3.8.62 \(\int \frac {-8 x-2 x \log (4)+2 \log (\frac {3}{x})+(4 x+x \log (4)) \log ^2(\frac {3}{x})+(-2+\log ^2(\frac {3}{x})) \log (\frac {1}{6} e^{-4 x-x \log (4)} (-2+\log ^2(\frac {3}{x})))}{(-2 x+x \log ^2(\frac {3}{x})) \log (\frac {1}{6} e^{-4 x-x \log (4)} (-2+\log ^2(\frac {3}{x})))} \, dx\)

Optimal. Leaf size=29 \[ \log \left (\frac {x}{\log \left (\frac {1}{6} e^{-x (4+\log (4))} \left (-2+\log ^2\left (\frac {3}{x}\right )\right )\right )}\right ) \]

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Rubi [F]  time = 2.79, 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 {-8 x-2 x \log (4)+2 \log \left (\frac {3}{x}\right )+(4 x+x \log (4)) \log ^2\left (\frac {3}{x}\right )+\left (-2+\log ^2\left (\frac {3}{x}\right )\right ) \log \left (\frac {1}{6} e^{-4 x-x \log (4)} \left (-2+\log ^2\left (\frac {3}{x}\right )\right )\right )}{\left (-2 x+x \log ^2\left (\frac {3}{x}\right )\right ) \log \left (\frac {1}{6} e^{-4 x-x \log (4)} \left (-2+\log ^2\left (\frac {3}{x}\right )\right )\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-8*x - 2*x*Log[4] + 2*Log[3/x] + (4*x + x*Log[4])*Log[3/x]^2 + (-2 + Log[3/x]^2)*Log[(E^(-4*x - x*Log[4])
*(-2 + Log[3/x]^2))/6])/((-2*x + x*Log[3/x]^2)*Log[(E^(-4*x - x*Log[4])*(-2 + Log[3/x]^2))/6]),x]

[Out]

