3.72.29 \(\int (-3-2 \log (\frac {3 x^2}{\log ^2(2 e^3)})) \, dx\)

Optimal. Leaf size=19 \[ x-2 x \log \left (\frac {3 x^2}{\log ^2\left (2 e^3\right )}\right ) \]

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Rubi [A]  time = 0.01, antiderivative size = 17, normalized size of antiderivative = 0.89, number of steps used = 2, number of rules used = 1, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used = {2295} \begin {gather*} x-2 x \log \left (\frac {3 x^2}{(3+\log (2))^2}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

x - 2*x*Log[(3*x^2)/(3 + Log[2])^2]

Rule 2295

Int[Log[(c_.)*(x_)^(n_.)], x_Symbol] :> Simp[x*Log[c*x^n], x] - Simp[n*x, x] /; FreeQ[{c, n}, x]

Rubi steps

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

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Mathematica [A]  time = 0.00, size = 17, normalized size = 0.89 \begin {gather*} x-2 x \log \left (\frac {3 x^2}{(3+\log (2))^2}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

x - 2*x*Log[(3*x^2)/(3 + Log[2])^2]

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fricas [A]  time = 0.61, size = 23, normalized size = 1.21 \begin {gather*} -2 \, x \log \left (\frac {3 \, x^{2}}{\log \relax (2)^{2} + 6 \, \log \relax (2) + 9}\right ) + x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*x*log(3*x^2/(log(2)^2 + 6*log(2) + 9)) + x

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giac [A]  time = 0.13, size = 18, normalized size = 0.95 \begin {gather*} -2 \, x \log \left (\frac {3 \, x^{2}}{\log \left (2 \, e^{3}\right )^{2}}\right ) + x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*x*log(3*x^2/log(2*e^3)^2) + x

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maple [A]  time = 0.02, size = 18, normalized size = 0.95

method	result	size
risch	\(-2 \ln \left (\frac {3 x^{2}}{\left (3+\ln \relax (2)\right )^{2}}\right ) x +x\)	\(18\)
norman	\(x -2 x \ln \left (\frac {3 x^{2}}{\ln \left (2 \,{\mathrm e}^{3}\right )^{2}}\right )\)	\(19\)
default	\(x +4 \ln \left (3+\ln \relax (2)\right ) x -2 x \ln \relax (3)-2 x \ln \left (x^{2}\right )\)	\(23\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.35, size = 18, normalized size = 0.95 \begin {gather*} -2 \, x \log \left (\frac {3 \, x^{2}}{\log \left (2 \, e^{3}\right )^{2}}\right ) + x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*x*log(3*x^2/log(2*e^3)^2) + x

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mupad [B]  time = 4.18, size = 28, normalized size = 1.47 \begin {gather*} -x\,\left (2\,\ln \left (x^2\right )-2\,\ln \left (6\,\ln \relax (2)+{\ln \relax (2)}^2+9\right )+2\,\ln \relax (3)-1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-x*(2*log(x^2) - 2*log(6*log(2) + log(2)^2 + 9) + 2*log(3) - 1)

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sympy [A]  time = 0.11, size = 19, normalized size = 1.00 \begin {gather*} - 2 x \log {\left (\frac {3 x^{2}}{\log {\left (2 e^{3} \right )}^{2}} \right )} + x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(-2*ln(3*x**2/ln(2*exp(3))**2)-3,x)

[Out]

-2*x*log(3*x**2/log(2*exp(3))**2) + x

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