3.43.65 \(\int \frac {-36+54 x+(-12+63 x-27 x^2) \log (x)+(30-36 x+27 x^2) \log ^2(x)}{((6 x-9 x^2) \log (x)+(10 x-21 x^2+9 x^3) \log ^2(x)) \log (\frac {(-6+9 x) \log ^2(x)}{-3 x+(-5 x+3 x^2) \log (x)}) \log ^2(\log (\frac {(-6+9 x) \log ^2(x)}{-3 x+(-5 x+3 x^2) \log (x)}))} \, dx\)

Optimal. Leaf size=32 \[ \frac {3}{\log \left (\log \left (\frac {3 \log (x)}{x+\frac {x+\frac {x}{\log (x)}}{\frac {2}{3}-x}}\right )\right )} \]

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Rubi [F]  time = 25.63, 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 {-36+54 x+\left (-12+63 x-27 x^2\right ) \log (x)+\left (30-36 x+27 x^2\right ) \log ^2(x)}{\left (\left (6 x-9 x^2\right ) \log (x)+\left (10 x-21 x^2+9 x^3\right ) \log ^2(x)\right ) \log \left (\frac {(-6+9 x) \log ^2(x)}{-3 x+\left (-5 x+3 x^2\right ) \log (x)}\right ) \log ^2\left (\log \left (\frac {(-6+9 x) \log ^2(x)}{-3 x+\left (-5 x+3 x^2\right ) \log (x)}\right )\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-36 + 54*x + (-12 + 63*x - 27*x^2)*Log[x] + (30 - 36*x + 27*x^2)*Log[x]^2)/(((6*x - 9*x^2)*Log[x] + (10*x
- 21*x^2 + 9*x^3)*Log[x]^2)*Log[((-6 + 9*x)*Log[x]^2)/(-3*x + (-5*x + 3*x^2)*Log[x])]*Log[Log[((-6 + 9*x)*Log
[x]^2)/(-3*x + (-5*x + 3*x^2)*Log[x])]]^2),x]

[Out]

