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

Optimal. Leaf size=24 \[ 5+\log \left (\left (-3 x+\frac {\log (x) \left (x-\log \left (\log \left (x^2\right )\right )\right )}{x^2}\right )^2\right ) \]

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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 {4 \log (x)+\left (-2 x+6 x^3+2 x \log (x)\right ) \log \left (x^2\right )+(2-4 \log (x)) \log \left (x^2\right ) \log \left (\log \left (x^2\right )\right )}{\left (3 x^4-x^2 \log (x)\right ) \log \left (x^2\right )+x \log (x) \log \left (x^2\right ) \log \left (\log \left (x^2\right )\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

-4*Log[x] + 2*Log[Log[x]] + 6*Defer[Int][x^2/(Log[x]*(-3*x^3 + x*Log[x] - Log[x]*Log[Log[x^2]])), x] + 2*Defer
[Int][Log[x]/(-3*x^3 + x*Log[x] - Log[x]*Log[Log[x^2]]), x] + 18*Defer[Int][x^2/(3*x^3 - x*Log[x] + Log[x]*Log
[Log[x^2]]), x] + 4*Defer[Int][Log[x]/(x*Log[x^2]*(3*x^3 - x*Log[x] + Log[x]*Log[Log[x^2]])), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {4 \log (x)+\left (-2 x+6 x^3+2 x \log (x)\right ) \log \left (x^2\right )+(2-4 \log (x)) \log \left (x^2\right ) \log \left (\log \left (x^2\right )\right )}{x \log \left (x^2\right ) \left (3 x^3-x \log (x)+\log (x) \log \left (\log \left (x^2\right )\right )\right )} \, dx\\ &=\int \left (-\frac {2 (-1+2 \log (x))}{x \log (x)}+\frac {2 \left (2 \log ^2(x)-3 x^3 \log \left (x^2\right )+9 x^3 \log (x) \log \left (x^2\right )-x \log ^2(x) \log \left (x^2\right )\right )}{x \log (x) \log \left (x^2\right ) \left (3 x^3-x \log (x)+\log (x) \log \left (\log \left (x^2\right )\right )\right )}\right ) \, dx\\ &=-\left (2 \int \frac {-1+2 \log (x)}{x \log (x)} \, dx\right )+2 \int \frac {2 \log ^2(x)-3 x^3 \log \left (x^2\right )+9 x^3 \log (x) \log \left (x^2\right )-x \log ^2(x) \log \left (x^2\right )}{x \log (x) \log \left (x^2\right ) \left (3 x^3-x \log (x)+\log (x) \log \left (\log \left (x^2\right )\right )\right )} \, dx\\ &=2 \int \left (\frac {3 x^2}{\log (x) \left (-3 x^3+x \log (x)-\log (x) \log \left (\log \left (x^2\right )\right )\right )}+\frac {\log (x)}{-3 x^3+x \log (x)-\log (x) \log \left (\log \left (x^2\right )\right )}+\frac {9 x^2}{3 x^3-x \log (x)+\log (x) \log \left (\log \left (x^2\right )\right )}+\frac {2 \log (x)}{x \log \left (x^2\right ) \left (3 x^3-x \log (x)+\log (x) \log \left (\log \left (x^2\right )\right )\right )}\right ) \, dx-2 \operatorname {Subst}\left (\int \frac {-1+2 x}{x} \, dx,x,\log (x)\right )\\ &=2 \int \frac {\log (x)}{-3 x^3+x \log (x)-\log (x) \log \left (\log \left (x^2\right )\right )} \, dx-2 \operatorname {Subst}\left (\int \left (2-\frac {1}{x}\right ) \, dx,x,\log (x)\right )+4 \int \frac {\log (x)}{x \log \left (x^2\right ) \left (3 x^3-x \log (x)+\log (x) \log \left (\log \left (x^2\right )\right )\right )} \, dx+6 \int \frac {x^2}{\log (x) \left (-3 x^3+x \log (x)-\log (x) \log \left (\log \left (x^2\right )\right )\right )} \, dx+18 \int \frac {x^2}{3 x^3-x \log (x)+\log (x) \log \left (\log \left (x^2\right )\right )} \, dx\\ &=-4 \log (x)+2 \log (\log (x))+2 \int \frac {\log (x)}{-3 x^3+x \log (x)-\log (x) \log \left (\log \left (x^2\right )\right )} \, dx+4 \int \frac {\log (x)}{x \log \left (x^2\right ) \left (3 x^3-x \log (x)+\log (x) \log \left (\log \left (x^2\right )\right )\right )} \, dx+6 \int \frac {x^2}{\log (x) \left (-3 x^3+x \log (x)-\log (x) \log \left (\log \left (x^2\right )\right )\right )} \, dx+18 \int \frac {x^2}{3 x^3-x \log (x)+\log (x) \log \left (\log \left (x^2\right )\right )} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [B]  time = 0.39, size = 63, normalized size = 2.62 \begin {gather*} 2 \left (-2 \log (x)+\log \left (-6 x^3+2 x \left (\log (x)-\frac {\log \left (x^2\right )}{2}\right )+x \log \left (x^2\right )-2 \left (\log (x)-\frac {\log \left (x^2\right )}{2}\right ) \log \left (\log \left (x^2\right )\right )-\log \left (x^2\right ) \log \left (\log \left (x^2\right )\right )\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

