3.32.65 \(\int \frac {-1+6 x-x^2+3 \log (x)}{(12 x^2-4 x^3+4 x \log (x)) \log (\frac {2 x^3}{3 x-x^2+\log (x)})+(3 x^2-x^3+x \log (x)) \log (\frac {2 x^3}{3 x-x^2+\log (x)}) \log (\log (\frac {2 x^3}{3 x-x^2+\log (x)}))} \, dx\) [3165]

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

[Out]

ln(-12-3*ln(ln(2*x^2/(3+ln(x)/x-x))))

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Rubi [A]
time = 0.27, 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 = {6820, 6816} \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 6816

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

Rule 6820

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 = 0.09, 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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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 10.41, size = 191, normalized size = 7.07

method	result	size
default	\(\ln \left (\ln \left (\ln \left (2\right )+3 \ln \left (x \right )-\ln \left (\ln \left (x \right )-x^{2}+3 x \right )-\frac {i \pi \,\mathrm {csgn}\left (i x^{2}\right ) \left (-\mathrm {csgn}\left (i x^{2}\right )+\mathrm {csgn}\left (i x \right )\right )^{2}}{2}-\frac {i \pi \,\mathrm {csgn}\left (i x^{3}\right ) \left (-\mathrm {csgn}\left (i x^{3}\right )+\mathrm {csgn}\left (i x^{2}\right )\right ) \left (-\mathrm {csgn}\left (i x^{3}\right )+\mathrm {csgn}\left (i x \right )\right )}{2}-\frac {i \pi \,\mathrm {csgn}\left (\frac {i x^{3}}{\ln \left (x \right )-x^{2}+3 x}\right ) \left (-\mathrm {csgn}\left (\frac {i x^{3}}{\ln \left (x \right )-x^{2}+3 x}\right )+\mathrm {csgn}\left (i x^{3}\right )\right ) \left (-\mathrm {csgn}\left (\frac {i x^{3}}{\ln \left (x \right )-x^{2}+3 x}\right )+\mathrm {csgn}\left (\frac {i}{\ln \left (x \right )-x^{2}+3 x}\right )\right )}{2}\right )+4\right )\)	\(191\)
risch	\(\ln \left (\ln \left (\ln \left (2\right )+i \pi +3 \ln \left (x \right )-\ln \left (-\ln \left (x \right )+x^{2}-3 x \right )-\frac {i \pi \,\mathrm {csgn}\left (i x^{2}\right ) \left (-\mathrm {csgn}\left (i x^{2}\right )+\mathrm {csgn}\left (i x \right )\right )^{2}}{2}-\frac {i \pi \,\mathrm {csgn}\left (i x^{3}\right ) \left (-\mathrm {csgn}\left (i x^{3}\right )+\mathrm {csgn}\left (i x^{2}\right )\right ) \left (-\mathrm {csgn}\left (i x^{3}\right )+\mathrm {csgn}\left (i x \right )\right )}{2}+\frac {i \pi \,\mathrm {csgn}\left (\frac {i x^{3}}{\ln \left (x \right )-x^{2}+3 x}\right ) \left (\mathrm {csgn}\left (\frac {i x^{3}}{\ln \left (x \right )-x^{2}+3 x}\right )+\mathrm {csgn}\left (i x^{3}\right )\right ) \left (\mathrm {csgn}\left (\frac {i x^{3}}{\ln \left (x \right )-x^{2}+3 x}\right )-\mathrm {csgn}\left (\frac {i}{\ln \left (x \right )-x^{2}+3 x}\right )\right )}{2}+i \pi \mathrm {csgn}\left (\frac {i x^{3}}{\ln \left (x \right )-x^{2}+3 x}\right )^{2} \left (-\mathrm {csgn}\left (\frac {i x^{3}}{\ln \left (x \right )-x^{2}+3 x}\right )-1\right )\right )+4\right )\)	\(243\)

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)+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)*(-c
sgn(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))))+4)

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Maxima [A]
time = 0.52, size = 25, normalized size = 0.93 \begin {gather*} \log \left (\log \left (\log \left (2\right ) - \log \left (-x^{2} + 3 \, x + \log \left (x\right )\right ) + 3 \, \log \left (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="maxima")

[Out]

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

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Fricas [A]
time = 0.50, size = 23, normalized size = 0.85 \begin {gather*} \log \left (\log \left (\log \left (-\frac {2 \, x^{3}}{x^{2} - 3 \, x - \log \left (x\right )}\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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Sympy [A]
time = 1.10, size = 20, normalized size = 0.74 \begin {gather*} \log {\left (\log {\left (\log {\left (\frac {2 x^{3}}{- x^{2} + 3 x + \log {\left (x \right )}} \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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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \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]

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

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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 \left (x\right )-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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