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

Optimal. Leaf size=20 \[ \frac {1+\log \left (-\frac {2}{3}+x^2+\frac {2}{\log (x)}\right )}{x} \]

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Rubi [F]  time = 1.46, 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 {-6-6 \log (x)+\left (2+3 x^2\right ) \log ^2(x)+\left (-6 \log (x)+\left (2-3 x^2\right ) \log ^2(x)\right ) \log \left (\frac {6+\left (-2+3 x^2\right ) \log (x)}{3 \log (x)}\right )}{6 x^2 \log (x)+\left (-2 x^2+3 x^4\right ) \log ^2(x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

x^(-1) - Sqrt[6]*ArcTanh[Sqrt[3/2]*x] - ExpIntegralEi[-Log[x]] + 3*Defer[Int][(6 - 2*Log[x] + 3*x^2*Log[x])^(-
1), x] - 2*Defer[Int][1/(x^2*(6 - 2*Log[x] + 3*x^2*Log[x])), x] + 9*Sqrt[2]*Defer[Int][1/((Sqrt[2] - Sqrt[3]*x
)*(6 - 2*Log[x] + 3*x^2*Log[x])), x] + 9*Sqrt[2]*Defer[Int][1/((Sqrt[2] + Sqrt[3]*x)*(6 - 2*Log[x] + 3*x^2*Log
[x])), x] - Defer[Int][Log[-2/3 + x^2 + 2/Log[x]]/x^2, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-6-6 \log (x)+\left (2+3 x^2\right ) \log ^2(x)+\left (-6 \log (x)+\left (2-3 x^2\right ) \log ^2(x)\right ) \log \left (\frac {6+\left (-2+3 x^2\right ) \log (x)}{3 \log (x)}\right )}{x^2 \log (x) \left (6-2 \log (x)+3 x^2 \log (x)\right )} \, dx\\ &=\int \left (\frac {-6-6 \log (x)+2 \log ^2(x)+3 x^2 \log ^2(x)}{x^2 \log (x) \left (6-2 \log (x)+3 x^2 \log (x)\right )}-\frac {\log \left (-\frac {2}{3}+x^2+\frac {2}{\log (x)}\right )}{x^2}\right ) \, dx\\ &=\int \frac {-6-6 \log (x)+2 \log ^2(x)+3 x^2 \log ^2(x)}{x^2 \log (x) \left (6-2 \log (x)+3 x^2 \log (x)\right )} \, dx-\int \frac {\log \left (-\frac {2}{3}+x^2+\frac {2}{\log (x)}\right )}{x^2} \, dx\\ &=\int \left (\frac {2+3 x^2}{x^2 \left (-2+3 x^2\right )}-\frac {1}{x^2 \log (x)}-\frac {36}{\left (-2+3 x^2\right ) \left (6-2 \log (x)+3 x^2 \log (x)\right )}+\frac {-2+3 x^2}{x^2 \left (6-2 \log (x)+3 x^2 \log (x)\right )}\right ) \, dx-\int \frac {\log \left (-\frac {2}{3}+x^2+\frac {2}{\log (x)}\right )}{x^2} \, dx\\ &=-\left (36 \int \frac {1}{\left (-2+3 x^2\right ) \left (6-2 \log (x)+3 x^2 \log (x)\right )} \, dx\right )+\int \frac {2+3 x^2}{x^2 \left (-2+3 x^2\right )} \, dx-\int \frac {1}{x^2 \log (x)} \, dx+\int \frac {-2+3 x^2}{x^2 \left (6-2 \log (x)+3 x^2 \log (x)\right )} \, dx-\int \frac {\log \left (-\frac {2}{3}+x^2+\frac {2}{\log (x)}\right )}{x^2} \, dx\\ &=\frac {1}{x}+6 \int \frac {1}{-2+3 x^2} \, dx-36 \int \left (-\frac {1}{2 \sqrt {2} \left (\sqrt {2}-\sqrt {3} x\right ) \left (6-2 \log (x)+3 x^2 \log (x)\right )}-\frac {1}{2 \sqrt {2} \left (\sqrt {2}+\sqrt {3} x\right ) \left (6-2 \log (x)+3 x^2 \log (x)\right )}\right ) \, dx+\int \left (\frac {3}{6-2 \log (x)+3 x^2 \log (x)}-\frac {2}{x^2 \left (6-2 \log (x)+3 x^2 \log (x)\right )}\right ) \, dx-\int \frac {\log \left (-\frac {2}{3}+x^2+\frac {2}{\log (x)}\right )}{x^2} \, dx-\operatorname {Subst}\left (\int \frac {e^{-x}}{x} \, dx,x,\log (x)\right )\\ &=\frac {1}{x}-\sqrt {6} \tanh ^{-1}\left (\sqrt {\frac {3}{2}} x\right )-\text {Ei}(-\log (x))-2 \int \frac {1}{x^2 \left (6-2 \log (x)+3 x^2 \log (x)\right )} \, dx+3 \int \frac {1}{6-2 \log (x)+3 x^2 \log (x)} \, dx+\left (9 \sqrt {2}\right ) \int \frac {1}{\left (\sqrt {2}-\sqrt {3} x\right ) \left (6-2 \log (x)+3 x^2 \log (x)\right )} \, dx+\left (9 \sqrt {2}\right ) \int \frac {1}{\left (\sqrt {2}+\sqrt {3} x\right ) \left (6-2 \log (x)+3 x^2 \log (x)\right )} \, dx-\int \frac {\log \left (-\frac {2}{3}+x^2+\frac {2}{\log (x)}\right )}{x^2} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.29, size = 22, normalized size = 1.10 \begin {gather*} \frac {1}{x}+\frac {\log \left (-\frac {2}{3}+x^2+\frac {2}{\log (x)}\right )}{x} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

