3.83.45 \(\int \frac {(4+20 x-8 \log (x)) \log (\frac {100 x^2-40 x \log (x)+4 \log ^2(x)}{x^4})+(10 x-2 \log (x)) \log ^2(\frac {100 x^2-40 x \log (x)+4 \log ^2(x)}{x^4})}{-5 x^4+x^3 \log (x)} \, dx\)

Optimal. Leaf size=22 \[ \frac {\log ^2\left (\frac {4 \left (-5+\frac {\log (x)}{x}\right )^2}{x^2}\right )}{x^2} \]

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Rubi [F]  time = 2.41, 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 {(4+20 x-8 \log (x)) \log \left (\frac {100 x^2-40 x \log (x)+4 \log ^2(x)}{x^4}\right )+(10 x-2 \log (x)) \log ^2\left (\frac {100 x^2-40 x \log (x)+4 \log ^2(x)}{x^4}\right )}{-5 x^4+x^3 \log (x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[((4 + 20*x - 8*Log[x])*Log[(100*x^2 - 40*x*Log[x] + 4*Log[x]^2)/x^4] + (10*x - 2*Log[x])*Log[(100*x^2 - 40
*x*Log[x] + 4*Log[x]^2)/x^4]^2)/(-5*x^4 + x^3*Log[x]),x]

[Out]

-4*Defer[Int][Log[(4*(-5*x + Log[x])^2)/x^4]/(x^3*(5*x - Log[x])), x] - 20*Defer[Int][Log[(4*(-5*x + Log[x])^2
)/x^4]/(x^2*(5*x - Log[x])), x] + 8*Defer[Int][(Log[x]*Log[(4*(-5*x + Log[x])^2)/x^4])/(x^3*(5*x - Log[x])), x
] - 2*Defer[Int][Log[(4*(-5*x + Log[x])^2)/x^4]^2/x^3, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {(4+20 x-8 \log (x)) \log \left (\frac {100 x^2-40 x \log (x)+4 \log ^2(x)}{x^4}\right )+(10 x-2 \log (x)) \log ^2\left (\frac {100 x^2-40 x \log (x)+4 \log ^2(x)}{x^4}\right )}{x^3 (-5 x+\log (x))} \, dx\\ &=\int \frac {2 \log \left (\frac {4 (-5 x+\log (x))^2}{x^4}\right ) \left (-2-10 x+4 \log (x)-5 x \log \left (\frac {4 (-5 x+\log (x))^2}{x^4}\right )+\log (x) \log \left (\frac {4 (-5 x+\log (x))^2}{x^4}\right )\right )}{x^3 (5 x-\log (x))} \, dx\\ &=2 \int \frac {\log \left (\frac {4 (-5 x+\log (x))^2}{x^4}\right ) \left (-2-10 x+4 \log (x)-5 x \log \left (\frac {4 (-5 x+\log (x))^2}{x^4}\right )+\log (x) \log \left (\frac {4 (-5 x+\log (x))^2}{x^4}\right )\right )}{x^3 (5 x-\log (x))} \, dx\\ &=2 \int \left (-\frac {2 (1+5 x-2 \log (x)) \log \left (\frac {4 (-5 x+\log (x))^2}{x^4}\right )}{x^3 (5 x-\log (x))}-\frac {\log ^2\left (\frac {4 (-5 x+\log (x))^2}{x^4}\right )}{x^3}\right ) \, dx\\ &=-\left (2 \int \frac {\log ^2\left (\frac {4 (-5 x+\log (x))^2}{x^4}\right )}{x^3} \, dx\right )-4 \int \frac {(1+5 x-2 \log (x)) \log \left (\frac {4 (-5 x+\log (x))^2}{x^4}\right )}{x^3 (5 x-\log (x))} \, dx\\ &=-\left (2 \int \frac {\log ^2\left (\frac {4 (-5 x+\log (x))^2}{x^4}\right )}{x^3} \, dx\right )-4 \int \left (\frac {\log \left (\frac {4 (-5 x+\log (x))^2}{x^4}\right )}{x^3 (5 x-\log (x))}+\frac {5 \log \left (\frac {4 (-5 x+\log (x))^2}{x^4}\right )}{x^2 (5 x-\log (x))}-\frac {2 \log (x) \log \left (\frac {4 (-5 x+\log (x))^2}{x^4}\right )}{x^3 (5 x-\log (x))}\right ) \, dx\\ &=-\left (2 \int \frac {\log ^2\left (\frac {4 (-5 x+\log (x))^2}{x^4}\right )}{x^3} \, dx\right )-4 \int \frac {\log \left (\frac {4 (-5 x+\log (x))^2}{x^4}\right )}{x^3 (5 x-\log (x))} \, dx+8 \int \frac {\log (x) \log \left (\frac {4 (-5 x+\log (x))^2}{x^4}\right )}{x^3 (5 x-\log (x))} \, dx-20 \int \frac {\log \left (\frac {4 (-5 x+\log (x))^2}{x^4}\right )}{x^2 (5 x-\log (x))} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [F]  time = 0.67, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(4+20 x-8 \log (x)) \log \left (\frac {100 x^2-40 x \log (x)+4 \log ^2(x)}{x^4}\right )+(10 x-2 \log (x)) \log ^2\left (\frac {100 x^2-40 x \log (x)+4 \log ^2(x)}{x^4}\right )}{-5 x^4+x^3 \log (x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[((4 + 20*x - 8*Log[x])*Log[(100*x^2 - 40*x*Log[x] + 4*Log[x]^2)/x^4] + (10*x - 2*Log[x])*Log[(100*x^
2 - 40*x*Log[x] + 4*Log[x]^2)/x^4]^2)/(-5*x^4 + x^3*Log[x]),x]

