3.23.56 \(\int \frac {(-12-6 x-4 x^2+4 x^3+2 x^2 \log (x)) \log (\log (\frac {54-18 x-27 x^2+9 x^3+(-27 x+9 x^2) \log (x)}{x}))}{(54 x-18 x^2-27 x^3+9 x^4+(-27 x^2+9 x^3) \log (x)) \log (\frac {54-18 x-27 x^2+9 x^3+(-27 x+9 x^2) \log (x)}{x})} \, dx\)

Optimal. Leaf size=24 \[ \frac {1}{9} \log ^2\left (\log \left (\frac {9 (-3+x) (-2+x (x+\log (x)))}{x}\right )\right ) \]

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Rubi [F]  time = 9.62, 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 (-12-6 x-4 x^2+4 x^3+2 x^2 \log (x)\right ) \log \left (\log \left (\frac {54-18 x-27 x^2+9 x^3+\left (-27 x+9 x^2\right ) \log (x)}{x}\right )\right )}{\left (54 x-18 x^2-27 x^3+9 x^4+\left (-27 x^2+9 x^3\right ) \log (x)\right ) \log \left (\frac {54-18 x-27 x^2+9 x^3+\left (-27 x+9 x^2\right ) \log (x)}{x}\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[((-12 - 6*x - 4*x^2 + 4*x^3 + 2*x^2*Log[x])*Log[Log[(54 - 18*x - 27*x^2 + 9*x^3 + (-27*x + 9*x^2)*Log[x])/
x]])/((54*x - 18*x^2 - 27*x^3 + 9*x^4 + (-27*x^2 + 9*x^3)*Log[x])*Log[(54 - 18*x - 27*x^2 + 9*x^3 + (-27*x + 9
*x^2)*Log[x])/x]),x]

[Out]

(8*Defer[Int][Log[Log[(9*(-3 + x)*(-2 + x^2 + x*Log[x]))/x]]/((-2 + x^2 + x*Log[x])*Log[(9*(-3 + x)*(-2 + x^2
+ x*Log[x]))/x]), x])/9 + (14*Defer[Int][Log[Log[(9*(-3 + x)*(-2 + x^2 + x*Log[x]))/x]]/((-3 + x)*(-2 + x^2 +
x*Log[x])*Log[(9*(-3 + x)*(-2 + x^2 + x*Log[x]))/x]), x])/9 + (4*Defer[Int][Log[Log[(9*(-3 + x)*(-2 + x^2 + x*
Log[x]))/x]]/(x*(-2 + x^2 + x*Log[x])*Log[(9*(-3 + x)*(-2 + x^2 + x*Log[x]))/x]), x])/9 + (4*Defer[Int][(x*Log
[Log[(9*(-3 + x)*(-2 + x^2 + x*Log[x]))/x]])/((-2 + x^2 + x*Log[x])*Log[(9*(-3 + x)*(-2 + x^2 + x*Log[x]))/x])
, x])/9 + (2*Defer[Int][(Log[x]*Log[Log[(9*(-3 + x)*(-2 + x^2 + x*Log[x]))/x]])/((-2 + x^2 + x*Log[x])*Log[(9*
(-3 + x)*(-2 + x^2 + x*Log[x]))/x]), x])/9 + (2*Defer[Int][(Log[x]*Log[Log[(9*(-3 + x)*(-2 + x^2 + x*Log[x]))/
x]])/((-3 + x)*(-2 + x^2 + x*Log[x])*Log[(9*(-3 + x)*(-2 + x^2 + x*Log[x]))/x]), x])/3