2*(-2*Log[x] + Log[-6*x^3 + 2*x*(Log[x] - Log[x^2]/2) + x*Log[x^2] - 2*(Log[x] - Log[x^2]/2)*Log[Log[x^2]] - L
og[x^2]*Log[Log[x^2]]])

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fricas [A]  time = 0.69, size = 39, normalized size = 1.62 \begin {gather*} -4 \, \log \relax (x) + 2 \, \log \left (2 \, \log \relax (x)\right ) + 2 \, \log \left (\frac {3 \, x^{3} - x \log \relax (x) + \log \relax (x) \log \left (2 \, \log \relax (x)\right )}{\log \relax (x)}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-4*log(x)+2)*log(x^2)*log(log(x^2))+(2*x*log(x)+6*x^3-2*x)*log(x^2)+4*log(x))/(x*log(x)*log(x^2)*l
og(log(x^2))+(-x^2*log(x)+3*x^4)*log(x^2)),x, algorithm="fricas")

[Out]

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

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int -\frac {2 \, {\left ({\left (2 \, \log \relax (x) - 1\right )} \log \left (x^{2}\right ) \log \left (\log \left (x^{2}\right )\right ) - {\left (3 \, x^{3} + x \log \relax (x) - x\right )} \log \left (x^{2}\right ) - 2 \, \log \relax (x)\right )}}{x \log \left (x^{2}\right ) \log \relax (x) \log \left (\log \left (x^{2}\right )\right ) + {\left (3 \, x^{4} - x^{2} \log \relax (x)\right )} \log \left (x^{2}\right )}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-4*log(x)+2)*log(x^2)*log(log(x^2))+(2*x*log(x)+6*x^3-2*x)*log(x^2)+4*log(x))/(x*log(x)*log(x^2)*l
og(log(x^2))+(-x^2*log(x)+3*x^4)*log(x^2)),x, algorithm="giac")

[Out]

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

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maple [C]  time = 0.10, size = 65, normalized size = 2.71

method	result	size
risch	\(-4 \ln \relax (x )+2 \ln \left (\ln \relax (x )\right )+2 \ln \left (\ln \left (2 \ln \relax (x )-\frac {i \pi \,\mathrm {csgn}\left (i x^{2}\right ) \left (-\mathrm {csgn}\left (i x^{2}\right )+\mathrm {csgn}\left (i x \right )\right )^{2}}{2}\right )+\frac {x \left (3 x^{2}-\ln \relax (x )\right )}{\ln \relax (x )}\right )\)	\(65\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.58, size = 40, normalized size = 1.67 \begin {gather*} -4 \, \log \relax (x) + 2 \, \log \left (\frac {3 \, x^{3} - {\left (x - \log \relax (2)\right )} \log \relax (x) + \log \relax (x) \log \left (\log \relax (x)\right )}{\log \relax (x)}\right ) + 2 \, \log \left (\log \relax (x)\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-4*log(x)+2)*log(x^2)*log(log(x^2))+(2*x*log(x)+6*x^3-2*x)*log(x^2)+4*log(x))/(x*log(x)*log(x^2)*l
og(log(x^2))+(-x^2*log(x)+3*x^4)*log(x^2)),x, algorithm="maxima")

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \int -\frac {4\,\ln \relax (x)+\ln \left (x^2\right )\,\left (2\,x\,\ln \relax (x)-2\,x+6\,x^3\right )-\ln \left (x^2\right )\,\ln \left (\ln \left (x^2\right )\right )\,\left (4\,\ln \relax (x)-2\right )}{\ln \left (x^2\right )\,\left (x^2\,\ln \relax (x)-3\,x^4\right )-x\,\ln \left (x^2\right )\,\ln \left (\ln \left (x^2\right )\right )\,\ln \relax (x)} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(4*log(x) + log(x^2)*(2*x*log(x) - 2*x + 6*x^3) - log(x^2)*log(log(x^2))*(4*log(x) - 2))/(log(x^2)*(x^2*l
og(x) - 3*x^4) - x*log(x^2)*log(log(x^2))*log(x)),x)

[Out]

int(-(4*log(x) + log(x^2)*(2*x*log(x) - 2*x + 6*x^3) - log(x^2)*log(log(x^2))*(4*log(x) - 2))/(log(x^2)*(x^2*l
og(x) - 3*x^4) - x*log(x^2)*log(log(x^2))*log(x)), x)

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sympy [A]  time = 0.44, size = 34, normalized size = 1.42 \begin {gather*} - 4 \log {\relax (x )} + 2 \log {\left (\frac {3 x^{3} - x \log {\relax (x )}}{\log {\relax (x )}} + \log {\left (2 \log {\relax (x )} \right )} \right )} + 2 \log {\left (\log {\relax (x )} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-4*ln(x)+2)*ln(x**2)*ln(ln(x**2))+(2*x*ln(x)+6*x**3-2*x)*ln(x**2)+4*ln(x))/(x*ln(x)*ln(x**2)*ln(ln
(x**2))+(-x**2*ln(x)+3*x**4)*ln(x**2)),x)

[Out]

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

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