x^(-1) + Log[-2/3 + x^2 + 2/Log[x]]/x

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fricas [A]  time = 0.54, size = 25, normalized size = 1.25 \begin {gather*} \frac {\log \left (\frac {{\left (3 \, x^{2} - 2\right )} \log \relax (x) + 6}{3 \, \log \relax (x)}\right ) + 1}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 0.22, size = 32, normalized size = 1.60 \begin {gather*} \frac {\log \left (3 \, x^{2} \log \relax (x) - 2 \, \log \relax (x) + 6\right )}{x} - \frac {\log \left (3 \, \log \relax (x)\right )}{x} + \frac {1}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [C]  time = 0.11, size = 181, normalized size = 9.05

method	result	size
risch	\(\frac {\ln \left (x^{2} \ln \relax (x )-\frac {2 \ln \relax (x )}{3}+2\right )}{x}-\frac {i \pi \,\mathrm {csgn}\left (\frac {i}{\ln \relax (x )}\right ) \mathrm {csgn}\left (i \left (x^{2} \ln \relax (x )-\frac {2 \ln \relax (x )}{3}+2\right )\right ) \mathrm {csgn}\left (\frac {i \left (x^{2} \ln \relax (x )-\frac {2 \ln \relax (x )}{3}+2\right )}{\ln \relax (x )}\right )-i \pi \,\mathrm {csgn}\left (\frac {i}{\ln \relax (x )}\right ) \mathrm {csgn}\left (\frac {i \left (x^{2} \ln \relax (x )-\frac {2 \ln \relax (x )}{3}+2\right )}{\ln \relax (x )}\right )^{2}-i \pi \,\mathrm {csgn}\left (i \left (x^{2} \ln \relax (x )-\frac {2 \ln \relax (x )}{3}+2\right )\right ) \mathrm {csgn}\left (\frac {i \left (x^{2} \ln \relax (x )-\frac {2 \ln \relax (x )}{3}+2\right )}{\ln \relax (x )}\right )^{2}+i \pi \mathrm {csgn}\left (\frac {i \left (x^{2} \ln \relax (x )-\frac {2 \ln \relax (x )}{3}+2\right )}{\ln \relax (x )}\right )^{3}+2 \ln \left (\ln \relax (x )\right )-2}{2 x}\)	\(181\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.49, size = 27, normalized size = 1.35 \begin {gather*} -\frac {\log \relax (3) - \log \left ({\left (3 \, x^{2} - 2\right )} \log \relax (x) + 6\right ) + \log \left (\log \relax (x)\right ) - 1}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 10.79, size = 25, normalized size = 1.25 \begin {gather*} \frac {\ln \left (\frac {\frac {\ln \relax (x)\,\left (3\,x^2-2\right )}{3}+2}{\ln \relax (x)}\right )+1}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 0.47, size = 22, normalized size = 1.10 \begin {gather*} \frac {\log {\left (\frac {\frac {\left (3 x^{2} - 2\right ) \log {\relax (x )}}{3} + 2}{\log {\relax (x )}} \right )}}{x} + \frac {1}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-3*x**2+2)*ln(x)**2-6*ln(x))*ln(1/3*((3*x**2-2)*ln(x)+6)/ln(x))+(3*x**2+2)*ln(x)**2-6*ln(x)-6)/((
3*x**4-2*x**2)*ln(x)**2+6*x**2*ln(x)),x)

[Out]

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

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