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

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {x (-8-2 \log (4))+2 \log \left (\frac {3}{x}\right )+(4 x+x \log (4)) \log ^2\left (\frac {3}{x}\right )+\left (-2+\log ^2\left (\frac {3}{x}\right )\right ) \log \left (\frac {1}{6} e^{-4 x-x \log (4)} \left (-2+\log ^2\left (\frac {3}{x}\right )\right )\right )}{\left (-2 x+x \log ^2\left (\frac {3}{x}\right )\right ) \log \left (\frac {1}{6} e^{-4 x-x \log (4)} \left (-2+\log ^2\left (\frac {3}{x}\right )\right )\right )} \, dx\\ &=\int \frac {x (-8-2 \log (4))+2 \log \left (\frac {3}{x}\right )+(4 x+x \log (4)) \log ^2\left (\frac {3}{x}\right )+\left (-2+\log ^2\left (\frac {3}{x}\right )\right ) \log \left (\frac {1}{6} e^{-4 x-x \log (4)} \left (-2+\log ^2\left (\frac {3}{x}\right )\right )\right )}{x \left (-2+\log ^2\left (\frac {3}{x}\right )\right ) \log \left (\frac {1}{6} e^{-4 x-x \log (4)} \left (-2+\log ^2\left (\frac {3}{x}\right )\right )\right )} \, dx\\ &=\int \frac {-x (-8-2 \log (4))-2 \log \left (\frac {3}{x}\right )-(4 x+x \log (4)) \log ^2\left (\frac {3}{x}\right )-\left (-2+\log ^2\left (\frac {3}{x}\right )\right ) \log \left (\frac {1}{6} e^{-4 x-x \log (4)} \left (-2+\log ^2\left (\frac {3}{x}\right )\right )\right )}{x \left (2-\log ^2\left (\frac {3}{x}\right )\right ) \log \left (\frac {1}{6} e^{-x (4+\log (4))} \left (-2+\log ^2\left (\frac {3}{x}\right )\right )\right )} \, dx\\ &=\int \left (\frac {1}{x}+\frac {8 x \left (1+\frac {\log (2)}{2}\right )-2 \log \left (\frac {3}{x}\right )-4 x \left (1+\frac {\log (2)}{2}\right ) \log ^2\left (\frac {3}{x}\right )}{x \left (2-\log ^2\left (\frac {3}{x}\right )\right ) \log \left (\frac {1}{6} e^{-x (4+\log (4))} \left (-2+\log ^2\left (\frac {3}{x}\right )\right )\right )}\right ) \, dx\\ &=\log (x)+\int \frac {8 x \left (1+\frac {\log (2)}{2}\right )-2 \log \left (\frac {3}{x}\right )-4 x \left (1+\frac {\log (2)}{2}\right ) \log ^2\left (\frac {3}{x}\right )}{x \left (2-\log ^2\left (\frac {3}{x}\right )\right ) \log \left (\frac {1}{6} e^{-x (4+\log (4))} \left (-2+\log ^2\left (\frac {3}{x}\right )\right )\right )} \, dx\\ &=\log (x)+\int \frac {2 \left (4 x \left (1+\frac {\log (2)}{2}\right )-\log \left (\frac {3}{x}\right )-2 x \left (1+\frac {\log (2)}{2}\right ) \log ^2\left (\frac {3}{x}\right )\right )}{x \left (2-\log ^2\left (\frac {3}{x}\right )\right ) \log \left (\frac {1}{6} e^{-x (4+\log (4))} \left (-2+\log ^2\left (\frac {3}{x}\right )\right )\right )} \, dx\\ &=\log (x)+2 \int \frac {4 x \left (1+\frac {\log (2)}{2}\right )-\log \left (\frac {3}{x}\right )-2 x \left (1+\frac {\log (2)}{2}\right ) \log ^2\left (\frac {3}{x}\right )}{x \left (2-\log ^2\left (\frac {3}{x}\right )\right ) \log \left (\frac {1}{6} e^{-x (4+\log (4))} \left (-2+\log ^2\left (\frac {3}{x}\right )\right )\right )} \, dx\\ &=\log (x)+2 \int \left (-\frac {2 (2+\log (2))}{\left (-2+\log ^2\left (\frac {3}{x}\right )\right ) \log \left (\frac {1}{6} e^{-x (4+\log (4))} \left (-2+\log ^2\left (\frac {3}{x}\right )\right )\right )}+\frac {\log \left (\frac {3}{x}\right )}{x \left (-2+\log ^2\left (\frac {3}{x}\right )\right ) \log \left (\frac {1}{6} e^{-x (4+\log (4))} \left (-2+\log ^2\left (\frac {3}{x}\right )\right )\right )}+\frac {(2+\log (2)) \log ^2\left (\frac {3}{x}\right )}{\left (-2+\log ^2\left (\frac {3}{x}\right )\right ) \log \left (\frac {1}{6} e^{-x (4+\log (4))} \left (-2+\log ^2\left (\frac {3}{x}\right )\right )\right )}\right ) \, dx\\ &=\log (x)+2 \int \frac {\log \left (\frac {3}{x}\right )}{x \left (-2+\log ^2\left (\frac {3}{x}\right )\right ) \log \left (\frac {1}{6} e^{-x (4+\log (4))} \left (-2+\log ^2\left (\frac {3}{x}\right )\right )\right )} \, dx+(2 (2+\log (2))) \int \frac {\log ^2\left (\frac {3}{x}\right )}{\left (-2+\log ^2\left (\frac {3}{x}\right )\right ) \log \left (\frac {1}{6} e^{-x (4+\log (4))} \left (-2+\log ^2\left (\frac {3}{x}\right )\right )\right )} \, dx-(4 (2+\log (2))) \int \frac {1}{\left (-2+\log ^2\left (\frac {3}{x}\right )\right ) \log \left (\frac {1}{6} e^{-x (4+\log (4))} \left (-2+\log ^2\left (\frac {3}{x}\right )\right )\right )} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.07, size = 45, normalized size = 1.55 \begin {gather*} \log (x)-\log \left (-4 x-x \log (4)+x (4+\log (4))+\log \left (\frac {1}{6} e^{-x (4+\log (4))} \left (-2+\log ^2\left (\frac {3}{x}\right )\right )\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-8*x - 2*x*Log[4] + 2*Log[3/x] + (4*x + x*Log[4])*Log[3/x]^2 + (-2 + Log[3/x]^2)*Log[(E^(-4*x - x*L
og[4])*(-2 + Log[3/x]^2))/6])/((-2*x + x*Log[3/x]^2)*Log[(E^(-4*x - x*Log[4])*(-2 + Log[3/x]^2))/6]),x]

[Out]

Log[x] - Log[-4*x - x*Log[4] + x*(4 + Log[4]) + Log[(-2 + Log[3/x]^2)/(6*E^(x*(4 + Log[4])))]]

