3.45.72 \(\int \frac {(4 x^3+(32 x^3-4 x^3 \log (x)) \log (16-2 \log (x))+(-16+18 \log (x)-2 \log ^2(x)) \log ^3(16-2 \log (x))) \log (\frac {x^3+(-x-\log (x)) \log ^2(16-2 \log (x))}{x \log ^2(16-2 \log (x))})}{(8 x^4-x^4 \log (x)) \log (16-2 \log (x))+(-8 x^2+(-8 x+x^2) \log (x)+x \log ^2(x)) \log ^3(16-2 \log (x))} \, dx\)

Optimal. Leaf size=28 \[ \log ^2\left (-\frac {x+\log (x)}{x}+\frac {x^2}{\log ^2(2 (8-\log (x)))}\right ) \]

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Rubi [F]  time = 33.62, 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 {\left (4 x^3+\left (32 x^3-4 x^3 \log (x)\right ) \log (16-2 \log (x))+\left (-16+18 \log (x)-2 \log ^2(x)\right ) \log ^3(16-2 \log (x))\right ) \log \left (\frac {x^3+(-x-\log (x)) \log ^2(16-2 \log (x))}{x \log ^2(16-2 \log (x))}\right )}{\left (8 x^4-x^4 \log (x)\right ) \log (16-2 \log (x))+\left (-8 x^2+\left (-8 x+x^2\right ) \log (x)+x \log ^2(x)\right ) \log ^3(16-2 \log (x))} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[((4*x^3 + (32*x^3 - 4*x^3*Log[x])*Log[16 - 2*Log[x]] + (-16 + 18*Log[x] - 2*Log[x]^2)*Log[16 - 2*Log[x]]^3
)*Log[(x^3 + (-x - Log[x])*Log[16 - 2*Log[x]]^2)/(x*Log[16 - 2*Log[x]]^2)])/((8*x^4 - x^4*Log[x])*Log[16 - 2*L
og[x]] + (-8*x^2 + (-8*x + x^2)*Log[x] + x*Log[x]^2)*Log[16 - 2*Log[x]]^3),x]

[Out]