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {2 \left (-6-3 x-2 x^2+2 x^3+x^2 \log (x)\right ) \log \left (\log \left (\frac {9 (-3+x) \left (-2+x^2+x \log (x)\right )}{x}\right )\right )}{9 (3-x) x \left (2-x^2-x \log (x)\right ) \log \left (\frac {9 (-3+x) \left (-2+x^2+x \log (x)\right )}{x}\right )} \, dx\\ &=\frac {2}{9} \int \frac {\left (-6-3 x-2 x^2+2 x^3+x^2 \log (x)\right ) \log \left (\log \left (\frac {9 (-3+x) \left (-2+x^2+x \log (x)\right )}{x}\right )\right )}{(3-x) x \left (2-x^2-x \log (x)\right ) \log \left (\frac {9 (-3+x) \left (-2+x^2+x \log (x)\right )}{x}\right )} \, dx\\ &=\frac {2}{9} \int \left (\frac {\left (-6-3 x-2 x^2+2 x^3+x^2 \log (x)\right ) \log \left (\log \left (\frac {9 (-3+x) \left (-2+x^2+x \log (x)\right )}{x}\right )\right )}{3 (-3+x) \left (-2+x^2+x \log (x)\right ) \log \left (\frac {9 (-3+x) \left (-2+x^2+x \log (x)\right )}{x}\right )}-\frac {\left (-6-3 x-2 x^2+2 x^3+x^2 \log (x)\right ) \log \left (\log \left (\frac {9 (-3+x) \left (-2+x^2+x \log (x)\right )}{x}\right )\right )}{3 x \left (-2+x^2+x \log (x)\right ) \log \left (\frac {9 (-3+x) \left (-2+x^2+x \log (x)\right )}{x}\right )}\right ) \, dx\\ &=\frac {2}{27} \int \frac {\left (-6-3 x-2 x^2+2 x^3+x^2 \log (x)\right ) \log \left (\log \left (\frac {9 (-3+x) \left (-2+x^2+x \log (x)\right )}{x}\right )\right )}{(-3+x) \left (-2+x^2+x \log (x)\right ) \log \left (\frac {9 (-3+x) \left (-2+x^2+x \log (x)\right )}{x}\right )} \, dx-\frac {2}{27} \int \frac {\left (-6-3 x-2 x^2+2 x^3+x^2 \log (x)\right ) \log \left (\log \left (\frac {9 (-3+x) \left (-2+x^2+x \log (x)\right )}{x}\right )\right )}{x \left (-2+x^2+x \log (x)\right ) \log \left (\frac {9 (-3+x) \left (-2+x^2+x \log (x)\right )}{x}\right )} \, dx\\ &=\text {Rest of rules removed due to large latex content} \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.04, size = 25, normalized size = 1.04 \begin {gather*} \frac {1}{9} \log ^2\left (\log \left (\frac {9 (-3+x) \left (-2+x^2+x \log (x)\right )}{x}\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((-12 - 6*x - 4*x^2 + 4*x^3 + 2*x^2*Log[x])*Log[Log[(54 - 18*x - 27*x^2 + 9*x^3 + (-27*x + 9*x^2)*Lo
g[x])/x]])/((54*x - 18*x^2 - 27*x^3 + 9*x^4 + (-27*x^2 + 9*x^3)*Log[x])*Log[(54 - 18*x - 27*x^2 + 9*x^3 + (-27
*x + 9*x^2)*Log[x])/x]),x]

[Out]

Log[Log[(9*(-3 + x)*(-2 + x^2 + x*Log[x]))/x]]^2/9

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fricas [A]  time = 0.79, size = 34, normalized size = 1.42 \begin {gather*} \frac {1}{9} \, \log \left (\log \left (\frac {9 \, {\left (x^{3} - 3 \, x^{2} + {\left (x^{2} - 3 \, x\right )} \log \relax (x) - 2 \, x + 6\right )}}{x}\right )\right )^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*x^2*log(x)+4*x^3-4*x^2-6*x-12)*log(log(((9*x^2-27*x)*log(x)+9*x^3-27*x^2-18*x+54)/x))/((9*x^3-27*
x^2)*log(x)+9*x^4-27*x^3-18*x^2+54*x)/log(((9*x^2-27*x)*log(x)+9*x^3-27*x^2-18*x+54)/x),x, algorithm="fricas")

[Out]

1/9*log(log(9*(x^3 - 3*x^2 + (x^2 - 3*x)*log(x) - 2*x + 6)/x))^2

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giac [A]  time = 0.43, size = 38, normalized size = 1.58 \begin {gather*} \frac {1}{9} \, \log \left (\log \left (9 \, x^{3} + 9 \, x^{2} \log \relax (x) - 27 \, x^{2} - 27 \, x \log \relax (x) - 18 \, x + 54\right ) - \log \relax (x)\right )^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*x^2*log(x)+4*x^3-4*x^2-6*x-12)*log(log(((9*x^2-27*x)*log(x)+9*x^3-27*x^2-18*x+54)/x))/((9*x^3-27*
x^2)*log(x)+9*x^4-27*x^3-18*x^2+54*x)/log(((9*x^2-27*x)*log(x)+9*x^3-27*x^2-18*x+54)/x),x, algorithm="giac")

[Out]

