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

Optimal. Leaf size=22 \[ \frac {60 x}{2-x+\frac {\log ^2\left (\frac {x}{4}\right )}{x}} \]

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Rubi [F]  time = 0.92, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \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 \end {gather*}

Verification is not applicable to the result.

[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 {gather*} \begin {aligned} \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 {aligned} \end {gather*}

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Mathematica [A]  time = 0.31, size = 24, normalized size = 1.09 \begin {gather*} -\frac {120 x^2}{2 (-2+x) x-2 \log ^2\left (\frac {x}{4}\right )} \end {gather*}

Antiderivative was successfully verified.

[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]

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

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fricas [A]  time = 0.59, size = 22, normalized size = 1.00 \begin {gather*} -\frac {60 \, x^{2}}{x^{2} - \log \left (\frac {1}{4} \, x\right )^{2} - 2 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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giac [A]  time = 0.25, size = 22, normalized size = 1.00 \begin {gather*} -\frac {60 \, x^{2}}{x^{2} - \log \left (\frac {1}{4} \, x\right )^{2} - 2 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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maple [A]  time = 0.05, size = 23, normalized size = 1.05

method	result	size
norman	\(-\frac {60 x^{2}}{x^{2}-\ln \left (\frac {x}{4}\right )^{2}-2 x}\)	\(23\)
risch	\(-\frac {60 x^{2}}{x^{2}-\ln \left (\frac {x}{4}\right )^{2}-2 x}\)	\(23\)

Verification of antiderivative is not currently implemented for this CAS.

[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/(x^2-ln(1/4*x)^2-2*x)

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maxima [A]  time = 0.48, size = 32, normalized size = 1.45 \begin {gather*} -\frac {60 \, x^{2}}{x^{2} - 4 \, \log \relax (2)^{2} + 4 \, \log \relax (2) \log \relax (x) - \log \relax (x)^{2} - 2 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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mupad [B]  time = 6.35, size = 22, normalized size = 1.00 \begin {gather*} \frac {60\,x^2}{-x^2+2\,x+{\ln \left (\frac {x}{4}\right )}^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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sympy [A]  time = 0.15, size = 17, normalized size = 0.77 \begin {gather*} \frac {60 x^{2}}{- x^{2} + 2 x + \log {\left (\frac {x}{4} \right )}^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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