3.32.65 \(\int \frac {-1+6 x-x^2+3 \log (x)}{(12 x^2-4 x^3+4 x \log (x)) \log (\frac {2 x^3}{3 x-x^2+\log (x)})+(3 x^2-x^3+x \log (x)) \log (\frac {2 x^3}{3 x-x^2+\log (x)}) \log (\log (\frac {2 x^3}{3 x-x^2+\log (x)}))} \, dx\)

Optimal. Leaf size=27 \[ \log \left (3 \left (-4-\log \left (\log \left (\frac {2 x^2}{3-x+\frac {\log (x)}{x}}\right )\right )\right )\right ) \]

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Rubi [A]  time = 0.25, antiderivative size = 22, normalized size of antiderivative = 0.81, number of steps used = 2, number of rules used = 2, integrand size = 109, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.018, Rules used = {6688, 6684} \begin {gather*} \log \left (\log \left (\log \left (\frac {2 x^3}{(3-x) x+\log (x)}\right )\right )+4\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-1 + 6*x - x^2 + 3*Log[x])/((12*x^2 - 4*x^3 + 4*x*Log[x])*Log[(2*x^3)/(3*x - x^2 + Log[x])] + (3*x^2 - x^
3 + x*Log[x])*Log[(2*x^3)/(3*x - x^2 + Log[x])]*Log[Log[(2*x^3)/(3*x - x^2 + Log[x])]]),x]

[Out]

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

Rule 6684

Int[(u_)/(y_), x_Symbol] :> With[{q = DerivativeDivides[y, u, x]}, Simp[q*Log[RemoveContent[y, x]], x] /;  !Fa
lseQ[q]]

Rule 6688

Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; SimplerIntegrandQ[v, u, x]]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {1-6 x+x^2-3 \log (x)}{x ((-3+x) x-\log (x)) \log \left (\frac {2 x^3}{-((-3+x) x)+\log (x)}\right ) \left (4+\log \left (\log \left (\frac {2 x^3}{-((-3+x) x)+\log (x)}\right )\right )\right )} \, dx\\ &=\log \left (4+\log \left (\log \left (\frac {2 x^3}{(3-x) x+\log (x)}\right )\right )\right )\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.40, size = 21, normalized size = 0.78 \begin {gather*} \log \left (4+\log \left (\log \left (\frac {2 x^3}{-((-3+x) x)+\log (x)}\right )\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-1 + 6*x - x^2 + 3*Log[x])/((12*x^2 - 4*x^3 + 4*x*Log[x])*Log[(2*x^3)/(3*x - x^2 + Log[x])] + (3*x^
2 - x^3 + x*Log[x])*Log[(2*x^3)/(3*x - x^2 + Log[x])]*Log[Log[(2*x^3)/(3*x - x^2 + Log[x])]]),x]

[Out]

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

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fricas [A]  time = 0.67, size = 23, normalized size = 0.85 \begin {gather*} \log \left (\log \left (\log \left (-\frac {2 \, x^{3}}{x^{2} - 3 \, x - \log \relax (x)}\right )\right ) + 4\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3*log(x)-x^2+6*x-1)/((x*log(x)-x^3+3*x^2)*log(2*x^3/(log(x)-x^2+3*x))*log(log(2*x^3/(log(x)-x^2+3*x
)))+(4*x*log(x)-4*x^3+12*x^2)*log(2*x^3/(log(x)-x^2+3*x))),x, algorithm="fricas")

[Out]

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

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giac [C]  time = 0.62, size = 28, normalized size = 1.04 \begin {gather*} \log \left (\log \left (i \, \pi + \log \relax (2) - \log \left (x^{2} - 3 \, x - \log \relax (x)\right ) + 3 \, \log \relax (x)\right ) + 4\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3*log(x)-x^2+6*x-1)/((x*log(x)-x^3+3*x^2)*log(2*x^3/(log(x)-x^2+3*x))*log(log(2*x^3/(log(x)-x^2+3*x
)))+(4*x*log(x)-4*x^3+12*x^2)*log(2*x^3/(log(x)-x^2+3*x))),x, algorithm="giac")

[Out]

log(log(I*pi + log(2) - log(x^2 - 3*x - log(x)) + 3*log(x)) + 4)