1/9*log(log(9*x^3 + 9*x^2*log(x) - 27*x^2 - 27*x*log(x) - 18*x + 54) - log(x))^2

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maple [C]  time = 0.15, size = 163, normalized size = 6.79

method	result	size
risch	\(\frac {\ln \left (2 \ln \relax (3)-\ln \relax (x )+\ln \left (6+x^{3}+\left (\ln \relax (x )-3\right ) x^{2}+\left (-3 \ln \relax (x )-2\right ) x \right )-\frac {i \pi \,\mathrm {csgn}\left (\frac {i \left (6+x^{3}+\left (\ln \relax (x )-3\right ) x^{2}+\left (-3 \ln \relax (x )-2\right ) x \right )}{x}\right ) \left (-\mathrm {csgn}\left (\frac {i \left (6+x^{3}+\left (\ln \relax (x )-3\right ) x^{2}+\left (-3 \ln \relax (x )-2\right ) x \right )}{x}\right )+\mathrm {csgn}\left (\frac {i}{x}\right )\right ) \left (-\mathrm {csgn}\left (\frac {i \left (6+x^{3}+\left (\ln \relax (x )-3\right ) x^{2}+\left (-3 \ln \relax (x )-2\right ) x \right )}{x}\right )+\mathrm {csgn}\left (i \left (6+x^{3}+\left (\ln \relax (x )-3\right ) x^{2}+\left (-3 \ln \relax (x )-2\right ) x \right )\right )\right )}{2}\right )^{2}}{9}\)	\(163\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*x^2*ln(x)+4*x^3-4*x^2-6*x-12)*ln(ln(((9*x^2-27*x)*ln(x)+9*x^3-27*x^2-18*x+54)/x))/((9*x^3-27*x^2)*ln(x)
+9*x^4-27*x^3-18*x^2+54*x)/ln(((9*x^2-27*x)*ln(x)+9*x^3-27*x^2-18*x+54)/x),x,method=_RETURNVERBOSE)

[Out]

1/9*ln(2*ln(3)-ln(x)+ln(6+x^3+(ln(x)-3)*x^2+(-3*ln(x)-2)*x)-1/2*I*Pi*csgn(I/x*(6+x^3+(ln(x)-3)*x^2+(-3*ln(x)-2
)*x))*(-csgn(I/x*(6+x^3+(ln(x)-3)*x^2+(-3*ln(x)-2)*x))+csgn(I/x))*(-csgn(I/x*(6+x^3+(ln(x)-3)*x^2+(-3*ln(x)-2)
*x))+csgn(I*(6+x^3+(ln(x)-3)*x^2+(-3*ln(x)-2)*x))))^2

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maxima [A]  time = 0.56, size = 28, normalized size = 1.17 \begin {gather*} \frac {1}{9} \, \log \left (2 \, \log \relax (3) + \log \left (x^{2} + x \log \relax (x) - 2\right ) + \log \left (x - 3\right ) - \log \relax (x)\right )^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*x^2*log(x)+4*x^3-4*x^2-6*x-12)*log(log(((9*x^2-27*x)*log(x)+9*x^3-27*x^2-18*x+54)/x))/((9*x^3-27*
x^2)*log(x)+9*x^4-27*x^3-18*x^2+54*x)/log(((9*x^2-27*x)*log(x)+9*x^3-27*x^2-18*x+54)/x),x, algorithm="maxima")

[Out]

1/9*log(2*log(3) + log(x^2 + x*log(x) - 2) + log(x - 3) - log(x))^2

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mupad [B]  time = 2.84, size = 38, normalized size = 1.58 \begin {gather*} \frac {{\ln \left (\ln \left (-\frac {18\,x+\ln \relax (x)\,\left (27\,x-9\,x^2\right )+27\,x^2-9\,x^3-54}{x}\right )\right )}^2}{9} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((log(log(-(18*x + log(x)*(27*x - 9*x^2) + 27*x^2 - 9*x^3 - 54)/x))*(6*x - 2*x^2*log(x) + 4*x^2 - 4*x^3 + 1
2))/(log(-(18*x + log(x)*(27*x - 9*x^2) + 27*x^2 - 9*x^3 - 54)/x)*(log(x)*(27*x^2 - 9*x^3) - 54*x + 18*x^2 + 2
7*x^3 - 9*x^4)),x)

[Out]

log(log(-(18*x + log(x)*(27*x - 9*x^2) + 27*x^2 - 9*x^3 - 54)/x))^2/9

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sympy [A]  time = 3.60, size = 34, normalized size = 1.42 \begin {gather*} \frac {\log {\left (\log {\left (\frac {9 x^{3} - 27 x^{2} - 18 x + \left (9 x^{2} - 27 x\right ) \log {\relax (x )} + 54}{x} \right )} \right )}^{2}}{9} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*x**2*ln(x)+4*x**3-4*x**2-6*x-12)*ln(ln(((9*x**2-27*x)*ln(x)+9*x**3-27*x**2-18*x+54)/x))/((9*x**3-
27*x**2)*ln(x)+9*x**4-27*x**3-18*x**2+54*x)/ln(((9*x**2-27*x)*ln(x)+9*x**3-27*x**2-18*x+54)/x),x)

[Out]

log(log((9*x**3 - 27*x**2 - 18*x + (9*x**2 - 27*x)*log(x) + 54)/x))**2/9

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