3.45.65 \(\int \frac {1-\log (x)-2 \log (x) \log (\frac {\log (x)}{x})}{x \log (x) \log (\frac {\log (x)}{x})} \, dx\)

Optimal. Leaf size=13 \[ \log \left (\frac {4 \log \left (\frac {\log (x)}{x}\right )}{x^2}\right ) \]

________________________________________________________________________________________

Rubi [A]  time = 0.20, antiderivative size = 13, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 34, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, Rules used = {6742, 6684} \begin {gather*} \log \left (\log \left (\frac {\log (x)}{x}\right )\right )-2 \log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(1 - Log[x] - 2*Log[x]*Log[Log[x]/x])/(x*Log[x]*Log[Log[x]/x]),x]

[Out]

-2*Log[x] + Log[Log[Log[x]/x]]

Rule 6684

Int[(u_)/(y_), x_Symbol] :> With[{q = DerivativeDivides[y, u, x]}, Simp[q*Log[RemoveContent[y, x]], x] /;  !Fa
lseQ[q]]

Rule 6742

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]

Rubi steps

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

________________________________________________________________________________________

Mathematica [A]  time = 0.06, size = 13, normalized size = 1.00 \begin {gather*} -2 \log (x)+\log \left (\log \left (\frac {\log (x)}{x}\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(1 - Log[x] - 2*Log[x]*Log[Log[x]/x])/(x*Log[x]*Log[Log[x]/x]),x]

[Out]

-2*Log[x] + Log[Log[Log[x]/x]]

________________________________________________________________________________________

fricas [A]  time = 0.68, size = 13, normalized size = 1.00 \begin {gather*} -2 \, \log \relax (x) + \log \left (\log \left (\frac {\log \relax (x)}{x}\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*log(x) + log(log(log(x)/x))

________________________________________________________________________________________

giac [A]  time = 0.15, size = 14, normalized size = 1.08 \begin {gather*} -2 \, \log \relax (x) + \log \left (\log \relax (x) - \log \left (\log \relax (x)\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*log(x) + log(log(x) - log(log(x)))

________________________________________________________________________________________

maple [A]  time = 0.06, size = 14, normalized size = 1.08

method	result	size
norman	\(-2 \ln \relax (x )+\ln \left (\ln \left (\frac {\ln \relax (x )}{x}\right )\right )\)	\(14\)
risch	\(-2 \ln \relax (x )+\ln \left (\ln \left (\ln \relax (x )\right )-\frac {i \left (\pi \,\mathrm {csgn}\left (\frac {i}{x}\right ) \mathrm {csgn}\left (i \ln \relax (x )\right ) \mathrm {csgn}\left (\frac {i \ln \relax (x )}{x}\right )-\pi \,\mathrm {csgn}\left (\frac {i}{x}\right ) \mathrm {csgn}\left (\frac {i \ln \relax (x )}{x}\right )^{2}-\pi \,\mathrm {csgn}\left (i \ln \relax (x )\right ) \mathrm {csgn}\left (\frac {i \ln \relax (x )}{x}\right )^{2}+\pi \mathrm {csgn}\left (\frac {i \ln \relax (x )}{x}\right )^{3}-2 i \ln \relax (x )\right )}{2}\right )\)	\(98\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-2*ln(x)*ln(ln(x)/x)+1-ln(x))/x/ln(x)/ln(ln(x)/x),x,method=_RETURNVERBOSE)

[Out]

-2*ln(x)+ln(ln(ln(x)/x))

________________________________________________________________________________________

maxima [A]  time = 0.38, size = 14, normalized size = 1.08 \begin {gather*} -2 \, \log \relax (x) + \log \left (-\log \relax (x) + \log \left (\log \relax (x)\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*log(x) + log(-log(x) + log(log(x)))

________________________________________________________________________________________

mupad [B]  time = 3.44, size = 13, normalized size = 1.00 \begin {gather*} \ln \left (\ln \left (\frac {\ln \relax (x)}{x}\right )\right )-2\,\ln \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(log(x) + 2*log(log(x)/x)*log(x) - 1)/(x*log(log(x)/x)*log(x)),x)

[Out]

log(log(log(x)/x)) - 2*log(x)

________________________________________________________________________________________

sympy [A]  time = 0.28, size = 12, normalized size = 0.92 \begin {gather*} - 2 \log {\relax (x )} + \log {\left (\log {\left (\frac {\log {\relax (x )}}{x} \right )} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*log(x) + log(log(log(x)/x))

________________________________________________________________________________________