3.40.80 \(\int \frac {-14-4 \log (\frac {4}{x})+\log (x)}{-9 x-4 x \log (\frac {4}{x})+x \log (x)} \, dx\)

Optimal. Leaf size=26 \[ \log \left (\frac {3 x}{4 \left (3+\log \left (\frac {4}{x}\right )+\frac {1}{4} (-3-\log (x))\right )}\right ) \]

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Rubi [A]  time = 0.10, antiderivative size = 20, normalized size of antiderivative = 0.77, number of steps used = 3, number of rules used = 2, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {6742, 6684} \begin {gather*} \log (x)-\log \left (4 \log \left (\frac {4}{x}\right )-\log (x)+9\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-14 - 4*Log[4/x] + Log[x])/(-9*x - 4*x*Log[4/x] + x*Log[x]),x]

[Out]

Log[x] - Log[9 + 4*Log[4/x] - Log[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 {1}{x}+\frac {5}{x \left (9+4 \log \left (\frac {4}{x}\right )-\log (x)\right )}\right ) \, dx\\ &=\log (x)+5 \int \frac {1}{x \left (9+4 \log \left (\frac {4}{x}\right )-\log (x)\right )} \, dx\\ &=\log (x)-\log \left (9+4 \log \left (\frac {4}{x}\right )-\log (x)\right )\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.07, size = 32, normalized size = 1.23 \begin {gather*} -\frac {4}{5} \log \left (\frac {4}{x}\right )+\frac {\log (x)}{5}-\log \left (-9-4 \log \left (\frac {4}{x}\right )+\log (x)\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-14 - 4*Log[4/x] + Log[x])/(-9*x - 4*x*Log[4/x] + x*Log[x]),x]

[Out]

(-4*Log[4/x])/5 + Log[x]/5 - Log[-9 - 4*Log[4/x] + Log[x]]

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fricas [A]  time = 0.56, size = 26, normalized size = 1.00 \begin {gather*} -\log \left (\frac {4}{x}\right ) - \log \left (-2 \, \log \relax (2) + 5 \, \log \left (\frac {4}{x}\right ) + 9\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((log(x)-4*log(4/x)-14)/(x*log(x)-4*x*log(4/x)-9*x),x, algorithm="fricas")

[Out]

-log(4/x) - log(-2*log(2) + 5*log(4/x) + 9)

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giac [A]  time = 0.19, size = 31, normalized size = 1.19 \begin {gather*} -\frac {1}{2} \, \log \left (\frac {25}{4} \, \pi ^{2} {\left (\mathrm {sgn}\relax (x) - 1\right )}^{2} + {\left (8 \, \log \relax (2) - 5 \, \log \left ({\left | x \right |}\right ) + 9\right )}^{2}\right ) + \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((log(x)-4*log(4/x)-14)/(x*log(x)-4*x*log(4/x)-9*x),x, algorithm="giac")

[Out]

-1/2*log(25/4*pi^2*(sgn(x) - 1)^2 + (8*log(2) - 5*log(abs(x)) + 9)^2) + log(x)

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maple [A]  time = 0.07, size = 19, normalized size = 0.73

method	result	size
norman	\(\ln \relax (x )-\ln \left (\ln \relax (x )-4 \ln \left (\frac {4}{x}\right )-9\right )\)	\(19\)
risch	\(\ln \relax (x )-\ln \left (\ln \relax (x )+\frac {i \left (8 i \ln \relax (2)+9 i\right )}{5}\right )\)	\(21\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((ln(x)-4*ln(4/x)-14)/(x*ln(x)-4*x*ln(4/x)-9*x),x,method=_RETURNVERBOSE)

[Out]

ln(x)-ln(ln(x)-4*ln(4/x)-9)

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maxima [B]  time = 0.58, size = 70, normalized size = 2.69 \begin {gather*} \frac {1}{5} \, {\left (8 \, \log \relax (2) - 5 \, \log \relax (x) + 9\right )} \log \left (-\frac {8}{5} \, \log \relax (2) + \log \relax (x) - \frac {9}{5}\right ) + \frac {1}{5} \, \log \relax (x) \log \left (-\frac {8}{5} \, \log \relax (2) + \log \relax (x) - \frac {9}{5}\right ) - \frac {4}{5} \, \log \left (\frac {4}{x}\right ) \log \left (-\frac {8}{5} \, \log \relax (2) + \log \relax (x) - \frac {9}{5}\right ) - \frac {8}{5} \, \log \relax (2) + \log \relax (x) - \frac {14}{5} \, \log \left (-\frac {8}{5} \, \log \relax (2) + \log \relax (x) - \frac {9}{5}\right ) - \frac {9}{5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((log(x)-4*log(4/x)-14)/(x*log(x)-4*x*log(4/x)-9*x),x, algorithm="maxima")

[Out]

1/5*(8*log(2) - 5*log(x) + 9)*log(-8/5*log(2) + log(x) - 9/5) + 1/5*log(x)*log(-8/5*log(2) + log(x) - 9/5) - 4
/5*log(4/x)*log(-8/5*log(2) + log(x) - 9/5) - 8/5*log(2) + log(x) - 14/5*log(-8/5*log(2) + log(x) - 9/5) - 9/5

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mupad [B]  time = 2.76, size = 20, normalized size = 0.77 \begin {gather*} \ln \relax (x)-\ln \left (4\,\ln \left (\frac {4}{x}\right )-\ln \relax (x)+9\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((4*log(4/x) - log(x) + 14)/(9*x + 4*x*log(4/x) - x*log(x)),x)

[Out]

log(x) - log(4*log(4/x) - log(x) + 9)

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sympy [A]  time = 0.31, size = 17, normalized size = 0.65 \begin {gather*} \log {\relax (x )} - \log {\left (\log {\relax (x )} - \frac {9}{5} - \frac {8 \log {\relax (2 )}}{5} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((ln(x)-4*ln(4/x)-14)/(x*ln(x)-4*x*ln(4/x)-9*x),x)

[Out]

log(x) - log(log(x) - 9/5 - 8*log(2)/5)

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