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fricas [A]  time = 0.80, size = 46, normalized size = 1.59 \begin {gather*} -\log \left (\frac {3}{x}\right ) - \log \left (\log \left (\frac {1}{6} \, e^{\left (-2 \, x \log \relax (2) - 4 \, x\right )} \log \left (\frac {3}{x}\right )^{2} - \frac {1}{3} \, e^{\left (-2 \, x \log \relax (2) - 4 \, x\right )}\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((log(3/x)^2-2)*log(1/6*(log(3/x)^2-2)/exp(2*x*log(2)+4*x))+(2*x*log(2)+4*x)*log(3/x)^2+2*log(3/x)-4
*x*log(2)-8*x)/(x*log(3/x)^2-2*x)/log(1/6*(log(3/x)^2-2)/exp(2*x*log(2)+4*x)),x, algorithm="fricas")

[Out]

-log(3/x) - log(log(1/6*e^(-2*x*log(2) - 4*x)*log(3/x)^2 - 1/3*e^(-2*x*log(2) - 4*x)))

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giac [A]  time = 0.38, size = 38, normalized size = 1.31 \begin {gather*} -\log \left (2 \, x \log \relax (2) + 4 \, x + \log \relax (3) + \log \relax (2) - \log \left (\log \relax (3)^{2} - 2 \, \log \relax (3) \log \relax (x) + \log \relax (x)^{2} - 2\right )\right ) + \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((log(3/x)^2-2)*log(1/6*(log(3/x)^2-2)/exp(2*x*log(2)+4*x))+(2*x*log(2)+4*x)*log(3/x)^2+2*log(3/x)-4
*x*log(2)-8*x)/(x*log(3/x)^2-2*x)/log(1/6*(log(3/x)^2-2)/exp(2*x*log(2)+4*x)),x, algorithm="giac")

[Out]

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

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maple [C]  time = 0.64, size = 317, normalized size = 10.93

method	result	size
risch	\(\ln \relax (x )-\ln \left (\ln \left (4^{x} {\mathrm e}^{4 x}\right )+\frac {i \left (2 \pi \mathrm {csgn}\left (i 4^{-x} {\mathrm e}^{-4 x} \left (-8-8 \ln \relax (3) \ln \relax (x )+4 \ln \relax (x )^{2}+4 \ln \relax (3)^{2}\right )\right )^{2}+\pi \,\mathrm {csgn}\left (i \left (-8-8 \ln \relax (3) \ln \relax (x )+4 \ln \relax (x )^{2}+4 \ln \relax (3)^{2}\right )\right ) \mathrm {csgn}\left (i 4^{-x} {\mathrm e}^{-4 x}\right ) \mathrm {csgn}\left (i 4^{-x} {\mathrm e}^{-4 x} \left (-8-8 \ln \relax (3) \ln \relax (x )+4 \ln \relax (x )^{2}+4 \ln \relax (3)^{2}\right )\right )+\pi \,\mathrm {csgn}\left (i \left (-8-8 \ln \relax (3) \ln \relax (x )+4 \ln \relax (x )^{2}+4 \ln \relax (3)^{2}\right )\right ) \mathrm {csgn}\left (i 4^{-x} {\mathrm e}^{-4 x} \left (-8-8 \ln \relax (3) \ln \relax (x )+4 \ln \relax (x )^{2}+4 \ln \relax (3)^{2}\right )\right )^{2}-\pi \,\mathrm {csgn}\left (i 4^{-x} {\mathrm e}^{-4 x}\right ) \mathrm {csgn}\left (i 4^{-x} {\mathrm e}^{-4 x} \left (-8-8 \ln \relax (3) \ln \relax (x )+4 \ln \relax (x )^{2}+4 \ln \relax (3)^{2}\right )\right )^{2}+\pi \mathrm {csgn}\left (i 4^{-x} {\mathrm e}^{-4 x} \left (-8-8 \ln \relax (3) \ln \relax (x )+4 \ln \relax (x )^{2}+4 \ln \relax (3)^{2}\right )\right )^{3}-6 i \ln \relax (2)-2 i \ln \relax (3)+2 i \ln \left (8+8 \ln \relax (3) \ln \relax (x )-4 \ln \relax (x )^{2}-4 \ln \relax (3)^{2}\right )-2 \pi \right )}{2}\right )\)	\(317\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((ln(3/x)^2-2)*ln(1/6*(ln(3/x)^2-2)/exp(2*x*ln(2)+4*x))+(2*x*ln(2)+4*x)*ln(3/x)^2+2*ln(3/x)-4*x*ln(2)-8*x)
/(x*ln(3/x)^2-2*x)/ln(1/6*(ln(3/x)^2-2)/exp(2*x*ln(2)+4*x)),x,method=_RETURNVERBOSE)

