3.40.55 \(\int \frac {-6 \log (4)+(6 x^2-9 x^3) \log ^2(x)+(6 \log (4) \log (x)+(3 x^2-3 x^3) \log ^2(x)) \log (\frac {4 \log (4)+(2 x^2-2 x^3) \log (x)}{\log (x)}) \log (\log (\frac {4 \log (4)+(2 x^2-2 x^3) \log (x)}{\log (x)}))}{(-2 x^2 \log (4) \log (x)+(-x^4+x^5) \log ^2(x)) \log (\frac {4 \log (4)+(2 x^2-2 x^3) \log (x)}{\log (x)}) \log ^2(\log (\frac {4 \log (4)+(2 x^2-2 x^3) \log (x)}{\log (x)}))} \, dx\)

Optimal. Leaf size=32 \[ \frac {3}{x \log \left (\log \left (x \log \left (e^{2 x-2 x^2}\right )+\frac {4 \log (4)}{\log (x)}\right )\right )} \]

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Rubi [F]  time = 11.64, 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 {-6 \log (4)+\left (6 x^2-9 x^3\right ) \log ^2(x)+\left (6 \log (4) \log (x)+\left (3 x^2-3 x^3\right ) \log ^2(x)\right ) \log \left (\frac {4 \log (4)+\left (2 x^2-2 x^3\right ) \log (x)}{\log (x)}\right ) \log \left (\log \left (\frac {4 \log (4)+\left (2 x^2-2 x^3\right ) \log (x)}{\log (x)}\right )\right )}{\left (-2 x^2 \log (4) \log (x)+\left (-x^4+x^5\right ) \log ^2(x)\right ) \log \left (\frac {4 \log (4)+\left (2 x^2-2 x^3\right ) \log (x)}{\log (x)}\right ) \log ^2\left (\log \left (\frac {4 \log (4)+\left (2 x^2-2 x^3\right ) \log (x)}{\log (x)}\right )\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-6*Log[4] + (6*x^2 - 9*x^3)*Log[x]^2 + (6*Log[4]*Log[x] + (3*x^2 - 3*x^3)*Log[x]^2)*Log[(4*Log[4] + (2*x^
2 - 2*x^3)*Log[x])/Log[x]]*Log[Log[(4*Log[4] + (2*x^2 - 2*x^3)*Log[x])/Log[x]]])/((-2*x^2*Log[4]*Log[x] + (-x^
4 + x^5)*Log[x]^2)*Log[(4*Log[4] + (2*x^2 - 2*x^3)*Log[x])/Log[x]]*Log[Log[(4*Log[4] + (2*x^2 - 2*x^3)*Log[x])
/Log[x]]]^2),x]

[Out]

