3.74.59 \(\int \frac {\log (\log (75) \log (x))-\log (x) \log ^2(\log (75) \log (x))}{324 x^3 \log (x)} \, dx\)

Optimal. Leaf size=15 \[ \frac {\log ^2(\log (75) \log (x))}{648 x^2} \]

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Rubi [F]  time = 0.48, 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 {\log (\log (75) \log (x))-\log (x) \log ^2(\log (75) \log (x))}{324 x^3 \log (x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(Log[Log[75]*Log[x]] - Log[x]*Log[Log[75]*Log[x]]^2)/(324*x^3*Log[x]),x]

[Out]

Defer[Int][Log[Log[75]*Log[x]]/(x^3*Log[x]), x]/324 - Defer[Int][Log[Log[75]*Log[x]]^2/x^3, x]/324

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\frac {1}{324} \int \frac {\log (\log (75) \log (x))-\log (x) \log ^2(\log (75) \log (x))}{x^3 \log (x)} \, dx\\ &=\frac {1}{324} \int \frac {\log (\log (75) \log (x)) (1-\log (x) \log (\log (75) \log (x)))}{x^3 \log (x)} \, dx\\ &=\frac {1}{324} \int \left (\frac {\log (\log (75) \log (x))}{x^3 \log (x)}-\frac {\log ^2(\log (75) \log (x))}{x^3}\right ) \, dx\\ &=\frac {1}{324} \int \frac {\log (\log (75) \log (x))}{x^3 \log (x)} \, dx-\frac {1}{324} \int \frac {\log ^2(\log (75) \log (x))}{x^3} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.05, size = 15, normalized size = 1.00 \begin {gather*} \frac {\log ^2(\log (75) \log (x))}{648 x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(Log[Log[75]*Log[x]] - Log[x]*Log[Log[75]*Log[x]]^2)/(324*x^3*Log[x]),x]

[Out]

Log[Log[75]*Log[x]]^2/(648*x^2)

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fricas [A]  time = 0.93, size = 13, normalized size = 0.87 \begin {gather*} \frac {\log \left (\log \left (75\right ) \log \relax (x)\right )^{2}}{648 \, x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/324*(-log(x)*log(log(75)*log(x))^2+log(log(75)*log(x)))/x^3/log(x),x, algorithm="fricas")

[Out]

1/648*log(log(75)*log(x))^2/x^2

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giac [A]  time = 0.15, size = 13, normalized size = 0.87 \begin {gather*} \frac {\log \left (\log \left (75\right ) \log \relax (x)\right )^{2}}{648 \, x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/324*(-log(x)*log(log(75)*log(x))^2+log(log(75)*log(x)))/x^3/log(x),x, algorithm="giac")

[Out]

1/648*log(log(75)*log(x))^2/x^2

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maple [A]  time = 0.03, size = 19, normalized size = 1.27

method	result	size
risch	\(\frac {\ln \left (\left (\ln \relax (3)+2 \ln \relax (5)\right ) \ln \relax (x )\right )^{2}}{648 x^{2}}\)	\(19\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/324*(-ln(x)*ln(ln(75)*ln(x))^2+ln(ln(75)*ln(x)))/x^3/ln(x),x,method=_RETURNVERBOSE)

[Out]

1/648/x^2*ln((ln(3)+2*ln(5))*ln(x))^2

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maxima [B]  time = 0.48, size = 34, normalized size = 2.27 \begin {gather*} \frac {\log \left (2 \, \log \relax (5) + \log \relax (3)\right )^{2} + 2 \, \log \left (2 \, \log \relax (5) + \log \relax (3)\right ) \log \left (\log \relax (x)\right ) + \log \left (\log \relax (x)\right )^{2}}{648 \, x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/324*(-log(x)*log(log(75)*log(x))^2+log(log(75)*log(x)))/x^3/log(x),x, algorithm="maxima")

[Out]

1/648*(log(2*log(5) + log(3))^2 + 2*log(2*log(5) + log(3))*log(log(x)) + log(log(x))^2)/x^2

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mupad [B]  time = 6.28, size = 14, normalized size = 0.93 \begin {gather*} \frac {{\left (\ln \left (\ln \relax (x)\right )+\ln \left (\ln \left (75\right )\right )\right )}^2}{648\,x^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((log(log(75)*log(x))/324 - (log(log(75)*log(x))^2*log(x))/324)/(x^3*log(x)),x)

[Out]

(log(log(x)) + log(log(75)))^2/(648*x^2)

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sympy [A]  time = 0.30, size = 14, normalized size = 0.93 \begin {gather*} \frac {\log {\left (\log {\left (75 \right )} \log {\relax (x )} \right )}^{2}}{648 x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/324*(-ln(x)*ln(ln(75)*ln(x))**2+ln(ln(75)*ln(x)))/x**3/ln(x),x)

[Out]

log(log(75)*log(x))**2/(648*x**2)

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