-9*Defer[Int][1/((-3 - 5*Log[x] + 3*x*Log[x])*Log[(3*(-2 + 3*x)*Log[x]^2)/(x*(-3 + (-5 + 3*x)*Log[x]))]*Log[Lo
g[(3*(-2 + 3*x)*Log[x]^2)/(x*(-3 + (-5 + 3*x)*Log[x]))]]^2), x] + 6*Defer[Int][1/(x*(-3 - 5*Log[x] + 3*x*Log[x
])*Log[(3*(-2 + 3*x)*Log[x]^2)/(x*(-3 + (-5 + 3*x)*Log[x]))]*Log[Log[(3*(-2 + 3*x)*Log[x]^2)/(x*(-3 + (-5 + 3*
x)*Log[x]))]]^2), x] + 27*Defer[Int][1/((-2 + 3*x)*(-3 - 5*Log[x] + 3*x*Log[x])*Log[(3*(-2 + 3*x)*Log[x]^2)/(x
*(-3 + (-5 + 3*x)*Log[x]))]*Log[Log[(3*(-2 + 3*x)*Log[x]^2)/(x*(-3 + (-5 + 3*x)*Log[x]))]]^2), x] + 18*Defer[I
nt][1/(x*Log[x]*(-3 - 5*Log[x] + 3*x*Log[x])*Log[(3*(-2 + 3*x)*Log[x]^2)/(x*(-3 + (-5 + 3*x)*Log[x]))]*Log[Log
[(3*(-2 + 3*x)*Log[x]^2)/(x*(-3 + (-5 + 3*x)*Log[x]))]]^2), x] + 9*Defer[Int][Log[x]/((-3 - 5*Log[x] + 3*x*Log
[x])*Log[(3*(-2 + 3*x)*Log[x]^2)/(x*(-3 + (-5 + 3*x)*Log[x]))]*Log[Log[(3*(-2 + 3*x)*Log[x]^2)/(x*(-3 + (-5 +
3*x)*Log[x]))]]^2), x] - 15*Defer[Int][Log[x]/(x*(-3 - 5*Log[x] + 3*x*Log[x])*Log[(3*(-2 + 3*x)*Log[x]^2)/(x*(
-3 + (-5 + 3*x)*Log[x]))]*Log[Log[(3*(-2 + 3*x)*Log[x]^2)/(x*(-3 + (-5 + 3*x)*Log[x]))]]^2), x] + 27*Defer[Int
][Log[x]/((-2 + 3*x)*(-3 - 5*Log[x] + 3*x*Log[x])*Log[(3*(-2 + 3*x)*Log[x]^2)/(x*(-3 + (-5 + 3*x)*Log[x]))]*Lo
g[Log[(3*(-2 + 3*x)*Log[x]^2)/(x*(-3 + (-5 + 3*x)*Log[x]))]]^2), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-36+54 x+\left (-12+63 x-27 x^2\right ) \log (x)+\left (30-36 x+27 x^2\right ) \log ^2(x)}{(2-3 x) x \log (x) (3+5 \log (x)-3 x \log (x)) \log \left (\frac {(-6+9 x) \log ^2(x)}{-3 x+\left (-5 x+3 x^2\right ) \log (x)}\right ) \log ^2\left (\log \left (\frac {(-6+9 x) \log ^2(x)}{-3 x+\left (-5 x+3 x^2\right ) \log (x)}\right )\right )} \, dx\\ &=\int \left (-\frac {3 \left (-12+18 x-4 \log (x)+21 x \log (x)-9 x^2 \log (x)+10 \log ^2(x)-12 x \log ^2(x)+9 x^2 \log ^2(x)\right )}{2 x \log (x) (-3-5 \log (x)+3 x \log (x)) \log \left (\frac {3 (-2+3 x) \log ^2(x)}{x (-3+(-5+3 x) \log (x))}\right ) \log ^2\left (\log \left (\frac {3 (-2+3 x) \log ^2(x)}{x (-3+(-5+3 x) \log (x))}\right )\right )}+\frac {9 \left (-12+18 x-4 \log (x)+21 x \log (x)-9 x^2 \log (x)+10 \log ^2(x)-12 x \log ^2(x)+9 x^2 \log ^2(x)\right )}{2 (-2+3 x) \log (x) (-3-5 \log (x)+3 x \log (x)) \log \left (\frac {3 (-2+3 x) \log ^2(x)}{x (-3+(-5+3 x) \log (x))}\right ) \log ^2\left (\log \left (\frac {3 (-2+3 x) \log ^2(x)}{x (-3+(-5+3 x) \log (x))}\right )\right )}\right ) \, dx\\ &=-\left (\frac {3}{2} \int \frac {-12+18 x-4 \log (x)+21 x \log (x)-9 x^2 \log (x)+10 \log ^2(x)-12 x \log ^2(x)+9 x^2 \log ^2(x)}{x \log (x) (-3-5 \log (x)+3 x \log (x)) \log \left (\frac {3 (-2+3 x) \log ^2(x)}{x (-3+(-5+3 x) \log (x))}\right ) \log ^2\left (\log \left (\frac {3 (-2+3 x) \log ^2(x)}{x (-3+(-5+3 x) \log (x))}\right )\right )} \, dx\right )+\frac {9}{2} \int \frac {-12+18 x-4 \log (x)+21 x \log (x)-9 x^2 \log (x)+10 \log ^2(x)-12 x \log ^2(x)+9 x^2 \log ^2(x)}{(-2+3 x) \log (x) (-3-5 \log (x)+3 x \log (x)) \log \left (\frac {3 (-2+3 x) \log ^2(x)}{x (-3+(-5+3 x) \log (x))}\right ) \log ^2\left (\log \left (\frac {3 (-2+3 x) \log ^2(x)}{x (-3+(-5+3 x) \log (x))}\right )\right )} \, dx\\ &=\text {Rest of rules removed due to large latex content} \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.08, size = 32, normalized size = 1.00 \begin {gather*} \frac {3}{\log \left (\log \left (\frac {3 (-2+3 x) \log ^2(x)}{x (-3+(-5+3 x) \log (x))}\right )\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-36 + 54*x + (-12 + 63*x - 27*x^2)*Log[x] + (30 - 36*x + 27*x^2)*Log[x]^2)/(((6*x - 9*x^2)*Log[x] +
(10*x - 21*x^2 + 9*x^3)*Log[x]^2)*Log[((-6 + 9*x)*Log[x]^2)/(-3*x + (-5*x + 3*x^2)*Log[x])]*Log[Log[((-6 + 9*
x)*Log[x]^2)/(-3*x + (-5*x + 3*x^2)*Log[x])]]^2),x]