32*Defer[Int][(x^2*Log[-1 - Log[x]/x + x^2/Log[-2*(-8 + Log[x])]^2])/((-8 + Log[x])*(-x^3 + x*Log[-2*(-8 + Log
[x])]^2 + Log[x]*Log[-2*(-8 + Log[x])]^2)), x] - 4*Defer[Int][(x^2*Log[x]*Log[-1 - Log[x]/x + x^2/Log[-2*(-8 +
Log[x])]^2])/((-8 + Log[x])*(-x^3 + x*Log[-2*(-8 + Log[x])]^2 + Log[x]*Log[-2*(-8 + Log[x])]^2)), x] + 4*Defe
r[Int][(x^2*Log[-1 - Log[x]/x + x^2/Log[-2*(-8 + Log[x])]^2])/((-8 + Log[x])*Log[-2*(-8 + Log[x])]*(-x^3 + x*L
og[-2*(-8 + Log[x])]^2 + Log[x]*Log[-2*(-8 + Log[x])]^2)), x] + 16*Defer[Int][(Log[-2*(-8 + Log[x])]^2*Log[-1
- Log[x]/x + x^2/Log[-2*(-8 + Log[x])]^2])/(x*(-8 + Log[x])*(x^3 - (x + Log[x])*Log[-2*(-8 + Log[x])]^2)), x]
- 18*Defer[Int][(Log[x]*Log[-2*(-8 + Log[x])]^2*Log[-1 - Log[x]/x + x^2/Log[-2*(-8 + Log[x])]^2])/(x*(-8 + Log
[x])*(x^3 - (x + Log[x])*Log[-2*(-8 + Log[x])]^2)), x] + 2*Defer[Int][(Log[x]^2*Log[-2*(-8 + Log[x])]^2*Log[-1
- Log[x]/x + x^2/Log[-2*(-8 + Log[x])]^2])/(x*(-8 + Log[x])*(x^3 - (x + Log[x])*Log[-2*(-8 + Log[x])]^2)), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {\left (4 x^3+\left (32 x^3-4 x^3 \log (x)\right ) \log (16-2 \log (x))+\left (-16+18 \log (x)-2 \log ^2(x)\right ) \log ^3(16-2 \log (x))\right ) \log \left (\frac {x^3+(-x-\log (x)) \log ^2(16-2 \log (x))}{x \log ^2(-2 (-8+\log (x)))}\right )}{\left (8 x^4-x^4 \log (x)\right ) \log (16-2 \log (x))+\left (-8 x^2+\left (-8 x+x^2\right ) \log (x)+x \log ^2(x)\right ) \log ^3(16-2 \log (x))} \, dx\\ &=\int \left (\frac {16 \log ^2(-2 (-8+\log (x))) \log \left (-1-\frac {\log (x)}{x}+\frac {x^2}{\log ^2(-2 (-8+\log (x)))}\right )}{x (-8+\log (x)) \left (x^3-x \log ^2(-2 (-8+\log (x)))-\log (x) \log ^2(-2 (-8+\log (x)))\right )}-\frac {18 \log (x) \log ^2(-2 (-8+\log (x))) \log \left (-1-\frac {\log (x)}{x}+\frac {x^2}{\log ^2(-2 (-8+\log (x)))}\right )}{x (-8+\log (x)) \left (x^3-x \log ^2(-2 (-8+\log (x)))-\log (x) \log ^2(-2 (-8+\log (x)))\right )}+\frac {2 \log ^2(x) \log ^2(-2 (-8+\log (x))) \log \left (-1-\frac {\log (x)}{x}+\frac {x^2}{\log ^2(-2 (-8+\log (x)))}\right )}{x (-8+\log (x)) \left (x^3-x \log ^2(-2 (-8+\log (x)))-\log (x) \log ^2(-2 (-8+\log (x)))\right )}+\frac {32 x^2 \log \left (-1-\frac {\log (x)}{x}+\frac {x^2}{\log ^2(-2 (-8+\log (x)))}\right )}{(-8+\log (x)) \left (-x^3+x \log ^2(-2 (-8+\log (x)))+\log (x) \log ^2(-2 (-8+\log (x)))\right )}-\frac {4 x^2 \log (x) \log \left (-1-\frac {\log (x)}{x}+\frac {x^2}{\log ^2(-2 (-8+\log (x)))}\right )}{(-8+\log (x)) \left (-x^3+x \log ^2(-2 (-8+\log (x)))+\log (x) \log ^2(-2 (-8+\log (x)))\right )}+\frac {4 x^2 \log \left (-1-\frac {\log (x)}{x}+\frac {x^2}{\log ^2(-2 (-8+\log (x)))}\right )}{(-8+\log (x)) \log (-2 (-8+\log (x))) \left (-x^3+x \log ^2(-2 (-8+\log (x)))+\log (x) \log ^2(-2 (-8+\log (x)))\right )}\right ) \, dx\\ &=2 \int \frac {\log ^2(x) \log ^2(-2 (-8+\log (x))) \log \left (-1-\frac {\log (x)}{x}+\frac {x^2}{\log ^2(-2 (-8+\log (x)))}\right )}{x (-8+\log (x)) \left (x^3-x \log ^2(-2 (-8+\log (x)))-\log (x) \log ^2(-2 (-8+\log (x)))\right )} \, dx-4 \int \frac {x^2 \log (x) \log \left (-1-\frac {\log (x)}{x}+\frac {x^2}{\log ^2(-2 (-8+\log (x)))}\right )}{(-8+\log (x)) \left (-x^3+x \log ^2(-2 (-8+\log (x)))+\log (x) \log ^2(-2 (-8+\log (x)))\right )} \, dx+4 \int \frac {x^2 \log \left (-1-\frac {\log (x)}{x}+\frac {x^2}{\log ^2(-2 (-8+\log (x)))}\right )}{(-8+\log (x)) \log (-2 (-8+\log (x))) \left (-x^3+x \log ^2(-2 (-8+\log (x)))+\log (x) \log ^2(-2 (-8+\log (x)))\right )} \, dx+16 \int \frac {\log ^2(-2 (-8+\log (x))) \log \left (-1-\frac {\log (x)}{x}+\frac {x^2}{\log ^2(-2 (-8+\log (x)))}\right )}{x (-8+\log (x)) \left (x^3-x \log ^2(-2 (-8+\log (x)))-\log (x) \log ^2(-2 (-8+\log (x)))\right )} \, dx-18 \int \frac {\log (x) \log ^2(-2 (-8+\log (x))) \log \left (-1-\frac {\log (x)}{x}+\frac {x^2}{\log ^2(-2 (-8+\log (x)))}\right )}{x (-8+\log (x)) \left (x^3-x \log ^2(-2 (-8+\log (x)))-\log (x) \log ^2(-2 (-8+\log (x)))\right )} \, dx+32 \int \frac {x^2 \log \left (-1-\frac {\log (x)}{x}+\frac {x^2}{\log ^2(-2 (-8+\log (x)))}\right )}{(-8+\log (x)) \left (-x^3+x \log ^2(-2 (-8+\log (x)))+\log (x) \log ^2(-2 (-8+\log (x)))\right )} \, dx\\ &=2 \int \frac {\log ^2(x) \log ^2(-2 (-8+\log (x))) \log \left (-1-\frac {\log (x)}{x}+\frac {x^2}{\log ^2(-2 (-8+\log (x)))}\right )}{x (-8+\log (x)) \left (x^3-(x+\log (x)) \log ^2(-2 (-8+\log (x)))\right )} \, dx-4 \int \frac {x^2 \log (x) \log \left (-1-\frac {\log (x)}{x}+\frac {x^2}{\log ^2(-2 (-8+\log (x)))}\right )}{(-8+\log (x)) \left (-x^3+x \log ^2(-2 (-8+\log (x)))+\log (x) \log ^2(-2 (-8+\log (x)))\right )} \, dx+4 \int \frac {x^2 \log \left (-1-\frac {\log (x)}{x}+\frac {x^2}{\log ^2(-2 (-8+\log (x)))}\right )}{(-8+\log (x)) \log (-2 (-8+\log (x))) \left (-x^3+x \log ^2(-2 (-8+\log (x)))+\log (x) \log ^2(-2 (-8+\log (x)))\right )} \, dx+16 \int \frac {\log ^2(-2 (-8+\log (x))) \log \left (-1-\frac {\log (x)}{x}+\frac {x^2}{\log ^2(-2 (-8+\log (x)))}\right )}{x (-8+\log (x)) \left (x^3-(x+\log (x)) \log ^2(-2 (-8+\log (x)))\right )} \, dx-18 \int \frac {\log (x) \log ^2(-2 (-8+\log (x))) \log \left (-1-\frac {\log (x)}{x}+\frac {x^2}{\log ^2(-2 (-8+\log (x)))}\right )}{x (-8+\log (x)) \left (x^3-(x+\log (x)) \log ^2(-2 (-8+\log (x)))\right )} \, dx+32 \int \frac {x^2 \log \left (-1-\frac {\log (x)}{x}+\frac {x^2}{\log ^2(-2 (-8+\log (x)))}\right )}{(-8+\log (x)) \left (-x^3+x \log ^2(-2 (-8+\log (x)))+\log (x) \log ^2(-2 (-8+\log (x)))\right )} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.14, size = 25, normalized size = 0.89 \begin {gather*} \log ^2\left (-1-\frac {\log (x)}{x}+\frac {x^2}{\log ^2(-2 (-8+\log (x)))}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((4*x^3 + (32*x^3 - 4*x^3*Log[x])*Log[16 - 2*Log[x]] + (-16 + 18*Log[x] - 2*Log[x]^2)*Log[16 - 2*Log
[x]]^3)*Log[(x^3 + (-x - Log[x])*Log[16 - 2*Log[x]]^2)/(x*Log[16 - 2*Log[x]]^2)])/((8*x^4 - x^4*Log[x])*Log[16
- 2*Log[x]] + (-8*x^2 + (-8*x + x^2)*Log[x] + x*Log[x]^2)*Log[16 - 2*Log[x]]^3),x]

