\(\int \frac {2 \log (9)-\log (9) \log (\frac {3}{x^2})}{x \log (\frac {3}{x^2})} \, dx\) [2990]

Optimal result

Integrand size = 27, antiderivative size = 25 \[ \int \frac {2 \log (9)-\log (9) \log \left (\frac {3}{x^2}\right )}{x \log \left (\frac {3}{x^2}\right )} \, dx=\log (9) \log \left (\frac {2}{x \log (3) \log (5) \log \left (\frac {3}{x^2}\right )}\right ) \]

[Out]

2*ln(2/x/ln(3)/ln(5)/ln(3/x^2))*ln(3)

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.72, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.074, Rules used = {2412, 45} \[ \int \frac {2 \log (9)-\log (9) \log \left (\frac {3}{x^2}\right )}{x \log \left (\frac {3}{x^2}\right )} \, dx=-\log (9) \log \left (\log \left (\frac {3}{x^2}\right )\right )-\log (9) \log (x) \]

[In]

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

[Out]

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

Rule 45

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 2412

Int[(((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_.) + Log[(c_.)*(x_)^(n_.)]*(e_.))^(q_.))/(x_), x_Symbol]
:> Dist[1/n, Subst[Int[(a + b*x)^p*(d + e*x)^q, x], x, Log[c*x^n]], x] /; FreeQ[{a, b, c, d, e, n, p, q}, x]

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

Mathematica [A] (verified)

Time = 0.04 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.72 \[ \int \frac {2 \log (9)-\log (9) \log \left (\frac {3}{x^2}\right )}{x \log \left (\frac {3}{x^2}\right )} \, dx=-\log (9) \log (x)-\log (9) \log \left (\log \left (\frac {3}{x^2}\right )\right ) \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 0.14 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.76

method	result	size
risch	\(-2 \ln \left (3\right ) \ln \left (x \right )-2 \ln \left (3\right ) \ln \left (\ln \left (\frac {3}{x^{2}}\right )\right )\)	\(19\)
parts	\(-2 \ln \left (3\right ) \ln \left (x \right )-2 \ln \left (3\right ) \ln \left (\ln \left (\frac {3}{x^{2}}\right )\right )\)	\(19\)
norman	\(\ln \left (3\right ) \ln \left (\frac {3}{x^{2}}\right )-2 \ln \left (3\right ) \ln \left (\ln \left (\frac {3}{x^{2}}\right )\right )\)	\(22\)
parallelrisch	\(\ln \left (3\right ) \ln \left (\frac {3}{x^{2}}\right )-2 \ln \left (3\right ) \ln \left (\ln \left (\frac {3}{x^{2}}\right )\right )\)	\(22\)
derivativedivides	\(-\ln \left (3\right ) \left (-\ln \left (\frac {3}{x^{2}}\right )+2 \ln \left (\ln \left (\frac {3}{x^{2}}\right )\right )\right )\)	\(23\)
default	\(-\ln \left (3\right ) \left (-\ln \left (\frac {3}{x^{2}}\right )+2 \ln \left (\ln \left (\frac {3}{x^{2}}\right )\right )\right )\)	\(23\)

[In]

int((-2*ln(3)*ln(3/x^2)+4*ln(3))/x/ln(3/x^2),x,method=_RETURNVERBOSE)

[Out]

-2*ln(3)*ln(x)-2*ln(3)*ln(ln(3/x^2))

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.84 \[ \int \frac {2 \log (9)-\log (9) \log \left (\frac {3}{x^2}\right )}{x \log \left (\frac {3}{x^2}\right )} \, dx=\log \left (3\right ) \log \left (\frac {3}{x^{2}}\right ) - 2 \, \log \left (3\right ) \log \left (\log \left (\frac {3}{x^{2}}\right )\right ) \]

[In]

integrate((-2*log(3)*log(3/x^2)+4*log(3))/x/log(3/x^2),x, algorithm="fricas")

[Out]

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

Sympy [A] (verification not implemented)

Time = 0.06 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.88 \[ \int \frac {2 \log (9)-\log (9) \log \left (\frac {3}{x^2}\right )}{x \log \left (\frac {3}{x^2}\right )} \, dx=- 2 \log {\left (3 \right )} \log {\left (x \right )} - 2 \log {\left (3 \right )} \log {\left (\log {\left (\frac {3}{x^{2}} \right )} \right )} \]

[In]

integrate((-2*ln(3)*ln(3/x**2)+4*ln(3))/x/ln(3/x**2),x)

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.72 \[ \int \frac {2 \log (9)-\log (9) \log \left (\frac {3}{x^2}\right )}{x \log \left (\frac {3}{x^2}\right )} \, dx=-2 \, \log \left (3\right ) \log \left (x\right ) - 2 \, \log \left (3\right ) \log \left (\log \left (\frac {3}{x^{2}}\right )\right ) \]

[In]

integrate((-2*log(3)*log(3/x^2)+4*log(3))/x/log(3/x^2),x, algorithm="maxima")

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.84 \[ \int \frac {2 \log (9)-\log (9) \log \left (\frac {3}{x^2}\right )}{x \log \left (\frac {3}{x^2}\right )} \, dx=-2 \, \log \left (3\right ) \log \left (x\right ) - 2 \, \log \left (3\right ) \log \left (-\log \left (3\right ) + \log \left (x^{2}\right )\right ) \]

[In]

integrate((-2*log(3)*log(3/x^2)+4*log(3))/x/log(3/x^2),x, algorithm="giac")

[Out]

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

Mupad [B] (verification not implemented)

Time = 10.02 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.76 \[ \int \frac {2 \log (9)-\log (9) \log \left (\frac {3}{x^2}\right )}{x \log \left (\frac {3}{x^2}\right )} \, dx=\ln \left (\frac {1}{x^2}\right )\,\ln \left (3\right )-\ln \left (\ln \left (\frac {3}{x^2}\right )\right )\,\ln \left (9\right ) \]

[In]

int((4*log(3) - 2*log(3)*log(3/x^2))/(x*log(3/x^2)),x)

[Out]

log(1/x^2)*log(3) - log(log(3/x^2))*log(9)