-6*Defer[Int][Log[x]/((Log[16] + x^2*Log[x] - x^3*Log[x])*Log[-2*(-1 + x)*x^2 + Log[256]/Log[x]]*Log[Log[-2*(-
1 + x)*x^2 + Log[256]/Log[x]]]^2), x] - 3*Log[16]*Defer[Int][1/(x^2*Log[x]*(-Log[16] - x^2*Log[x] + x^3*Log[x]
)*Log[-2*(-1 + x)*x^2 + Log[256]/Log[x]]*Log[Log[-2*(-1 + x)*x^2 + Log[256]/Log[x]]]^2), x] - 9*Defer[Int][(x*
Log[x])/((-Log[16] - x^2*Log[x] + x^3*Log[x])*Log[-2*(-1 + x)*x^2 + Log[256]/Log[x]]*Log[Log[-2*(-1 + x)*x^2 +
Log[256]/Log[x]]]^2), x] - 3*Defer[Int][1/(x^2*Log[Log[-2*(-1 + x)*x^2 + Log[256]/Log[x]]]), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {6 \log (4)-\left (6 x^2-9 x^3\right ) \log ^2(x)-\left (6 \log (4) \log (x)+\left (3 x^2-3 x^3\right ) \log ^2(x)\right ) \log \left (\frac {4 \log (4)+\left (2 x^2-2 x^3\right ) \log (x)}{\log (x)}\right ) \log \left (\log \left (\frac {4 \log (4)+\left (2 x^2-2 x^3\right ) \log (x)}{\log (x)}\right )\right )}{x^2 \log (x) \left (2 \log (4)+x^2 \log (x)-x^3 \log (x)\right ) \log \left (\frac {4 \log (4)+\left (2 x^2-2 x^3\right ) \log (x)}{\log (x)}\right ) \log ^2\left (\log \left (\frac {4 \log (4)+\left (2 x^2-2 x^3\right ) \log (x)}{\log (x)}\right )\right )} \, dx\\ &=\int \frac {6 \log (4)+3 x^2 (-2+3 x) \log ^2(x)-\log (x) \left (6 \log (4)-3 (-1+x) x^2 \log (x)\right ) \log \left (-2 (-1+x) x^2+\frac {\log (256)}{\log (x)}\right ) \log \left (\log \left (-2 (-1+x) x^2+\frac {\log (256)}{\log (x)}\right )\right )}{x^2 \log (x) \left (\log (16)-(-1+x) x^2 \log (x)\right ) \log \left (-2 (-1+x) x^2+\frac {\log (256)}{\log (x)}\right ) \log ^2\left (\log \left (-2 (-1+x) x^2+\frac {\log (256)}{\log (x)}\right )\right )} \, dx\\ &=\int \left (-\frac {3 \left (2 \log (4)-2 x^2 \log ^2(x)+3 x^3 \log ^2(x)\right )}{x^2 \log (x) \left (-\log (16)-x^2 \log (x)+x^3 \log (x)\right ) \log \left (-2 (-1+x) x^2+\frac {\log (256)}{\log (x)}\right ) \log ^2\left (\log \left (-2 (-1+x) x^2+\frac {\log (256)}{\log (x)}\right )\right )}-\frac {3 \left (-2 \log (4)-x^2 \log (x)+x^3 \log (x)\right )}{x^2 \left (-\log (16)-x^2 \log (x)+x^3 \log (x)\right ) \log \left (\log \left (-2 (-1+x) x^2+\frac {\log (256)}{\log (x)}\right )\right )}\right ) \, dx\\ &=-\left (3 \int \frac {2 \log (4)-2 x^2 \log ^2(x)+3 x^3 \log ^2(x)}{x^2 \log (x) \left (-\log (16)-x^2 \log (x)+x^3 \log (x)\right ) \log \left (-2 (-1+x) x^2+\frac {\log (256)}{\log (x)}\right ) \log ^2\left (\log \left (-2 (-1+x) x^2+\frac {\log (256)}{\log (x)}\right )\right )} \, dx\right )-3 \int \frac {-2 \log (4)-x^2 \log (x)+x^3 \log (x)}{x^2 \left (-\log (16)-x^2 \log (x)+x^3 \log (x)\right ) \log \left (\log \left (-2 (-1+x) x^2+\frac {\log (256)}{\log (x)}\right )\right )} \, dx\\ &=-\left (3 \int \left (\frac {2 \log (x)}{\left (\log (16)+x^2 \log (x)-x^3 \log (x)\right ) \log \left (-2 (-1+x) x^2+\frac {\log (256)}{\log (x)}\right ) \log ^2\left (\log \left (-2 (-1+x) x^2+\frac {\log (256)}{\log (x)}\right )\right )}+\frac {\log (16)}{x^2 \log (x) \left (-\log (16)-x^2 \log (x)+x^3 \log (x)\right ) \log \left (-2 (-1+x) x^2+\frac {\log (256)}{\log (x)}\right ) \log ^2\left (\log \left (-2 (-1+x) x^2+\frac {\log (256)}{\log (x)}\right )\right )}+\frac {3 x \log (x)}{\left (-\log (16)-x^2 \log (x)+x^3 \log (x)\right ) \log \left (-2 (-1+x) x^2+\frac {\log (256)}{\log (x)}\right ) \log ^2\left (\log \left (-2 (-1+x) x^2+\frac {\log (256)}{\log (x)}\right )\right )}\right ) \, dx\right )-3 \int \frac {1}{x^2 \log \left (\log \left (-2 (-1+x) x^2+\frac {\log (256)}{\log (x)}\right )\right )} \, dx\\ &=-\left (3 \int \frac {1}{x^2 \log \left (\log \left (-2 (-1+x) x^2+\frac {\log (256)}{\log (x)}\right )\right )} \, dx\right )-6 \int \frac {\log (x)}{\left (\log (16)+x^2 \log (x)-x^3 \log (x)\right ) \log \left (-2 (-1+x) x^2+\frac {\log (256)}{\log (x)}\right ) \log ^2\left (\log \left (-2 (-1+x) x^2+\frac {\log (256)}{\log (x)}\right )\right )} \, dx-9 \int \frac {x \log (x)}{\left (-\log (16)-x^2 \log (x)+x^3 \log (x)\right ) \log \left (-2 (-1+x) x^2+\frac {\log (256)}{\log (x)}\right ) \log ^2\left (\log \left (-2 (-1+x) x^2+\frac {\log (256)}{\log (x)}\right )\right )} \, dx-(3 \log (16)) \int \frac {1}{x^2 \log (x) \left (-\log (16)-x^2 \log (x)+x^3 \log (x)\right ) \log \left (-2 (-1+x) x^2+\frac {\log (256)}{\log (x)}\right ) \log ^2\left (\log \left (-2 (-1+x) x^2+\frac {\log (256)}{\log (x)}\right )\right )} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [B]  time = 0.22, size = 89, normalized size = 2.78 \begin {gather*} -\frac {3 \left (\log (256)-2 (-1+x) x^2 \log (x)\right ) \left (\log (16)+x^2 (-2+3 x) \log ^2(x)\right )}{x \left (-2 \log (4)+(-1+x) x^2 \log (x)\right ) \left (\log (256)+2 x^2 (-2+3 x) \log ^2(x)\right ) \log \left (\log \left (-2 (-1+x) x^2+\frac {\log (256)}{\log (x)}\right )\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-6*Log[4] + (6*x^2 - 9*x^3)*Log[x]^2 + (6*Log[4]*Log[x] + (3*x^2 - 3*x^3)*Log[x]^2)*Log[(4*Log[4] +
(2*x^2 - 2*x^3)*Log[x])/Log[x]]*Log[Log[(4*Log[4] + (2*x^2 - 2*x^3)*Log[x])/Log[x]]])/((-2*x^2*Log[4]*Log[x]
+ (-x^4 + x^5)*Log[x]^2)*Log[(4*Log[4] + (2*x^2 - 2*x^3)*Log[x])/Log[x]]*Log[Log[(4*Log[4] + (2*x^2 - 2*x^3)*L
og[x])/Log[x]]]^2),x]