[Out]

3/Log[Log[(3*(-2 + 3*x)*Log[x]^2)/(x*(-3 + (-5 + 3*x)*Log[x]))]]

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fricas [A]  time = 0.72, size = 35, normalized size = 1.09 \begin {gather*} \frac {3}{\log \left (\log \left (\frac {3 \, {\left (3 \, x - 2\right )} \log \relax (x)^{2}}{{\left (3 \, x^{2} - 5 \, x\right )} \log \relax (x) - 3 \, x}\right )\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((27*x^2-36*x+30)*log(x)^2+(-27*x^2+63*x-12)*log(x)+54*x-36)/((9*x^3-21*x^2+10*x)*log(x)^2+(-9*x^2+6
*x)*log(x))/log((9*x-6)*log(x)^2/((3*x^2-5*x)*log(x)-3*x))/log(log((9*x-6)*log(x)^2/((3*x^2-5*x)*log(x)-3*x)))
^2,x, algorithm="fricas")

[Out]

3/log(log(3*(3*x - 2)*log(x)^2/((3*x^2 - 5*x)*log(x) - 3*x)))

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giac [A]  time = 1.10, size = 39, normalized size = 1.22 \begin {gather*} \frac {3}{\log \left (\log \left (9 \, x \log \relax (x)^{2} - 6 \, \log \relax (x)^{2}\right ) - \log \left (3 \, x \log \relax (x) - 5 \, \log \relax (x) - 3\right ) - \log \relax (x)\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((27*x^2-36*x+30)*log(x)^2+(-27*x^2+63*x-12)*log(x)+54*x-36)/((9*x^3-21*x^2+10*x)*log(x)^2+(-9*x^2+6
*x)*log(x))/log((9*x-6)*log(x)^2/((3*x^2-5*x)*log(x)-3*x))/log(log((9*x-6)*log(x)^2/((3*x^2-5*x)*log(x)-3*x)))
^2,x, algorithm="giac")

[Out]

3/log(log(9*x*log(x)^2 - 6*log(x)^2) - log(3*x*log(x) - 5*log(x) - 3) - log(x))

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maple [C]  time = 0.21, size = 380, normalized size = 11.88

