3.80.100 \(\int \frac {-3+(3-3 x) \log (\frac {3}{x})}{x^3 \log (\frac {3}{x})-2 x^2 \log (\frac {3}{x}) \log (9 x \log (\frac {3}{x}))+x \log (\frac {3}{x}) \log ^2(9 x \log (\frac {3}{x}))} \, dx\)

Optimal. Leaf size=22 \[ \frac {3}{4}+\frac {3}{x-\log \left (9 x \log \left (\frac {3}{x}\right )\right )} \]

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Rubi [A]  time = 0.16, antiderivative size = 18, normalized size of antiderivative = 0.82, number of steps used = 2, number of rules used = 2, integrand size = 69, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.029, Rules used = {6688, 6686} \begin {gather*} \frac {3}{x-\log \left (9 x \log \left (\frac {3}{x}\right )\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-3 + (3 - 3*x)*Log[3/x])/(x^3*Log[3/x] - 2*x^2*Log[3/x]*Log[9*x*Log[3/x]] + x*Log[3/x]*Log[9*x*Log[3/x]]^
2),x]

[Out]

3/(x - Log[9*x*Log[3/x]])

Rule 6686

Int[(u_)*(y_)^(m_.), x_Symbol] :> With[{q = DerivativeDivides[y, u, x]}, Simp[(q*y^(m + 1))/(m + 1), x] /;  !F
alseQ[q]] /; FreeQ[m, x] && NeQ[m, -1]

Rule 6688

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

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-3-3 (-1+x) \log \left (\frac {3}{x}\right )}{x \log \left (\frac {3}{x}\right ) \left (x-\log \left (9 x \log \left (\frac {3}{x}\right )\right )\right )^2} \, dx\\ &=\frac {3}{x-\log \left (9 x \log \left (\frac {3}{x}\right )\right )}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.02, size = 18, normalized size = 0.82 \begin {gather*} \frac {3}{x-\log \left (9 x \log \left (\frac {3}{x}\right )\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-3 + (3 - 3*x)*Log[3/x])/(x^3*Log[3/x] - 2*x^2*Log[3/x]*Log[9*x*Log[3/x]] + x*Log[3/x]*Log[9*x*Log[
3/x]]^2),x]

[Out]

3/(x - Log[9*x*Log[3/x]])

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fricas [A]  time = 0.59, size = 18, normalized size = 0.82 \begin {gather*} \frac {3}{x - \log \left (9 \, x \log \left (\frac {3}{x}\right )\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/(x - log(9*x*log(3/x)))

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giac [B]  time = 0.34, size = 284, normalized size = 12.91 \begin {gather*} \frac {3 \, {\left (x \log \relax (3) \log \left (\frac {3}{x}\right ) - x \log \relax (x) \log \left (\frac {3}{x}\right ) - \log \relax (3) \log \left (\frac {3}{x}\right ) + \log \relax (x) \log \left (\frac {3}{x}\right ) + \log \left (\frac {3}{x}\right )\right )}}{x^{2} \log \relax (3) \log \left (\frac {3}{x}\right ) - 2 \, x \log \relax (3)^{2} \log \left (\frac {3}{x}\right ) - x^{2} \log \relax (x) \log \left (\frac {3}{x}\right ) + x \log \relax (3) \log \relax (x) \log \left (\frac {3}{x}\right ) + x \log \relax (x)^{2} \log \left (\frac {3}{x}\right ) - x \log \relax (3) \log \left (\frac {3}{x}\right ) \log \left (\log \left (\frac {3}{x}\right )\right ) + x \log \relax (x) \log \left (\frac {3}{x}\right ) \log \left (\log \left (\frac {3}{x}\right )\right ) - x \log \relax (3) \log \left (\frac {3}{x}\right ) + 2 \, \log \relax (3)^{2} \log \left (\frac {3}{x}\right ) + x \log \relax (x) \log \left (\frac {3}{x}\right ) - \log \relax (3) \log \relax (x) \log \left (\frac {3}{x}\right ) - \log \relax (x)^{2} \log \left (\frac {3}{x}\right ) + \log \relax (3) \log \left (\frac {3}{x}\right ) \log \left (\log \left (\frac {3}{x}\right )\right ) - \log \relax (x) \log \left (\frac {3}{x}\right ) \log \left (\log \left (\frac {3}{x}\right )\right ) + x \log \relax (3) - 2 \, \log \relax (3)^{2} - x \log \relax (x) + \log \relax (3) \log \relax (x) + \log \relax (x)^{2} - \log \relax (3) \log \left (\log \left (\frac {3}{x}\right )\right ) + \log \relax (x) \log \left (\log \left (\frac {3}{x}\right )\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [F]  time = 0.02, size = 0, normalized size = 0.00 \[\int \frac {\left (-3 x +3\right ) \ln \left (\frac {3}{x}\right )-3}{x \ln \left (\frac {3}{x}\right ) \ln \left (9 x \ln \left (\frac {3}{x}\right )\right )^{2}-2 x^{2} \ln \left (\frac {3}{x}\right ) \ln \left (9 x \ln \left (\frac {3}{x}\right )\right )+x^{3} \ln \left (\frac {3}{x}\right )}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-3*x+3)*ln(3/x)-3)/(x*ln(3/x)*ln(9*x*ln(3/x))^2-2*x^2*ln(3/x)*ln(9*x*ln(3/x))+x^3*ln(3/x)),x)

[Out]

int(((-3*x+3)*ln(3/x)-3)/(x*ln(3/x)*ln(9*x*ln(3/x))^2-2*x^2*ln(3/x)*ln(9*x*ln(3/x))+x^3*ln(3/x)),x)

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maxima [C]  time = 0.49, size = 25, normalized size = 1.14 \begin {gather*} -\frac {3}{i \, \pi - x + 2 \, \log \relax (3) + \log \relax (x) + \log \left (-\log \relax (3) + \log \relax (x)\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-3/(I*pi - x + 2*log(3) + log(x) + log(-log(3) + log(x)))

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mupad [B]  time = 5.55, size = 18, normalized size = 0.82 \begin {gather*} \frac {3}{x-\ln \left (9\,x\,\ln \left (\frac {3}{x}\right )\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/(x - log(9*x*log(3/x)))

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sympy [A]  time = 0.30, size = 14, normalized size = 0.64 \begin {gather*} - \frac {3}{- x + \log {\left (9 x \log {\left (\frac {3}{x} \right )} \right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-3/(-x + log(9*x*log(3/x)))

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