Optimal. Leaf size=22 \[ e^{\frac {4 \left (\frac {2}{3}+x+\log (x)\right )}{x (-x+\log (3))}} \]
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Rubi [F]
time = 5.79, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps
used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {}
\begin {gather*} \int \frac {e^{\frac {8+12 x+12 \log (x)}{-3 x^2+3 x \log (3)}} \left (4 x+12 x^2+4 \log (3)+(24 x-12 \log (3)) \log (x)\right )}{3 x^4-6 x^3 \log (3)+3 x^2 \log ^2(3)} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {e^{\frac {8+12 x+12 \log (x)}{-3 x^2+3 x \log (3)}} \left (4 x+12 x^2+4 \log (3)+(24 x-12 \log (3)) \log (x)\right )}{x^2 \left (3 x^2-6 x \log (3)+3 \log ^2(3)\right )} \, dx\\ &=\int \frac {e^{\frac {8+12 x+12 \log (x)}{-3 x^2+3 x \log (3)}} \left (4 x+12 x^2+4 \log (3)+(24 x-12 \log (3)) \log (x)\right )}{3 x^2 (x-\log (3))^2} \, dx\\ &=\frac {1}{3} \int \frac {e^{\frac {8+12 x+12 \log (x)}{-3 x^2+3 x \log (3)}} \left (4 x+12 x^2+4 \log (3)+(24 x-12 \log (3)) \log (x)\right )}{x^2 (x-\log (3))^2} \, dx\\ &=\frac {1}{3} \int \frac {4 \exp \left (\frac {4 (2+3 x+3 \log (x))}{x (-3 x+\log (27))}\right ) \left (x+3 x^2+\log (3)+6 x \log (x)-3 \log (3) \log (x)\right )}{x^2 (x-\log (3))^2} \, dx\\ &=\frac {4}{3} \int \frac {\exp \left (\frac {4 (2+3 x+3 \log (x))}{x (-3 x+\log (27))}\right ) \left (x+3 x^2+\log (3)+6 x \log (x)-3 \log (3) \log (x)\right )}{x^2 (x-\log (3))^2} \, dx\\ &=\frac {4}{3} \int \left (\frac {\exp \left (\frac {4 (2+3 x+3 \log (x))}{x (-3 x+\log (27))}\right ) \left (x+3 x^2+\log (3)\right )}{x^2 (x-\log (3))^2}+\frac {3 \exp \left (\frac {4 (2+3 x+3 \log (x))}{x (-3 x+\log (27))}\right ) (2 x-\log (3)) \log (x)}{x^2 (x-\log (3))^2}\right ) \, dx\\ &=\frac {4}{3} \int \frac {\exp \left (\frac {4 (2+3 x+3 \log (x))}{x (-3 x+\log (27))}\right ) \left (x+3 x^2+\log (3)\right )}{x^2 (x-\log (3))^2} \, dx+4 \int \frac {\exp \left (\frac {4 (2+3 x+3 \log (x))}{x (-3 x+\log (27))}\right ) (2 x-\log (3)) \log (x)}{x^2 (x-\log (3))^2} \, dx\\ &=\frac {4}{3} \int \left (\frac {3 \exp \left (\frac {4 (2+3 x+3 \log (x))}{x (-3 x+\log (27))}\right )}{x \log ^2(3)}-\frac {3 \exp \left (\frac {4 (2+3 x+3 \log (x))}{x (-3 x+\log (27))}\right )}{(x-\log (3)) \log ^2(3)}+\frac {\exp \left (\frac {4 (2+3 x+3 \log (x))}{x (-3 x+\log (27))}\right )}{x^2 \log (3)}+\frac {\exp \left (\frac {4 (2+3 x+3 \log (x))}{x (-3 x+\log (27))}\right ) (2+\log (27))}{(x-\log (3))^2 \log (3)}\right ) \, dx+4 \int \left (-\frac {\exp \left (\frac {4 (2+3 x+3 \log (x))}{x (-3 x+\log (27))}\right ) \log (x)}{x^2 \log (3)}+\frac {\exp \left (\frac {4 (2+3 x+3 \log (x))}{x (-3 x+\log (27))}\right ) \log (x)}{(x-\log (3))^2 \log (3)}\right ) \, dx\\ &=\frac {4 \int \frac {\exp \left (\frac {4 (2+3 x+3 \log (x))}{x (-3 x+\log (27))}\right )}{x} \, dx}{\log ^2(3)}-\frac {4 \int \frac {\exp \left (\frac {4 (2+3 x+3 \log (x))}{x (-3 x+\log (27))}\right )}{x-\log (3)} \, dx}{\log ^2(3)}+\frac {4 \int \frac {\exp \left (\frac {4 (2+3 x+3 \log (x))}{x (-3 x+\log (27))}\right )}{x^2} \, dx}{3 \log (3)}-\frac {4 \int \frac {\exp \left (\frac {4 (2+3 x+3 \log (x))}{x (-3 x+\log (27))}\right ) \log (x)}{x^2} \, dx}{\log (3)}+\frac {4 \int \frac {\exp \left (\frac {4 (2+3 x+3 \log (x))}{x (-3 x+\log (27))}\right ) \log (x)}{(x-\log (3))^2} \, dx}{\log (3)}+\frac {(4 (2+\log (27))) \int \frac {\exp \left (\frac {4 (2+3 x+3 \log (x))}{x (-3 x+\log (27))}\right )}{(x-\log (3))^2} \, dx}{3 \log (3)}\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.71, size = 38, normalized size = 1.73 \begin {gather*} e^{-\frac {8+12 x}{3 x^2-x \log (27)}} x^{-\frac {4}{x^2-x \log (3)}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.52, size = 24, normalized size = 1.09
method | result | size |
risch | \({\mathrm e}^{\frac {4 \ln \left (x \right )+4 x +\frac {8}{3}}{x \left (\ln \left (3\right )-x \right )}}\) | \(24\) |
norman | \(\frac {x \ln \left (3\right ) {\mathrm e}^{\frac {12 \ln \left (x \right )+12 x +8}{3 x \ln \left (3\right )-3 x^{2}}}-x^{2} {\mathrm e}^{\frac {12 \ln \left (x \right )+12 x +8}{3 x \ln \left (3\right )-3 x^{2}}}}{x \left (\ln \left (3\right )-x \right )}\) | \(71\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 64 vs.
\(2 (23) = 46\).
time = 0.62, size = 64, normalized size = 2.91 \begin {gather*} e^{\left (-\frac {4 \, \log \left (x\right )}{x \log \left (3\right ) - \log \left (3\right )^{2}} - \frac {8}{3 \, {\left (x \log \left (3\right ) - \log \left (3\right )^{2}\right )}} - \frac {4}{x - \log \left (3\right )} + \frac {4 \, \log \left (x\right )}{x \log \left (3\right )} + \frac {8}{3 \, x \log \left (3\right )}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.35, size = 23, normalized size = 1.05 \begin {gather*} e^{\left (-\frac {4 \, {\left (3 \, x + 3 \, \log \left (x\right ) + 2\right )}}{3 \, {\left (x^{2} - x \log \left (3\right )\right )}}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.36, size = 22, normalized size = 1.00 \begin {gather*} e^{\frac {12 x + 12 \log {\left (x \right )} + 8}{- 3 x^{2} + 3 x \log {\left (3 \right )}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.43, size = 44, normalized size = 2.00 \begin {gather*} e^{\left (-\frac {4 \, x}{x^{2} - x \log \left (3\right )} - \frac {4 \, \log \left (x\right )}{x^{2} - x \log \left (3\right )} - \frac {8}{3 \, {\left (x^{2} - x \log \left (3\right )\right )}}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 4.82, size = 44, normalized size = 2.00 \begin {gather*} x^{\frac {4}{x\,\ln \left (3\right )-x^2}}\,{\mathrm {e}}^{-\frac {4}{x-\ln \left (3\right )}}\,{\mathrm {e}}^{\frac {8}{3\,x\,\ln \left (3\right )-3\,x^2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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