[Out]

ln(x)-ln(ln(4^x*exp(4*x))+1/2*I*(2*Pi*csgn(I/(4^x)*exp(-4*x)*(-8-8*ln(3)*ln(x)+4*ln(x)^2+4*ln(3)^2))^2+Pi*csgn
(I*(-8-8*ln(3)*ln(x)+4*ln(x)^2+4*ln(3)^2))*csgn(I/(4^x)*exp(-4*x))*csgn(I/(4^x)*exp(-4*x)*(-8-8*ln(3)*ln(x)+4*
ln(x)^2+4*ln(3)^2))+Pi*csgn(I*(-8-8*ln(3)*ln(x)+4*ln(x)^2+4*ln(3)^2))*csgn(I/(4^x)*exp(-4*x)*(-8-8*ln(3)*ln(x)
+4*ln(x)^2+4*ln(3)^2))^2-Pi*csgn(I/(4^x)*exp(-4*x))*csgn(I/(4^x)*exp(-4*x)*(-8-8*ln(3)*ln(x)+4*ln(x)^2+4*ln(3)
^2))^2+Pi*csgn(I/(4^x)*exp(-4*x)*(-8-8*ln(3)*ln(x)+4*ln(x)^2+4*ln(3)^2))^3-6*I*ln(2)-2*I*ln(3)+2*I*ln(8+8*ln(3
)*ln(x)-4*ln(x)^2-4*ln(3)^2)-2*Pi))

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maxima [A]  time = 0.74, size = 39, normalized size = 1.34 \begin {gather*} -\log \left (-2 \, x {\left (\log \relax (2) + 2\right )} - \log \relax (3) - \log \relax (2) + \log \left (\log \relax (3)^{2} - 2 \, \log \relax (3) \log \relax (x) + \log \relax (x)^{2} - 2\right )\right ) + \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((log(3/x)^2-2)*log(1/6*(log(3/x)^2-2)/exp(2*x*log(2)+4*x))+(2*x*log(2)+4*x)*log(3/x)^2+2*log(3/x)-4
*x*log(2)-8*x)/(x*log(3/x)^2-2*x)/log(1/6*(log(3/x)^2-2)/exp(2*x*log(2)+4*x)),x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 1.40, size = 31, normalized size = 1.07 \begin {gather*} \ln \relax (x)-\ln \left (\ln \left (\frac {{\mathrm {e}}^{-4\,x}\,\left (\frac {{\ln \left (\frac {3}{x}\right )}^2}{6}-\frac {1}{3}\right )}{2^{2\,x}}\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(2*log(3/x) - 8*x + log(exp(- 4*x - 2*x*log(2))*(log(3/x)^2/6 - 1/3))*(log(3/x)^2 - 2) - 4*x*log(2) + log
(3/x)^2*(4*x + 2*x*log(2)))/(log(exp(- 4*x - 2*x*log(2))*(log(3/x)^2/6 - 1/3))*(2*x - x*log(3/x)^2)),x)

[Out]

log(x) - log(log((exp(-4*x)*(log(3/x)^2/6 - 1/3))/2^(2*x)))

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sympy [A]  time = 0.98, size = 31, normalized size = 1.07 \begin {gather*} \log {\relax (x )} - \log {\left (\log {\left (\left (\frac {\log {\left (\frac {3}{x} \right )}^{2}}{6} - \frac {1}{3}\right ) e^{- 4 x - 2 x \log {\relax (2 )}} \right )} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((ln(3/x)**2-2)*ln(1/6*(ln(3/x)**2-2)/exp(2*x*ln(2)+4*x))+(2*x*ln(2)+4*x)*ln(3/x)**2+2*ln(3/x)-4*x*l
n(2)-8*x)/(x*ln(3/x)**2-2*x)/ln(1/6*(ln(3/x)**2-2)/exp(2*x*ln(2)+4*x)),x)

[Out]

log(x) - log(log((log(3/x)**2/6 - 1/3)*exp(-4*x - 2*x*log(2))))

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