

[Out]

(60*x^2)/(2*x + log(x/4)^2 - x^2)