3.103.9 \(\int \frac {-x+(-x+x^2) \log (x)+(-2 x^2 \log (x)+2 x \log (x) \log (x \log (x))) \log (8 x-8 \log (x \log (x)))}{(-x \log (x)+\log (x) \log (x \log (x))) \log ^2(8 x-8 \log (x \log (x)))} \, dx\)

Optimal. Leaf size=18 \[ \frac {x^2}{\log (8 (x-\log (x \log (x))))} \]

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Rubi [F]  time = 1.36, 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 {-x+\left (-x+x^2\right ) \log (x)+\left (-2 x^2 \log (x)+2 x \log (x) \log (x \log (x))\right ) \log (8 x-8 \log (x \log (x)))}{(-x \log (x)+\log (x) \log (x \log (x))) \log ^2(8 x-8 \log (x \log (x)))} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

Defer[Int][x/((x - Log[x*Log[x]])*Log[8*(x - Log[x*Log[x]])]^2), x] - Defer[Int][x^2/((x - Log[x*Log[x]])*Log[
8*(x - Log[x*Log[x]])]^2), x] + Defer[Int][x/(Log[x]*(x - Log[x*Log[x]])*Log[8*(x - Log[x*Log[x]])]^2), x] + 2
*Defer[Int][x^2/((x - Log[x*Log[x]])*Log[8*(x - Log[x*Log[x]])]), x] - 2*Defer[Int][(x*Log[x*Log[x]])/((x - Lo
g[x*Log[x]])*Log[8*(x - Log[x*Log[x]])]), x]

Rubi steps

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

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Mathematica [A]  time = 0.33, size = 18, normalized size = 1.00 \begin {gather*} \frac {x^2}{\log (8 (x-\log (x \log (x))))} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

x^2/Log[8*(x - Log[x*Log[x]])]

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fricas [A]  time = 0.60, size = 18, normalized size = 1.00 \begin {gather*} \frac {x^{2}}{\log \left (8 \, x - 8 \, \log \left (x \log \relax (x)\right )\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x*log(x)*log(x*log(x))-2*x^2*log(x))*log(-8*log(x*log(x))+8*x)+log(x)*(x^2-x)-x)/(log(x)*log(x*l
og(x))-x*log(x))/log(-8*log(x*log(x))+8*x)^2,x, algorithm="fricas")

[Out]

x^2/log(8*x - 8*log(x*log(x)))

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giac [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x*log(x)*log(x*log(x))-2*x^2*log(x))*log(-8*log(x*log(x))+8*x)+log(x)*(x^2-x)-x)/(log(x)*log(x*l
og(x))-x*log(x))/log(-8*log(x*log(x))+8*x)^2,x, algorithm="giac")

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:Simplification assuming t_nostep near 0Simplification assuming t_nostep near 0Simplification assuming t_nos
tep near 0S

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maple [C]  time = 0.10, size = 63, normalized size = 3.50

method	result	size
risch	\(\frac {x^{2}}{\ln \left (-8 \ln \relax (x )-8 \ln \left (\ln \relax (x )\right )+4 i \pi \,\mathrm {csgn}\left (i x \ln \relax (x )\right ) \left (-\mathrm {csgn}\left (i x \ln \relax (x )\right )+\mathrm {csgn}\left (i x \right )\right ) \left (-\mathrm {csgn}\left (i x \ln \relax (x )\right )+\mathrm {csgn}\left (i \ln \relax (x )\right )\right )+8 x \right )}\)	\(63\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((2*x*ln(x)*ln(x*ln(x))-2*x^2*ln(x))*ln(-8*ln(x*ln(x))+8*x)+ln(x)*(x^2-x)-x)/(ln(x)*ln(x*ln(x))-x*ln(x))/l
n(-8*ln(x*ln(x))+8*x)^2,x,method=_RETURNVERBOSE)

[Out]

x^2/ln(-8*ln(x)-8*ln(ln(x))+4*I*Pi*csgn(I*x*ln(x))*(-csgn(I*x*ln(x))+csgn(I*x))*(-csgn(I*x*ln(x))+csgn(I*ln(x)
))+8*x)

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maxima [C]  time = 0.48, size = 24, normalized size = 1.33 \begin {gather*} \frac {x^{2}}{i \, \pi + 3 \, \log \relax (2) + \log \left (-x + \log \relax (x) + \log \left (\log \relax (x)\right )\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x*log(x)*log(x*log(x))-2*x^2*log(x))*log(-8*log(x*log(x))+8*x)+log(x)*(x^2-x)-x)/(log(x)*log(x*l
og(x))-x*log(x))/log(-8*log(x*log(x))+8*x)^2,x, algorithm="maxima")

[Out]

x^2/(I*pi + 3*log(2) + log(-x + log(x) + log(log(x))))

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mupad [F]  time = 0.00, size = -1, normalized size = -0.06 \begin {gather*} \int -\frac {x+\ln \relax (x)\,\left (x-x^2\right )+\ln \left (8\,x-8\,\ln \left (x\,\ln \relax (x)\right )\right )\,\left (2\,x^2\,\ln \relax (x)-2\,x\,\ln \left (x\,\ln \relax (x)\right )\,\ln \relax (x)\right )}{{\ln \left (8\,x-8\,\ln \left (x\,\ln \relax (x)\right )\right )}^2\,\left (\ln \left (x\,\ln \relax (x)\right )\,\ln \relax (x)-x\,\ln \relax (x)\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 0.60, size = 15, normalized size = 0.83 \begin {gather*} \frac {x^{2}}{\log {\left (8 x - 8 \log {\left (x \log {\relax (x )} \right )} \right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x*ln(x)*ln(x*ln(x))-2*x**2*ln(x))*ln(-8*ln(x*ln(x))+8*x)+ln(x)*(x**2-x)-x)/(ln(x)*ln(x*ln(x))-x*
ln(x))/ln(-8*ln(x*ln(x))+8*x)**2,x)

[Out]

x**2/log(8*x - 8*log(x*log(x)))

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