method	result	size
risch	\(\frac {3}{\ln \left (\ln \relax (3)-\ln \relax (x )+2 \ln \left (\ln \relax (x )\right )+\ln \left (x -\frac {2}{3}\right )-\ln \left (x \ln \relax (x )-\frac {5 \ln \relax (x )}{3}-1\right )-\frac {i \pi \,\mathrm {csgn}\left (\frac {i \left (x -\frac {2}{3}\right )}{x \ln \relax (x )-\frac {5 \ln \relax (x )}{3}-1}\right ) \left (-\mathrm {csgn}\left (\frac {i \left (x -\frac {2}{3}\right )}{x \ln \relax (x )-\frac {5 \ln \relax (x )}{3}-1}\right )+\mathrm {csgn}\left (i \left (x -\frac {2}{3}\right )\right )\right ) \left (-\mathrm {csgn}\left (\frac {i \left (x -\frac {2}{3}\right )}{x \ln \relax (x )-\frac {5 \ln \relax (x )}{3}-1}\right )+\mathrm {csgn}\left (\frac {i}{x \ln \relax (x )-\frac {5 \ln \relax (x )}{3}-1}\right )\right )}{2}-\frac {i \pi \,\mathrm {csgn}\left (i \ln \relax (x )^{2}\right ) \left (-\mathrm {csgn}\left (i \ln \relax (x )^{2}\right )+\mathrm {csgn}\left (i \ln \relax (x )\right )\right )^{2}}{2}-\frac {i \pi \,\mathrm {csgn}\left (\frac {i \ln \relax (x )^{2} \left (x -\frac {2}{3}\right )}{x \ln \relax (x )-\frac {5 \ln \relax (x )}{3}-1}\right ) \left (-\mathrm {csgn}\left (\frac {i \ln \relax (x )^{2} \left (x -\frac {2}{3}\right )}{x \ln \relax (x )-\frac {5 \ln \relax (x )}{3}-1}\right )+\mathrm {csgn}\left (i \ln \relax (x )^{2}\right )\right ) \left (-\mathrm {csgn}\left (\frac {i \ln \relax (x )^{2} \left (x -\frac {2}{3}\right )}{x \ln \relax (x )-\frac {5 \ln \relax (x )}{3}-1}\right )+\mathrm {csgn}\left (\frac {i \left (x -\frac {2}{3}\right )}{x \ln \relax (x )-\frac {5 \ln \relax (x )}{3}-1}\right )\right )}{2}-\frac {i \pi \,\mathrm {csgn}\left (\frac {i \left (x -\frac {2}{3}\right ) \ln \relax (x )^{2}}{x \left (x \ln \relax (x )-\frac {5 \ln \relax (x )}{3}-1\right )}\right ) \left (-\mathrm {csgn}\left (\frac {i \left (x -\frac {2}{3}\right ) \ln \relax (x )^{2}}{x \left (x \ln \relax (x )-\frac {5 \ln \relax (x )}{3}-1\right )}\right )+\mathrm {csgn}\left (\frac {i}{x}\right )\right ) \left (-\mathrm {csgn}\left (\frac {i \left (x -\frac {2}{3}\right ) \ln \relax (x )^{2}}{x \left (x \ln \relax (x )-\frac {5 \ln \relax (x )}{3}-1\right )}\right )+\mathrm {csgn}\left (\frac {i \ln \relax (x )^{2} \left (x -\frac {2}{3}\right )}{x \ln \relax (x )-\frac {5 \ln \relax (x )}{3}-1}\right )\right )}{2}\right )}\)	\(380\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((27*x^2-36*x+30)*ln(x)^2+(-27*x^2+63*x-12)*ln(x)+54*x-36)/((9*x^3-21*x^2+10*x)*ln(x)^2+(-9*x^2+6*x)*ln(x)
)/ln((9*x-6)*ln(x)^2/((3*x^2-5*x)*ln(x)-3*x))/ln(ln((9*x-6)*ln(x)^2/((3*x^2-5*x)*ln(x)-3*x)))^2,x,method=_RETU
RNVERBOSE)

[Out]

