\(\int \frac {-3+(3-3 x) \log (\frac {3}{x})}{x^3 \log (\frac {3}{x})-2 x^2 \log (\frac {3}{x}) \log (9 x \log (\frac {3}{x}))+x \log (\frac {3}{x}) \log ^2(9 x \log (\frac {3}{x}))} \, dx\) [8000]

Optimal result

Integrand size = 69, antiderivative size = 22 \[ \int \frac {-3+(3-3 x) \log \left (\frac {3}{x}\right )}{x^3 \log \left (\frac {3}{x}\right )-2 x^2 \log \left (\frac {3}{x}\right ) \log \left (9 x \log \left (\frac {3}{x}\right )\right )+x \log \left (\frac {3}{x}\right ) \log ^2\left (9 x \log \left (\frac {3}{x}\right )\right )} \, dx=\frac {3}{4}+\frac {3}{x-\log \left (9 x \log \left (\frac {3}{x}\right )\right )} \]

[Out]

3/4+3/(x-ln(9*x*ln(3/x)))

Rubi [A] (verified)

Time = 0.10 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.82, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.029, Rules used = {6820, 6818} \[ \int \frac {-3+(3-3 x) \log \left (\frac {3}{x}\right )}{x^3 \log \left (\frac {3}{x}\right )-2 x^2 \log \left (\frac {3}{x}\right ) \log \left (9 x \log \left (\frac {3}{x}\right )\right )+x \log \left (\frac {3}{x}\right ) \log ^2\left (9 x \log \left (\frac {3}{x}\right )\right )} \, dx=\frac {3}{x-\log \left (9 x \log \left (\frac {3}{x}\right )\right )} \]

[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 6818

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 6820

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

Rubi steps \begin{align*} \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{align*}

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.82 \[ \int \frac {-3+(3-3 x) \log \left (\frac {3}{x}\right )}{x^3 \log \left (\frac {3}{x}\right )-2 x^2 \log \left (\frac {3}{x}\right ) \log \left (9 x \log \left (\frac {3}{x}\right )\right )+x \log \left (\frac {3}{x}\right ) \log ^2\left (9 x \log \left (\frac {3}{x}\right )\right )} \, dx=\frac {3}{x-\log \left (9 x \log \left (\frac {3}{x}\right )\right )} \]

[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]])

Maple [A] (verified)

Time = 5.04 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.86

method	result	size
parallelrisch	\(\frac {3}{x -\ln \left (9 x \ln \left (\frac {3}{x}\right )\right )}\)	\(19\)
default	\(-\frac {3}{x \left (\frac {2 \ln \left (3\right )}{x}+\frac {\ln \left (x \left (\ln \left (3\right )+\ln \left (\frac {1}{x}\right )\right )\right )}{x}-1\right )}\)	\(31\)
parts	\(-\frac {6 \left (\ln \left (3\right )+\ln \left (\frac {1}{x}\right )\right ) \left (\frac {1}{x}-1\right )}{\left (\frac {\ln \left (3\right )}{x}+\frac {\ln \left (\frac {1}{x}\right )}{x}-\ln \left (3\right )-\ln \left (\frac {1}{x}\right )-\frac {1}{x}\right ) x \left (-\frac {i \pi \,\operatorname {csgn}\left (i \left (\ln \left (3\right )+\ln \left (\frac {1}{x}\right )\right )\right ) \operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i x \left (\ln \left (3\right )+\ln \left (\frac {1}{x}\right )\right )\right )}{x}+\frac {i \pi \,\operatorname {csgn}\left (i \left (\ln \left (3\right )+\ln \left (\frac {1}{x}\right )\right )\right ) \operatorname {csgn}\left (i x \left (\ln \left (3\right )+\ln \left (\frac {1}{x}\right )\right )\right )^{2}}{x}+\frac {i \pi \,\operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i x \left (\ln \left (3\right )+\ln \left (\frac {1}{x}\right )\right )\right )^{2}}{x}-\frac {i \pi \operatorname {csgn}\left (i x \left (\ln \left (3\right )+\ln \left (\frac {1}{x}\right )\right )\right )^{3}}{x}+\frac {4 \ln \left (3\right )}{x}+\frac {2 \ln \left (\ln \left (3\right )+\ln \left (\frac {1}{x}\right )\right )}{x}-\frac {2 \ln \left (\frac {1}{x}\right )}{x}-2\right )}+\frac {6 i}{x^{2} \left (\frac {\ln \left (3\right )}{x}+\frac {\ln \left (\frac {1}{x}\right )}{x}-\ln \left (3\right )-\ln \left (\frac {1}{x}\right )-\frac {1}{x}\right ) \left (\frac {\pi \,\operatorname {csgn}\left (i \left (\ln \left (3\right )+\ln \left (\frac {1}{x}\right )\right )\right ) \operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i x \left (\ln \left (3\right )+\ln \left (\frac {1}{x}\right )\right )\right )}{x}-\frac {\pi \,\operatorname {csgn}\left (i \left (\ln \left (3\right )+\ln \left (\frac {1}{x}\right )\right )\right ) \operatorname {csgn}\left (i x \left (\ln \left (3\right )+\ln \left (\frac {1}{x}\right )\right )\right )^{2}}{x}-\frac {\pi \,\operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i x \left (\ln \left (3\right )+\ln \left (\frac {1}{x}\right )\right )\right )^{2}}{x}+\frac {\pi \operatorname {csgn}\left (i x \left (\ln \left (3\right )+\ln \left (\frac {1}{x}\right )\right )\right )^{3}}{x}+\frac {4 i \ln \left (3\right )}{x}+\frac {2 i \ln \left (\ln \left (3\right )+\ln \left (\frac {1}{x}\right )\right )}{x}-\frac {2 i \ln \left (\frac {1}{x}\right )}{x}-2 i\right )}\)	\(381\)

