3.82.51 \(\int \frac {9 x-3 x^4+(3-x^3) \log (16)+(-9+3 x^3) \log (3-x^3)+(-9 x-9 x^3+3 x^4) \log (x) \log (\frac {1}{\log (x)})}{(-9 x+3 x^4) \log (x)} \, dx\)

Optimal. Leaf size=24 \[ \left (x+\frac {\log (16)}{3}-\log \left (3-x^3\right )\right ) \log \left (\frac {1}{\log (x)}\right ) \]

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Rubi [F]  time = 0.85, 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 {9 x-3 x^4+\left (3-x^3\right ) \log (16)+\left (-9+3 x^3\right ) \log \left (3-x^3\right )+\left (-9 x-9 x^3+3 x^4\right ) \log (x) \log \left (\frac {1}{\log (x)}\right )}{\left (-9 x+3 x^4\right ) \log (x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(9*x - 3*x^4 + (3 - x^3)*Log[16] + (-9 + 3*x^3)*Log[3 - x^3] + (-9*x - 9*x^3 + 3*x^4)*Log[x]*Log[Log[x]^(-
1)])/((-9*x + 3*x^4)*Log[x]),x]

[Out]

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

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {9 x-3 x^4+\left (3-x^3\right ) \log (16)+\left (-9+3 x^3\right ) \log \left (3-x^3\right )+\left (-9 x-9 x^3+3 x^4\right ) \log (x) \log \left (\frac {1}{\log (x)}\right )}{x \left (-9+3 x^3\right ) \log (x)} \, dx\\ &=\int \left (-\frac {3 x+\log (16)-3 \log \left (3-x^3\right )}{3 x \log (x)}+\frac {\left (-3-3 x^2+x^3\right ) \log \left (\frac {1}{\log (x)}\right )}{-3+x^3}\right ) \, dx\\ &=-\left (\frac {1}{3} \int \frac {3 x+\log (16)-3 \log \left (3-x^3\right )}{x \log (x)} \, dx\right )+\int \frac {\left (-3-3 x^2+x^3\right ) \log \left (\frac {1}{\log (x)}\right )}{-3+x^3} \, dx\\ &=-\left (\frac {1}{3} \int \left (\frac {3 x+\log (16)}{x \log (x)}-\frac {3 \log \left (3-x^3\right )}{x \log (x)}\right ) \, dx\right )+\int \left (\log \left (\frac {1}{\log (x)}\right )-\frac {3 x^2 \log \left (\frac {1}{\log (x)}\right )}{-3+x^3}\right ) \, dx\\ &=-\left (\frac {1}{3} \int \frac {3 x+\log (16)}{x \log (x)} \, dx\right )-3 \int \frac {x^2 \log \left (\frac {1}{\log (x)}\right )}{-3+x^3} \, dx+\int \frac {\log \left (3-x^3\right )}{x \log (x)} \, dx+\int \log \left (\frac {1}{\log (x)}\right ) \, dx\\ &=x \log \left (\frac {1}{\log (x)}\right )-\frac {1}{3} \int \left (\frac {3}{\log (x)}+\frac {\log (16)}{x \log (x)}\right ) \, dx-3 \int \left (-\frac {\log \left (\frac {1}{\log (x)}\right )}{3 \left (-\sqrt [3]{-3}-x\right )}-\frac {\log \left (\frac {1}{\log (x)}\right )}{3 \left (\sqrt [3]{3}-x\right )}-\frac {\log \left (\frac {1}{\log (x)}\right )}{3 \left ((-1)^{2/3} \sqrt [3]{3}-x\right )}\right ) \, dx+\int \frac {1}{\log (x)} \, dx+\int \frac {\log \left (3-x^3\right )}{x \log (x)} \, dx\\ &=x \log \left (\frac {1}{\log (x)}\right )+\text {li}(x)-\frac {1}{3} \log (16) \int \frac {1}{x \log (x)} \, dx-\int \frac {1}{\log (x)} \, dx+\int \frac {\log \left (3-x^3\right )}{x \log (x)} \, dx+\int \frac {\log \left (\frac {1}{\log (x)}\right )}{-\sqrt [3]{-3}-x} \, dx+\int \frac {\log \left (\frac {1}{\log (x)}\right )}{\sqrt [3]{3}-x} \, dx+\int \frac {\log \left (\frac {1}{\log (x)}\right )}{(-1)^{2/3} \sqrt [3]{3}-x} \, dx\\ &=x \log \left (\frac {1}{\log (x)}\right )-\frac {1}{3} \log (16) \operatorname {Subst}\left (\int \frac {1}{x} \, dx,x,\log (x)\right )+\int \frac {\log \left (3-x^3\right )}{x \log (x)} \, dx+\int \frac {\log \left (\frac {1}{\log (x)}\right )}{-\sqrt [3]{-3}-x} \, dx+\int \frac {\log \left (\frac {1}{\log (x)}\right )}{\sqrt [3]{3}-x} \, dx+\int \frac {\log \left (\frac {1}{\log (x)}\right )}{(-1)^{2/3} \sqrt [3]{3}-x} \, dx\\ &=x \log \left (\frac {1}{\log (x)}\right )-\frac {1}{3} \log (16) \log (\log (x))+\int \frac {\log \left (3-x^3\right )}{x \log (x)} \, dx+\int \frac {\log \left (\frac {1}{\log (x)}\right )}{-\sqrt [3]{-3}-x} \, dx+\int \frac {\log \left (\frac {1}{\log (x)}\right )}{\sqrt [3]{3}-x} \, dx+\int \frac {\log \left (\frac {1}{\log (x)}\right )}{(-1)^{2/3} \sqrt [3]{3}-x} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.12, size = 28, normalized size = 1.17 \begin {gather*} \left (x-\log \left (3-x^3\right )\right ) \log \left (\frac {1}{\log (x)}\right )-\frac {1}{3} \log (16) \log (\log (x)) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(9*x - 3*x^4 + (3 - x^3)*Log[16] + (-9 + 3*x^3)*Log[3 - x^3] + (-9*x - 9*x^3 + 3*x^4)*Log[x]*Log[Log
[x]^(-1)])/((-9*x + 3*x^4)*Log[x]),x]