[Out]

Integrate[((4 + 20*x - 8*Log[x])*Log[(100*x^2 - 40*x*Log[x] + 4*Log[x]^2)/x^4] + (10*x - 2*Log[x])*Log[(100*x^
2 - 40*x*Log[x] + 4*Log[x]^2)/x^4]^2)/(-5*x^4 + x^3*Log[x]), x]

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fricas [A]  time = 0.61, size = 27, normalized size = 1.23 \begin {gather*} \frac {\log \left (\frac {4 \, {\left (25 \, x^{2} - 10 \, x \log \relax (x) + \log \relax (x)^{2}\right )}}{x^{4}}\right )^{2}}{x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-2*log(x)+10*x)*log((4*log(x)^2-40*x*log(x)+100*x^2)/x^4)^2+(-8*log(x)+20*x+4)*log((4*log(x)^2-40*
x*log(x)+100*x^2)/x^4))/(x^3*log(x)-5*x^4),x, algorithm="fricas")

[Out]

log(4*(25*x^2 - 10*x*log(x) + log(x)^2)/x^4)^2/x^2

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giac [B]  time = 1.14, size = 59, normalized size = 2.68 \begin {gather*} \frac {\log \left (100 \, x^{2} - 40 \, x \log \relax (x) + 4 \, \log \relax (x)^{2}\right )^{2}}{x^{2}} - \frac {8 \, \log \left (100 \, x^{2} - 40 \, x \log \relax (x) + 4 \, \log \relax (x)^{2}\right ) \log \relax (x)}{x^{2}} + \frac {16 \, \log \relax (x)^{2}}{x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-2*log(x)+10*x)*log((4*log(x)^2-40*x*log(x)+100*x^2)/x^4)^2+(-8*log(x)+20*x+4)*log((4*log(x)^2-40*
x*log(x)+100*x^2)/x^4))/(x^3*log(x)-5*x^4),x, algorithm="giac")

[Out]

log(100*x^2 - 40*x*log(x) + 4*log(x)^2)^2/x^2 - 8*log(100*x^2 - 40*x*log(x) + 4*log(x)^2)*log(x)/x^2 + 16*log(
x)^2/x^2

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maple [C]  time = 0.70, size = 7795, normalized size = 354.32

method	result	size
risch	\(\text {Expression too large to display}\)	\(7795\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-2*ln(x)+10*x)*ln((4*ln(x)^2-40*x*ln(x)+100*x^2)/x^4)^2+(-8*ln(x)+20*x+4)*ln((4*ln(x)^2-40*x*ln(x)+100*x
^2)/x^4))/(x^3*ln(x)-5*x^4),x,method=_RETURNVERBOSE)

[Out]

result too large to display

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maxima [B]  time = 0.51, size = 47, normalized size = 2.14 \begin {gather*} \frac {4 \, {\left (\log \relax (2)^{2} - 4 \, \log \relax (2) \log \relax (x) + 4 \, \log \relax (x)^{2} + 2 \, {\left (\log \relax (2) - 2 \, \log \relax (x)\right )} \log \left (-5 \, x + \log \relax (x)\right ) + \log \left (-5 \, x + \log \relax (x)\right )^{2}\right )}}{x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-2*log(x)+10*x)*log((4*log(x)^2-40*x*log(x)+100*x^2)/x^4)^2+(-8*log(x)+20*x+4)*log((4*log(x)^2-40*
x*log(x)+100*x^2)/x^4))/(x^3*log(x)-5*x^4),x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 5.22, size = 27, normalized size = 1.23 \begin {gather*} \frac {{\ln \left (\frac {4\,\left (25\,x^2-10\,x\,\ln \relax (x)+{\ln \relax (x)}^2\right )}{x^4}\right )}^2}{x^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((log((4*log(x)^2 - 40*x*log(x) + 100*x^2)/x^4)*(20*x - 8*log(x) + 4) + log((4*log(x)^2 - 40*x*log(x) + 100
*x^2)/x^4)^2*(10*x - 2*log(x)))/(x^3*log(x) - 5*x^4),x)

[Out]

log((4*(log(x)^2 - 10*x*log(x) + 25*x^2))/x^4)^2/x^2

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sympy [A]  time = 0.42, size = 27, normalized size = 1.23 \begin {gather*} \frac {\log {\left (\frac {100 x^{2} - 40 x \log {\relax (x )} + 4 \log {\relax (x )}^{2}}{x^{4}} \right )}^{2}}{x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-2*ln(x)+10*x)*ln((4*ln(x)**2-40*x*ln(x)+100*x**2)/x**4)**2+(-8*ln(x)+20*x+4)*ln((4*ln(x)**2-40*x*
ln(x)+100*x**2)/x**4))/(x**3*ln(x)-5*x**4),x)

[Out]

log((100*x**2 - 40*x*log(x) + 4*log(x)**2)/x**4)**2/x**2

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