[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,method=_RET
URNVERBOSE)

[Out]

3/(x-ln(9*x*ln(3/x)))

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.82 \[ \int \frac {-3+(3-3 x) \log \left (\frac {3}{x}\right )}{x^3 \log \left (\frac {3}{x}\right )-2 x^2 \log \left (\frac {3}{x}\right ) \log \left (9 x \log \left (\frac {3}{x}\right )\right )+x \log \left (\frac {3}{x}\right ) \log ^2\left (9 x \log \left (\frac {3}{x}\right )\right )} \, dx=\frac {3}{x - \log \left (9 \, x \log \left (\frac {3}{x}\right )\right )} \]

[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)))

Sympy [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.64 \[ \int \frac {-3+(3-3 x) \log \left (\frac {3}{x}\right )}{x^3 \log \left (\frac {3}{x}\right )-2 x^2 \log \left (\frac {3}{x}\right ) \log \left (9 x \log \left (\frac {3}{x}\right )\right )+x \log \left (\frac {3}{x}\right ) \log ^2\left (9 x \log \left (\frac {3}{x}\right )\right )} \, dx=- \frac {3}{- x + \log {\left (9 x \log {\left (\frac {3}{x} \right )} \right )}} \]

[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)))

Maxima [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.30 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.14 \[ \int \frac {-3+(3-3 x) \log \left (\frac {3}{x}\right )}{x^3 \log \left (\frac {3}{x}\right )-2 x^2 \log \left (\frac {3}{x}\right ) \log \left (9 x \log \left (\frac {3}{x}\right )\right )+x \log \left (\frac {3}{x}\right ) \log ^2\left (9 x \log \left (\frac {3}{x}\right )\right )} \, dx=-\frac {3}{i \, \pi - x + 2 \, \log \left (3\right ) + \log \left (x\right ) + \log \left (-\log \left (3\right ) + \log \left (x\right )\right )} \]

[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)))

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 284 vs. \(2 (20) = 40\).

Time = 0.39 (sec) , antiderivative size = 284, normalized size of antiderivative = 12.91 \[ \int \frac {-3+(3-3 x) \log \left (\frac {3}{x}\right )}{x^3 \log \left (\frac {3}{x}\right )-2 x^2 \log \left (\frac {3}{x}\right ) \log \left (9 x \log \left (\frac {3}{x}\right )\right )+x \log \left (\frac {3}{x}\right ) \log ^2\left (9 x \log \left (\frac {3}{x}\right )\right )} \, dx=\frac {3 \, {\left (x \log \left (3\right ) \log \left (\frac {3}{x}\right ) - x \log \left (x\right ) \log \left (\frac {3}{x}\right ) - \log \left (3\right ) \log \left (\frac {3}{x}\right ) + \log \left (x\right ) \log \left (\frac {3}{x}\right ) + \log \left (\frac {3}{x}\right )\right )}}{x^{2} \log \left (3\right ) \log \left (\frac {3}{x}\right ) - 2 \, x \log \left (3\right )^{2} \log \left (\frac {3}{x}\right ) - x^{2} \log \left (x\right ) \log \left (\frac {3}{x}\right ) + x \log \left (3\right ) \log \left (x\right ) \log \left (\frac {3}{x}\right ) + x \log \left (x\right )^{2} \log \left (\frac {3}{x}\right ) - x \log \left (3\right ) \log \left (\frac {3}{x}\right ) \log \left (\log \left (\frac {3}{x}\right )\right ) + x \log \left (x\right ) \log \left (\frac {3}{x}\right ) \log \left (\log \left (\frac {3}{x}\right )\right ) - x \log \left (3\right ) \log \left (\frac {3}{x}\right ) + 2 \, \log \left (3\right )^{2} \log \left (\frac {3}{x}\right ) + x \log \left (x\right ) \log \left (\frac {3}{x}\right ) - \log \left (3\right ) \log \left (x\right ) \log \left (\frac {3}{x}\right ) - \log \left (x\right )^{2} \log \left (\frac {3}{x}\right ) + \log \left (3\right ) \log \left (\frac {3}{x}\right ) \log \left (\log \left (\frac {3}{x}\right )\right ) - \log \left (x\right ) \log \left (\frac {3}{x}\right ) \log \left (\log \left (\frac {3}{x}\right )\right ) + x \log \left (3\right ) - 2 \, \log \left (3\right )^{2} - x \log \left (x\right ) + \log \left (3\right ) \log \left (x\right ) + \log \left (x\right )^{2} - \log \left (3\right ) \log \left (\log \left (\frac {3}{x}\right )\right ) + \log \left (x\right ) \log \left (\log \left (\frac {3}{x}\right )\right )} \]

[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)
))

Mupad [B] (verification not implemented)

Time = 13.90 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.82 \[ \int \frac {-3+(3-3 x) \log \left (\frac {3}{x}\right )}{x^3 \log \left (\frac {3}{x}\right )-2 x^2 \log \left (\frac {3}{x}\right ) \log \left (9 x \log \left (\frac {3}{x}\right )\right )+x \log \left (\frac {3}{x}\right ) \log ^2\left (9 x \log \left (\frac {3}{x}\right )\right )} \, dx=\frac {3}{x-\ln \left (9\,x\,\ln \left (\frac {3}{x}\right )\right )} \]

[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)))