[Out]

(x - Log[3 - x^3])*Log[Log[x]^(-1)] - (Log[16]*Log[Log[x]])/3

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fricas [A]  time = 0.49, size = 25, normalized size = 1.04 \begin {gather*} \frac {1}{3} \, {\left (3 \, x + 4 \, \log \relax (2) - 3 \, \log \left (-x^{3} + 3\right )\right )} \log \left (\frac {1}{\log \relax (x)}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((3*x^4-9*x^3-9*x)*log(x)*log(1/log(x))+(3*x^3-9)*log(-x^3+3)+4*(-x^3+3)*log(2)-3*x^4+9*x)/(3*x^4-9*
x)/log(x),x, algorithm="fricas")

[Out]

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

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giac [A]  time = 0.18, size = 26, normalized size = 1.08 \begin {gather*} -x \log \left (\log \relax (x)\right ) - \frac {4}{3} \, \log \relax (2) \log \left (\log \relax (x)\right ) + \log \left (-x^{3} + 3\right ) \log \left (\log \relax (x)\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((3*x^4-9*x^3-9*x)*log(x)*log(1/log(x))+(3*x^3-9)*log(-x^3+3)+4*(-x^3+3)*log(2)-3*x^4+9*x)/(3*x^4-9*
x)/log(x),x, algorithm="giac")

[Out]

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

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maple [A]  time = 0.32, size = 25, normalized size = 1.04

method	result	size
risch	\(\left (-x +\ln \left (-x^{3}+3\right )\right ) \ln \left (\ln \relax (x )\right )-\frac {4 \ln \relax (2) \ln \left (\ln \relax (x )\right )}{3}\)	\(25\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((3*x^4-9*x^3-9*x)*ln(x)*ln(1/ln(x))+(3*x^3-9)*ln(-x^3+3)+4*(-x^3+3)*ln(2)-3*x^4+9*x)/(3*x^4-9*x)/ln(x),x,
method=_RETURNVERBOSE)

[Out]

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

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maxima [A]  time = 0.51, size = 26, normalized size = 1.08 \begin {gather*} -\frac {1}{3} \, {\left (3 \, x + 4 \, \log \relax (2)\right )} \log \left (\log \relax (x)\right ) + \log \left (-x^{3} + 3\right ) \log \left (\log \relax (x)\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((3*x^4-9*x^3-9*x)*log(x)*log(1/log(x))+(3*x^3-9)*log(-x^3+3)+4*(-x^3+3)*log(2)-3*x^4+9*x)/(3*x^4-9*
x)/log(x),x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 6.61, size = 46, normalized size = 1.92 \begin {gather*} -\ln \left (\frac {1}{\ln \relax (x)}\right )\,\left (\ln \left (3-x^3\right )+\frac {3\,x^2-x^5}{x\,\left (x^3-3\right )}\right )-\frac {4\,\ln \left (\ln \relax (x)\right )\,\ln \relax (2)}{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((4*log(2)*(x^3 - 3) - 9*x + 3*x^4 - log(3 - x^3)*(3*x^3 - 9) + log(1/log(x))*log(x)*(9*x + 9*x^3 - 3*x^4))
/(log(x)*(9*x - 3*x^4)),x)

[Out]

- log(1/log(x))*(log(3 - x^3) + (3*x^2 - x^5)/(x*(x^3 - 3))) - (4*log(log(x))*log(2))/3

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sympy [A]  time = 0.57, size = 26, normalized size = 1.08 \begin {gather*} \left (x - \log {\left (3 - x^{3} \right )}\right ) \log {\left (\frac {1}{\log {\relax (x )}} \right )} - \frac {4 \log {\relax (2 )} \log {\left (\log {\relax (x )} \right )}}{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((3*x**4-9*x**3-9*x)*ln(x)*ln(1/ln(x))+(3*x**3-9)*ln(-x**3+3)+4*(-x**3+3)*ln(2)-3*x**4+9*x)/(3*x**4-
9*x)/ln(x),x)

[Out]

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

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