3.81.30 \(\int \frac {4 x \log (3 x^2)+2 x \log ^2(3 x^2)}{-9+x^2 \log ^2(3 x^2)} \, dx\)

Optimal. Leaf size=23 \[ 5-e^4+\log \left (9-x^2 \log ^2\left (3 x^2\right )\right ) \]

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Rubi [A]  time = 0.03, antiderivative size = 16, normalized size of antiderivative = 0.70, number of steps used = 1, number of rules used = 1, integrand size = 38, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.026, Rules used = {6684} \begin {gather*} \log \left (9-x^2 \log ^2\left (3 x^2\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(4*x*Log[3*x^2] + 2*x*Log[3*x^2]^2)/(-9 + x^2*Log[3*x^2]^2),x]

[Out]

Log[9 - x^2*Log[3*x^2]^2]

Rule 6684

Int[(u_)/(y_), x_Symbol] :> With[{q = DerivativeDivides[y, u, x]}, Simp[q*Log[RemoveContent[y, x]], x] /;  !Fa
lseQ[q]]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\log \left (9-x^2 \log ^2\left (3 x^2\right )\right )\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.09, size = 16, normalized size = 0.70 \begin {gather*} \log \left (-27+3 x^2 \log ^2\left (3 x^2\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(4*x*Log[3*x^2] + 2*x*Log[3*x^2]^2)/(-9 + x^2*Log[3*x^2]^2),x]

[Out]

Log[-27 + 3*x^2*Log[3*x^2]^2]

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fricas [A]  time = 0.64, size = 26, normalized size = 1.13 \begin {gather*} \log \left (3 \, x^{2}\right ) + \log \left (\frac {x^{2} \log \left (3 \, x^{2}\right )^{2} - 9}{x^{2}}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*x*log(3*x^2)^2+4*x*log(3*x^2))/(x^2*log(3*x^2)^2-9),x, algorithm="fricas")

[Out]

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

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giac [A]  time = 0.21, size = 16, normalized size = 0.70 \begin {gather*} \log \left (3 \, x^{2} \log \left (3 \, x^{2}\right )^{2} - 27\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*x*log(3*x^2)^2+4*x*log(3*x^2))/(x^2*log(3*x^2)^2-9),x, algorithm="giac")

[Out]

log(3*x^2*log(3*x^2)^2 - 27)

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maple [A]  time = 0.06, size = 16, normalized size = 0.70

method	result	size
derivativedivides	\(\ln \left (x^{2} \ln \left (3 x^{2}\right )^{2}-9\right )\)	\(16\)
risch	\(2 \ln \relax (x )+\ln \left (\ln \left (3 x^{2}\right )^{2}-\frac {9}{x^{2}}\right )\)	\(21\)
norman	\(\ln \left (x \ln \left (3 x^{2}\right )-3\right )+\ln \left (x \ln \left (3 x^{2}\right )+3\right )\)	\(24\)
default	\(\ln \left (x^{2} \ln \relax (3)^{2}+2 x^{2} \ln \relax (3) \ln \left (x^{2}\right )+x^{2} \ln \left (x^{2}\right )^{2}-9\right )\)	\(33\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*x*ln(3*x^2)^2+4*x*ln(3*x^2))/(x^2*ln(3*x^2)^2-9),x,method=_RETURNVERBOSE)

[Out]

ln(x^2*ln(3*x^2)^2-9)

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maxima [A]  time = 0.36, size = 15, normalized size = 0.65 \begin {gather*} \log \left (x^{2} \log \left (3 \, x^{2}\right )^{2} - 9\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*x*log(3*x^2)^2+4*x*log(3*x^2))/(x^2*log(3*x^2)^2-9),x, algorithm="maxima")

[Out]

log(x^2*log(3*x^2)^2 - 9)

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mupad [B]  time = 6.20, size = 23, normalized size = 1.00 \begin {gather*} \ln \left (x\,\ln \left (3\,x^2\right )-3\right )+\ln \left (x\,\ln \left (3\,x^2\right )+3\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((4*x*log(3*x^2) + 2*x*log(3*x^2)^2)/(x^2*log(3*x^2)^2 - 9),x)

[Out]

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

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sympy [A]  time = 0.17, size = 19, normalized size = 0.83 \begin {gather*} 2 \log {\relax (x )} + \log {\left (\log {\left (3 x^{2} \right )}^{2} - \frac {9}{x^{2}} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*x*ln(3*x**2)**2+4*x*ln(3*x**2))/(x**2*ln(3*x**2)**2-9),x)

[Out]

2*log(x) + log(log(3*x**2)**2 - 9/x**2)

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