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maple [C]  time = 0.43, size = 243, normalized size = 9.00

method	result	size
risch	\(\ln \left (\ln \left (\ln \relax (2)+i \pi +3 \ln \relax (x )-\ln \left (-\ln \relax (x )+x^{2}-3 x \right )-\frac {i \pi \,\mathrm {csgn}\left (i x^{2}\right ) \left (-\mathrm {csgn}\left (i x^{2}\right )+\mathrm {csgn}\left (i x \right )\right )^{2}}{2}-\frac {i \pi \,\mathrm {csgn}\left (i x^{3}\right ) \left (-\mathrm {csgn}\left (i x^{3}\right )+\mathrm {csgn}\left (i x^{2}\right )\right ) \left (-\mathrm {csgn}\left (i x^{3}\right )+\mathrm {csgn}\left (i x \right )\right )}{2}+\frac {i \pi \,\mathrm {csgn}\left (\frac {i x^{3}}{\ln \relax (x )-x^{2}+3 x}\right ) \left (\mathrm {csgn}\left (\frac {i x^{3}}{\ln \relax (x )-x^{2}+3 x}\right )+\mathrm {csgn}\left (i x^{3}\right )\right ) \left (\mathrm {csgn}\left (\frac {i x^{3}}{\ln \relax (x )-x^{2}+3 x}\right )-\mathrm {csgn}\left (\frac {i}{\ln \relax (x )-x^{2}+3 x}\right )\right )}{2}+i \pi \mathrm {csgn}\left (\frac {i x^{3}}{\ln \relax (x )-x^{2}+3 x}\right )^{2} \left (-\mathrm {csgn}\left (\frac {i x^{3}}{\ln \relax (x )-x^{2}+3 x}\right )-1\right )\right )+4\right )\)	\(243\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((3*ln(x)-x^2+6*x-1)/((x*ln(x)-x^3+3*x^2)*ln(2*x^3/(ln(x)-x^2+3*x))*ln(ln(2*x^3/(ln(x)-x^2+3*x)))+(4*x*ln(x
)-4*x^3+12*x^2)*ln(2*x^3/(ln(x)-x^2+3*x))),x,method=_RETURNVERBOSE)

[Out]

ln(ln(ln(2)+I*Pi+3*ln(x)-ln(-ln(x)+x^2-3*x)-1/2*I*Pi*csgn(I*x^2)*(-csgn(I*x^2)+csgn(I*x))^2-1/2*I*Pi*csgn(I*x^
3)*(-csgn(I*x^3)+csgn(I*x^2))*(-csgn(I*x^3)+csgn(I*x))+1/2*I*Pi*csgn(I*x^3/(ln(x)-x^2+3*x))*(csgn(I*x^3/(ln(x)
-x^2+3*x))+csgn(I*x^3))*(csgn(I*x^3/(ln(x)-x^2+3*x))-csgn(I/(ln(x)-x^2+3*x)))+I*Pi*csgn(I*x^3/(ln(x)-x^2+3*x))
^2*(-csgn(I*x^3/(ln(x)-x^2+3*x))-1))+4)

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maxima [A]  time = 0.86, size = 25, normalized size = 0.93 \begin {gather*} \log \left (\log \left (\log \relax (2) - \log \left (-x^{2} + 3 \, x + \log \relax (x)\right ) + 3 \, \log \relax (x)\right ) + 4\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3*log(x)-x^2+6*x-1)/((x*log(x)-x^3+3*x^2)*log(2*x^3/(log(x)-x^2+3*x))*log(log(2*x^3/(log(x)-x^2+3*x
)))+(4*x*log(x)-4*x^3+12*x^2)*log(2*x^3/(log(x)-x^2+3*x))),x, algorithm="maxima")

[Out]

log(log(log(2) - log(-x^2 + 3*x + log(x)) + 3*log(x)) + 4)

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mupad [B]  time = 3.03, size = 23, normalized size = 0.85 \begin {gather*} \ln \left (\ln \left (\ln \left (\frac {2\,x^3}{3\,x+\ln \relax (x)-x^2}\right )\right )+4\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((6*x + 3*log(x) - x^2 - 1)/(log((2*x^3)/(3*x + log(x) - x^2))*(4*x*log(x) + 12*x^2 - 4*x^3) + log((2*x^3)/
(3*x + log(x) - x^2))*log(log((2*x^3)/(3*x + log(x) - x^2)))*(x*log(x) + 3*x^2 - x^3)),x)

[Out]

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

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sympy [A]  time = 2.68, size = 20, normalized size = 0.74 \begin {gather*} \log {\left (\log {\left (\log {\left (\frac {2 x^{3}}{- x^{2} + 3 x + \log {\relax (x )}} \right )} \right )} + 4 \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3*ln(x)-x**2+6*x-1)/((x*ln(x)-x**3+3*x**2)*ln(2*x**3/(ln(x)-x**2+3*x))*ln(ln(2*x**3/(ln(x)-x**2+3*x
)))+(4*x*ln(x)-4*x**3+12*x**2)*ln(2*x**3/(ln(x)-x**2+3*x))),x)

[Out]

log(log(log(2*x**3/(-x**2 + 3*x + log(x)))) + 4)

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