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

Optimal. Leaf size=25 \[ \log (9) \log \left (\frac {2}{x \log (3) \log (5) \log \left (\frac {3}{x^2}\right )}\right ) \]

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Rubi [A]  time = 0.04, antiderivative size = 18, normalized size of antiderivative = 0.72, number of steps used = 3, number of rules used = 2, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.074, Rules used = {2365, 43} \begin {gather*} -\log (9) \log \left (\log \left (\frac {3}{x^2}\right )\right )-\log (9) \log (x) \end {gather*}

Antiderivative was successfully verified.

[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 43

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 2365

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 {gather*} \begin {aligned} \text {integral} &=-\left (\frac {1}{2} \operatorname {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} \operatorname {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 {aligned} \end {gather*}

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Mathematica [A]  time = 0.02, size = 18, normalized size = 0.72 \begin {gather*} -\log (9) \log (x)-\log (9) \log \left (\log \left (\frac {3}{x^2}\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[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]]

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fricas [A]  time = 5.42, size = 21, normalized size = 0.84 \begin {gather*} \log \relax (3) \log \left (\frac {3}{x^{2}}\right ) - 2 \, \log \relax (3) \log \left (\log \left (\frac {3}{x^{2}}\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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))

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giac [A]  time = 0.34, size = 21, normalized size = 0.84 \begin {gather*} -2 \, \log \relax (3) \log \relax (x) - 2 \, \log \relax (3) \log \left (-\log \relax (3) + \log \left (x^{2}\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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))

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maple [A]  time = 0.03, size = 19, normalized size = 0.76

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

Verification of antiderivative is not currently implemented for this CAS.

[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))

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maxima [A]  time = 0.75, size = 18, normalized size = 0.72 \begin {gather*} -2 \, \log \relax (3) \log \relax (x) - 2 \, \log \relax (3) \log \left (\log \left (\frac {3}{x^{2}}\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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))

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mupad [B]  time = 1.75, size = 19, normalized size = 0.76 \begin {gather*} \ln \left (\frac {1}{x^2}\right )\,\ln \relax (3)-\ln \left (\ln \left (\frac {3}{x^2}\right )\right )\,\ln \relax (9) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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sympy [A]  time = 0.11, size = 22, normalized size = 0.88 \begin {gather*} - 2 \log {\relax (3 )} \log {\relax (x )} - 2 \log {\relax (3 )} \log {\left (\log {\left (\frac {3}{x^{2}} \right )} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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))

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