3.28.89 \(\int \frac {8 x \log (x)+16 \log (\log (x))-16 \log (x) \log ^2(\log (x))}{5 x^3 \log (x)} \, dx\)

Optimal. Leaf size=16 \[ \frac {8 \left (-x+\log ^2(\log (x))\right )}{5 x^2} \]

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Rubi [F]  time = 0.26, 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 \log (x)+16 \log (\log (x))-16 \log (x) \log ^2(\log (x))}{5 x^3 \log (x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(8*x*Log[x] + 16*Log[Log[x]] - 16*Log[x]*Log[Log[x]]^2)/(5*x^3*Log[x]),x]

[Out]

-8/(5*x) + (16*Defer[Int][Log[Log[x]]/(x^3*Log[x]), x])/5 - (16*Defer[Int][Log[Log[x]]^2/x^3, x])/5

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\frac {1}{5} \int \frac {8 x \log (x)+16 \log (\log (x))-16 \log (x) \log ^2(\log (x))}{x^3 \log (x)} \, dx\\ &=\frac {1}{5} \int \frac {8 \left (x \log (x)+2 \log (\log (x))-2 \log (x) \log ^2(\log (x))\right )}{x^3 \log (x)} \, dx\\ &=\frac {8}{5} \int \frac {x \log (x)+2 \log (\log (x))-2 \log (x) \log ^2(\log (x))}{x^3 \log (x)} \, dx\\ &=\frac {8}{5} \int \left (\frac {1}{x^2}+\frac {2 \log (\log (x))}{x^3 \log (x)}-\frac {2 \log ^2(\log (x))}{x^3}\right ) \, dx\\ &=-\frac {8}{5 x}+\frac {16}{5} \int \frac {\log (\log (x))}{x^3 \log (x)} \, dx-\frac {16}{5} \int \frac {\log ^2(\log (x))}{x^3} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.07, size = 20, normalized size = 1.25 \begin {gather*} -\frac {8}{5 x}+\frac {8 \log ^2(\log (x))}{5 x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(8*x*Log[x] + 16*Log[Log[x]] - 16*Log[x]*Log[Log[x]]^2)/(5*x^3*Log[x]),x]

[Out]

-8/(5*x) + (8*Log[Log[x]]^2)/(5*x^2)

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fricas [A]  time = 0.72, size = 14, normalized size = 0.88 \begin {gather*} \frac {8 \, {\left (\log \left (\log \relax (x)\right )^{2} - x\right )}}{5 \, x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/5*(-16*log(x)*log(log(x))^2+16*log(log(x))+8*x*log(x))/x^3/log(x),x, algorithm="fricas")

[Out]

8/5*(log(log(x))^2 - x)/x^2

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giac [A]  time = 0.40, size = 16, normalized size = 1.00 \begin {gather*} \frac {8 \, \log \left (\log \relax (x)\right )^{2}}{5 \, x^{2}} - \frac {8}{5 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/5*(-16*log(x)*log(log(x))^2+16*log(log(x))+8*x*log(x))/x^3/log(x),x, algorithm="giac")

[Out]

8/5*log(log(x))^2/x^2 - 8/5/x

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maple [A]  time = 0.02, size = 17, normalized size = 1.06

method	result	size
risch	\(\frac {8 \ln \left (\ln \relax (x )\right )^{2}}{5 x^{2}}-\frac {8}{5 x}\)	\(17\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/5*(-16*ln(x)*ln(ln(x))^2+16*ln(ln(x))+8*x*ln(x))/x^3/ln(x),x,method=_RETURNVERBOSE)

[Out]

8/5/x^2*ln(ln(x))^2-8/5/x

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maxima [A]  time = 0.50, size = 16, normalized size = 1.00 \begin {gather*} \frac {8 \, \log \left (\log \relax (x)\right )^{2}}{5 \, x^{2}} - \frac {8}{5 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/5*(-16*log(x)*log(log(x))^2+16*log(log(x))+8*x*log(x))/x^3/log(x),x, algorithm="maxima")

[Out]

8/5*log(log(x))^2/x^2 - 8/5/x

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mupad [B]  time = 1.96, size = 14, normalized size = 0.88 \begin {gather*} -\frac {8\,\left (x-{\ln \left (\ln \relax (x)\right )}^2\right )}{5\,x^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((16*log(log(x)))/5 + (8*x*log(x))/5 - (16*log(log(x))^2*log(x))/5)/(x^3*log(x)),x)

[Out]

-(8*(x - log(log(x))^2))/(5*x^2)

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sympy [A]  time = 0.28, size = 17, normalized size = 1.06 \begin {gather*} - \frac {8}{5 x} + \frac {8 \log {\left (\log {\relax (x )} \right )}^{2}}{5 x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/5*(-16*ln(x)*ln(ln(x))**2+16*ln(ln(x))+8*x*ln(x))/x**3/ln(x),x)

[Out]

-8/(5*x) + 8*log(log(x))**2/(5*x**2)

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