Optimal. Leaf size=21 \[ \frac {1}{\log ^2\left (\frac {e+\frac {e^2}{x}-\frac {x}{3}}{x}\right )} \]
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Rubi [A]
time = 0.12, antiderivative size = 25, normalized size of antiderivative = 1.19, number of steps
used = 2, number of rules used = 2, integrand size = 56, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.036, Rules used = {1608, 6818}
\begin {gather*} \frac {1}{\log ^2\left (\frac {-x^2+3 e x+3 e^2}{3 x^2}\right )} \end {gather*}
Antiderivative was successfully verified.
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Rule 1608
Rule 6818
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {12 e^2+6 e x}{x \left (3 e^2+3 e x-x^2\right ) \log ^3\left (\frac {3 e^2+3 e x-x^2}{3 x^2}\right )} \, dx\\ &=\frac {1}{\log ^2\left (\frac {3 e^2+3 e x-x^2}{3 x^2}\right )}\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.02, size = 19, normalized size = 0.90 \begin {gather*} \frac {1}{\log ^2\left (-\frac {1}{3}+\frac {e^2}{x^2}+\frac {e}{x}\right )} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.24, size = 25, normalized size = 1.19
method | result | size |
norman | \(\frac {1}{\ln \left (\frac {3 \,{\mathrm e}^{2}+3 x \,{\mathrm e}-x^{2}}{3 x^{2}}\right )^{2}}\) | \(24\) |
risch | \(\frac {1}{\ln \left (\frac {3 \,{\mathrm e}^{2}+3 x \,{\mathrm e}-x^{2}}{3 x^{2}}\right )^{2}}\) | \(24\) |
derivativedivides | \(\frac {1}{\left (\ln \left (3\right )-\ln \left (\frac {3 \,{\mathrm e}^{2}}{x^{2}}+\frac {3 \,{\mathrm e}}{x}-1\right )\right )^{2}}\) | \(25\) |
default | \(\frac {1}{\left (\ln \left (3\right )-\ln \left (\frac {3 \,{\mathrm e}^{2}}{x^{2}}+\frac {3 \,{\mathrm e}}{x}-1\right )\right )^{2}}\) | \(25\) |
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. 62 vs.
\(2 (21) = 42\).
time = 0.50, size = 62, normalized size = 2.95 \begin {gather*} \frac {1}{\log \left (3\right )^{2} - 2 \, {\left (\log \left (3\right ) + 2 \, \log \left (x\right )\right )} \log \left (-x^{2} + 3 \, x e + 3 \, e^{2}\right ) + \log \left (-x^{2} + 3 \, x e + 3 \, e^{2}\right )^{2} + 4 \, \log \left (3\right ) \log \left (x\right ) + 4 \, \log \left (x\right )^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.37, size = 21, normalized size = 1.00 \begin {gather*} \frac {1}{\log \left (-\frac {x^{2} - 3 \, x e - 3 \, e^{2}}{3 \, x^{2}}\right )^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.06, size = 20, normalized size = 0.95 \begin {gather*} \frac {1}{\log {\left (\frac {- \frac {x^{2}}{3} + e x + e^{2}}{x^{2}} \right )}^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.41, size = 21, normalized size = 1.00 \begin {gather*} \frac {1}{\log \left (-\frac {x^{2} - 3 \, x e - 3 \, e^{2}}{3 \, x^{2}}\right )^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.92, size = 23, normalized size = 1.10 \begin {gather*} \frac {1}{{\ln \left (\frac {-x^2+3\,\mathrm {e}\,x+3\,{\mathrm {e}}^2}{3\,x^2}\right )}^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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