3/ln(ln(3)-ln(x)+2*ln(ln(x))+ln(x-2/3)-ln(x*ln(x)-5/3*ln(x)-1)-1/2*I*Pi*csgn(I*(x-2/3)/(x*ln(x)-5/3*ln(x)-1))*
(-csgn(I*(x-2/3)/(x*ln(x)-5/3*ln(x)-1))+csgn(I*(x-2/3)))*(-csgn(I*(x-2/3)/(x*ln(x)-5/3*ln(x)-1))+csgn(I/(x*ln(
x)-5/3*ln(x)-1)))-1/2*I*Pi*csgn(I*ln(x)^2)*(-csgn(I*ln(x)^2)+csgn(I*ln(x)))^2-1/2*I*Pi*csgn(I*ln(x)^2/(x*ln(x)
-5/3*ln(x)-1)*(x-2/3))*(-csgn(I*ln(x)^2/(x*ln(x)-5/3*ln(x)-1)*(x-2/3))+csgn(I*ln(x)^2))*(-csgn(I*ln(x)^2/(x*ln
(x)-5/3*ln(x)-1)*(x-2/3))+csgn(I*(x-2/3)/(x*ln(x)-5/3*ln(x)-1)))-1/2*I*Pi*csgn(I/x/(x*ln(x)-5/3*ln(x)-1)*(x-2/
3)*ln(x)^2)*(-csgn(I/x/(x*ln(x)-5/3*ln(x)-1)*(x-2/3)*ln(x)^2)+csgn(I/x))*(-csgn(I/x/(x*ln(x)-5/3*ln(x)-1)*(x-2
/3)*ln(x)^2)+csgn(I*ln(x)^2/(x*ln(x)-5/3*ln(x)-1)*(x-2/3))))

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maxima [A]  time = 0.53, size = 36, normalized size = 1.12 \begin {gather*} \frac {3}{\log \left (\log \relax (3) - \log \left ({\left (3 \, x - 5\right )} \log \relax (x) - 3\right ) + \log \left (3 \, x - 2\right ) - \log \relax (x) + 2 \, \log \left (\log \relax (x)\right )\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((27*x^2-36*x+30)*log(x)^2+(-27*x^2+63*x-12)*log(x)+54*x-36)/((9*x^3-21*x^2+10*x)*log(x)^2+(-9*x^2+6
*x)*log(x))/log((9*x-6)*log(x)^2/((3*x^2-5*x)*log(x)-3*x))/log(log((9*x-6)*log(x)^2/((3*x^2-5*x)*log(x)-3*x)))
^2,x, algorithm="maxima")

[Out]

3/log(log(3) - log((3*x - 5)*log(x) - 3) + log(3*x - 2) - log(x) + 2*log(log(x)))

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mupad [B]  time = 4.94, size = 35, normalized size = 1.09 \begin {gather*} \frac {3}{\ln \left (\ln \left (-\frac {{\ln \relax (x)}^2\,\left (9\,x-6\right )}{3\,x+\ln \relax (x)\,\left (5\,x-3\,x^2\right )}\right )\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((54*x + log(x)^2*(27*x^2 - 36*x + 30) - log(x)*(27*x^2 - 63*x + 12) - 36)/(log(-(log(x)^2*(9*x - 6))/(3*x
+ log(x)*(5*x - 3*x^2)))*log(log(-(log(x)^2*(9*x - 6))/(3*x + log(x)*(5*x - 3*x^2))))^2*(log(x)^2*(10*x - 21*x
^2 + 9*x^3) + log(x)*(6*x - 9*x^2))),x)

[Out]

3/log(log(-(log(x)^2*(9*x - 6))/(3*x + log(x)*(5*x - 3*x^2))))

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sympy [A]  time = 20.69, size = 29, normalized size = 0.91 \begin {gather*} \frac {3}{\log {\left (\log {\left (\frac {\left (9 x - 6\right ) \log {\relax (x )}^{2}}{- 3 x + \left (3 x^{2} - 5 x\right ) \log {\relax (x )}} \right )} \right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((27*x**2-36*x+30)*ln(x)**2+(-27*x**2+63*x-12)*ln(x)+54*x-36)/((9*x**3-21*x**2+10*x)*ln(x)**2+(-9*x*
*2+6*x)*ln(x))/ln((9*x-6)*ln(x)**2/((3*x**2-5*x)*ln(x)-3*x))/ln(ln((9*x-6)*ln(x)**2/((3*x**2-5*x)*ln(x)-3*x)))
**2,x)

[Out]

3/log(log((9*x - 6)*log(x)**2/(-3*x + (3*x**2 - 5*x)*log(x))))

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