[Out]

(-3*(Log[256] - 2*(-1 + x)*x^2*Log[x])*(Log[16] + x^2*(-2 + 3*x)*Log[x]^2))/(x*(-2*Log[4] + (-1 + x)*x^2*Log[x
])*(Log[256] + 2*x^2*(-2 + 3*x)*Log[x]^2)*Log[Log[-2*(-1 + x)*x^2 + Log[256]/Log[x]]])

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fricas [A]  time = 0.73, size = 32, normalized size = 1.00 \begin {gather*} \frac {3}{x \log \left (\log \left (-\frac {2 \, {\left ({\left (x^{3} - x^{2}\right )} \log \relax (x) - 4 \, \log \relax (2)\right )}}{\log \relax (x)}\right )\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-3*x^3+3*x^2)*log(x)^2+12*log(2)*log(x))*log(((-2*x^3+2*x^2)*log(x)+8*log(2))/log(x))*log(log(((-
2*x^3+2*x^2)*log(x)+8*log(2))/log(x)))+(-9*x^3+6*x^2)*log(x)^2-12*log(2))/((x^5-x^4)*log(x)^2-4*x^2*log(2)*log
(x))/log(((-2*x^3+2*x^2)*log(x)+8*log(2))/log(x))/log(log(((-2*x^3+2*x^2)*log(x)+8*log(2))/log(x)))^2,x, algor
ithm="fricas")

[Out]

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

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giac [A]  time = 0.71, size = 35, normalized size = 1.09 \begin {gather*} \frac {3}{x \log \left (\log \relax (2) + \log \left (-x^{3} \log \relax (x) + x^{2} \log \relax (x) + 4 \, \log \relax (2)\right ) - \log \left (\log \relax (x)\right )\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-3*x^3+3*x^2)*log(x)^2+12*log(2)*log(x))*log(((-2*x^3+2*x^2)*log(x)+8*log(2))/log(x))*log(log(((-
2*x^3+2*x^2)*log(x)+8*log(2))/log(x)))+(-9*x^3+6*x^2)*log(x)^2-12*log(2))/((x^5-x^4)*log(x)^2-4*x^2*log(2)*log
(x))/log(((-2*x^3+2*x^2)*log(x)+8*log(2))/log(x))/log(log(((-2*x^3+2*x^2)*log(x)+8*log(2))/log(x)))^2,x, algor
ithm="giac")

[Out]