[Out]

Log[-1 - Log[x]/x + x^2/Log[-2*(-8 + Log[x])]^2]^2

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fricas [A]  time = 0.54, size = 35, normalized size = 1.25 \begin {gather*} \log \left (\frac {x^{3} - {\left (x + \log \relax (x)\right )} \log \left (-2 \, \log \relax (x) + 16\right )^{2}}{x \log \left (-2 \, \log \relax (x) + 16\right )^{2}}\right )^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-2*log(x)^2+18*log(x)-16)*log(-2*log(x)+16)^3+(-4*x^3*log(x)+32*x^3)*log(-2*log(x)+16)+4*x^3)*log(
((-x-log(x))*log(-2*log(x)+16)^2+x^3)/x/log(-2*log(x)+16)^2)/((x*log(x)^2+(x^2-8*x)*log(x)-8*x^2)*log(-2*log(x
)+16)^3+(-x^4*log(x)+8*x^4)*log(-2*log(x)+16)),x, algorithm="fricas")

[Out]

log((x^3 - (x + log(x))*log(-2*log(x) + 16)^2)/(x*log(-2*log(x) + 16)^2))^2

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giac [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-2*log(x)^2+18*log(x)-16)*log(-2*log(x)+16)^3+(-4*x^3*log(x)+32*x^3)*log(-2*log(x)+16)+4*x^3)*log(
((-x-log(x))*log(-2*log(x)+16)^2+x^3)/x/log(-2*log(x)+16)^2)/((x*log(x)^2+(x^2-8*x)*log(x)-8*x^2)*log(-2*log(x
)+16)^3+(-x^4*log(x)+8*x^4)*log(-2*log(x)+16)),x, algorithm="giac")

[Out]

Timed out

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maple [F]  time = 0.03, size = 0, normalized size = 0.00 \[\int \frac {\left (\left (-2 \ln \relax (x )^{2}+18 \ln \relax (x )-16\right ) \ln \left (-2 \ln \relax (x )+16\right )^{3}+\left (-4 x^{3} \ln \relax (x )+32 x^{3}\right ) \ln \left (-2 \ln \relax (x )+16\right )+4 x^{3}\right ) \ln \left (\frac {\left (-x -\ln \relax (x )\right ) \ln \left (-2 \ln \relax (x )+16\right )^{2}+x^{3}}{x \ln \left (-2 \ln \relax (x )+16\right )^{2}}\right )}{\left (x \ln \relax (x )^{2}+\left (x^{2}-8 x \right ) \ln \relax (x )-8 x^{2}\right ) \ln \left (-2 \ln \relax (x )+16\right )^{3}+\left (-x^{4} \ln \relax (x )+8 x^{4}\right ) \ln \left (-2 \ln \relax (x )+16\right )}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-2*ln(x)^2+18*ln(x)-16)*ln(-2*ln(x)+16)^3+(-4*x^3*ln(x)+32*x^3)*ln(-2*ln(x)+16)+4*x^3)*ln(((-x-ln(x))*ln
(-2*ln(x)+16)^2+x^3)/x/ln(-2*ln(x)+16)^2)/((x*ln(x)^2+(x^2-8*x)*ln(x)-8*x^2)*ln(-2*ln(x)+16)^3+(-x^4*ln(x)+8*x
^4)*ln(-2*ln(x)+16)),x)

[Out]

int(((-2*ln(x)^2+18*ln(x)-16)*ln(-2*ln(x)+16)^3+(-4*x^3*ln(x)+32*x^3)*ln(-2*ln(x)+16)+4*x^3)*ln(((-x-ln(x))*ln
(-2*ln(x)+16)^2+x^3)/x/ln(-2*ln(x)+16)^2)/((x*ln(x)^2+(x^2-8*x)*ln(x)-8*x^2)*ln(-2*ln(x)+16)^3+(-x^4*ln(x)+8*x
^4)*ln(-2*ln(x)+16)),x)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} -2 \, \int \frac {{\left ({\left (\log \relax (x)^{2} - 9 \, \log \relax (x) + 8\right )} \log \left (-2 \, \log \relax (x) + 16\right )^{3} - 2 \, x^{3} + 2 \, {\left (x^{3} \log \relax (x) - 8 \, x^{3}\right )} \log \left (-2 \, \log \relax (x) + 16\right )\right )} \log \left (\frac {x^{3} - {\left (x + \log \relax (x)\right )} \log \left (-2 \, \log \relax (x) + 16\right )^{2}}{x \log \left (-2 \, \log \relax (x) + 16\right )^{2}}\right )}{{\left (x \log \relax (x)^{2} - 8 \, x^{2} + {\left (x^{2} - 8 \, x\right )} \log \relax (x)\right )} \log \left (-2 \, \log \relax (x) + 16\right )^{3} - {\left (x^{4} \log \relax (x) - 8 \, x^{4}\right )} \log \left (-2 \, \log \relax (x) + 16\right )}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-2*log(x)^2+18*log(x)-16)*log(-2*log(x)+16)^3+(-4*x^3*log(x)+32*x^3)*log(-2*log(x)+16)+4*x^3)*log(
((-x-log(x))*log(-2*log(x)+16)^2+x^3)/x/log(-2*log(x)+16)^2)/((x*log(x)^2+(x^2-8*x)*log(x)-8*x^2)*log(-2*log(x
)+16)^3+(-x^4*log(x)+8*x^4)*log(-2*log(x)+16)),x, algorithm="maxima")

[Out]

-2*integrate(((log(x)^2 - 9*log(x) + 8)*log(-2*log(x) + 16)^3 - 2*x^3 + 2*(x^3*log(x) - 8*x^3)*log(-2*log(x) +
16))*log((x^3 - (x + log(x))*log(-2*log(x) + 16)^2)/(x*log(-2*log(x) + 16)^2))/((x*log(x)^2 - 8*x^2 + (x^2 -
8*x)*log(x))*log(-2*log(x) + 16)^3 - (x^4*log(x) - 8*x^4)*log(-2*log(x) + 16)), x)

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mupad [B]  time = 4.87, size = 37, normalized size = 1.32 \begin {gather*} {\ln \left (-\frac {{\ln \left (16-2\,\ln \relax (x)\right )}^2\,\left (x+\ln \relax (x)\right )-x^3}{x\,{\ln \left (16-2\,\ln \relax (x)\right )}^2}\right )}^2 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((log(-(log(16 - 2*log(x))^2*(x + log(x)) - x^3)/(x*log(16 - 2*log(x))^2))*(log(16 - 2*log(x))^3*(2*log(x)^
2 - 18*log(x) + 16) + log(16 - 2*log(x))*(4*x^3*log(x) - 32*x^3) - 4*x^3))/(log(16 - 2*log(x))*(x^4*log(x) - 8
*x^4) + log(16 - 2*log(x))^3*(log(x)*(8*x - x^2) - x*log(x)^2 + 8*x^2)),x)

[Out]

log(-(log(16 - 2*log(x))^2*(x + log(x)) - x^3)/(x*log(16 - 2*log(x))^2))^2

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sympy [A]  time = 5.27, size = 34, normalized size = 1.21 \begin {gather*} \log {\left (\frac {x^{3} + \left (- x - \log {\relax (x )}\right ) \log {\left (16 - 2 \log {\relax (x )} \right )}^{2}}{x \log {\left (16 - 2 \log {\relax (x )} \right )}^{2}} \right )}^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-2*ln(x)**2+18*ln(x)-16)*ln(-2*ln(x)+16)**3+(-4*x**3*ln(x)+32*x**3)*ln(-2*ln(x)+16)+4*x**3)*ln(((-
x-ln(x))*ln(-2*ln(x)+16)**2+x**3)/x/ln(-2*ln(x)+16)**2)/((x*ln(x)**2+(x**2-8*x)*ln(x)-8*x**2)*ln(-2*ln(x)+16)*
*3+(-x**4*ln(x)+8*x**4)*ln(-2*ln(x)+16)),x)

[Out]

log((x**3 + (-x - log(x))*log(16 - 2*log(x))**2)/(x*log(16 - 2*log(x))**2))**2

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