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

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maple [C]  time = 0.16, size = 151, normalized size = 4.72

method	result	size
risch	\(\frac {3}{x \ln \left (3 \ln \relax (2)-\ln \left (\ln \relax (x )\right )+\ln \left (-\frac {x^{3} \ln \relax (x )}{4}+\frac {x^{2} \ln \relax (x )}{4}+\ln \relax (2)\right )-\frac {i \pi \,\mathrm {csgn}\left (\frac {i \left (-\frac {x^{3} \ln \relax (x )}{4}+\frac {x^{2} \ln \relax (x )}{4}+\ln \relax (2)\right )}{\ln \relax (x )}\right ) \left (-\mathrm {csgn}\left (\frac {i \left (-\frac {x^{3} \ln \relax (x )}{4}+\frac {x^{2} \ln \relax (x )}{4}+\ln \relax (2)\right )}{\ln \relax (x )}\right )+\mathrm {csgn}\left (\frac {i}{\ln \relax (x )}\right )\right ) \left (-\mathrm {csgn}\left (\frac {i \left (-\frac {x^{3} \ln \relax (x )}{4}+\frac {x^{2} \ln \relax (x )}{4}+\ln \relax (2)\right )}{\ln \relax (x )}\right )+\mathrm {csgn}\left (i \left (-\frac {x^{3} \ln \relax (x )}{4}+\frac {x^{2} \ln \relax (x )}{4}+\ln \relax (2)\right )\right )\right )}{2}\right )}\)	\(151\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((-3*x^3+3*x^2)*ln(x)^2+12*ln(2)*ln(x))*ln(((-2*x^3+2*x^2)*ln(x)+8*ln(2))/ln(x))*ln(ln(((-2*x^3+2*x^2)*ln
(x)+8*ln(2))/ln(x)))+(-9*x^3+6*x^2)*ln(x)^2-12*ln(2))/((x^5-x^4)*ln(x)^2-4*x^2*ln(2)*ln(x))/ln(((-2*x^3+2*x^2)
*ln(x)+8*ln(2))/ln(x))/ln(ln(((-2*x^3+2*x^2)*ln(x)+8*ln(2))/ln(x)))^2,x,method=_RETURNVERBOSE)

[Out]

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

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maxima [C]  time = 0.84, size = 37, normalized size = 1.16 \begin {gather*} \frac {3}{x \log \left (i \, \pi + \log \relax (2) + \log \left ({\left (x^{3} - x^{2}\right )} \log \relax (x) - 4 \, \log \relax (2)\right ) - \log \left (\log \relax (x)\right )\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-3*x^3+3*x^2)*log(x)^2+12*log(2)*log(x))*log(((-2*x^3+2*x^2)*log(x)+8*log(2))/log(x))*log(log(((-
2*x^3+2*x^2)*log(x)+8*log(2))/log(x)))+(-9*x^3+6*x^2)*log(x)^2-12*log(2))/((x^5-x^4)*log(x)^2-4*x^2*log(2)*log
(x))/log(((-2*x^3+2*x^2)*log(x)+8*log(2))/log(x))/log(log(((-2*x^3+2*x^2)*log(x)+8*log(2))/log(x)))^2,x, algor
ithm="maxima")

[Out]

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

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mupad [F(-1)]  time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \text {Hanged} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(log(x)^2*(6*x^2 - 9*x^3) - 12*log(2) + log((8*log(2) + log(x)*(2*x^2 - 2*x^3))/log(x))*log(log((8*log(2)
+ log(x)*(2*x^2 - 2*x^3))/log(x)))*(log(x)^2*(3*x^2 - 3*x^3) + 12*log(2)*log(x)))/(log((8*log(2) + log(x)*(2*
x^2 - 2*x^3))/log(x))*log(log((8*log(2) + log(x)*(2*x^2 - 2*x^3))/log(x)))^2*(log(x)^2*(x^4 - x^5) + 4*x^2*log
(2)*log(x))),x)

[Out]

\text{Hanged}

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sympy [A]  time = 7.16, size = 27, normalized size = 0.84 \begin {gather*} \frac {3}{x \log {\left (\log {\left (\frac {\left (- 2 x^{3} + 2 x^{2}\right ) \log {\relax (x )} + 8 \log {\relax (2 )}}{\log {\relax (x )}} \right )} \right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-3*x**3+3*x**2)*ln(x)**2+12*ln(2)*ln(x))*ln(((-2*x**3+2*x**2)*ln(x)+8*ln(2))/ln(x))*ln(ln(((-2*x*
*3+2*x**2)*ln(x)+8*ln(2))/ln(x)))+(-9*x**3+6*x**2)*ln(x)**2-12*ln(2))/((x**5-x**4)*ln(x)**2-4*x**2*ln(2)*ln(x)
)/ln(((-2*x**3+2*x**2)*ln(x)+8*ln(2))/ln(x))/ln(ln(((-2*x**3+2*x**2)*ln(x)+8*ln(2))/ln(x)))**2,x)

[Out]

3/(x*log(log(((-2*x**3 + 2*x**2)*log(x) + 8*log(2))/log(x))))

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