Optimal. Leaf size=17 \[ \log \left (\frac {x \log (x)}{1-e^2+3 x}\right ) \]
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Rubi [A]
time = 0.18, antiderivative size = 19, normalized size of antiderivative = 1.12, number of steps
used = 9, number of rules used = 7, integrand size = 37, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.189, Rules used = {6, 1607, 6874,
36, 31, 29, 2339} \begin {gather*} \log (x)-\log \left (3 x-e^2+1\right )+\log (\log (x)) \end {gather*}
Antiderivative was successfully verified.
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Rule 6
Rule 29
Rule 31
Rule 36
Rule 1607
Rule 2339
Rule 6874
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-1+e^2-3 x+\left (-1+e^2\right ) \log (x)}{\left (\left (-1+e^2\right ) x-3 x^2\right ) \log (x)} \, dx\\ &=\int \frac {-1+e^2-3 x+\left (-1+e^2\right ) \log (x)}{\left (-1+e^2-3 x\right ) x \log (x)} \, dx\\ &=\int \left (\frac {-1+e^2}{\left (-1+e^2-3 x\right ) x}+\frac {1}{x \log (x)}\right ) \, dx\\ &=\left (-1+e^2\right ) \int \frac {1}{\left (-1+e^2-3 x\right ) x} \, dx+\int \frac {1}{x \log (x)} \, dx\\ &=3 \int \frac {1}{-1+e^2-3 x} \, dx+\int \frac {1}{x} \, dx+\text {Subst}\left (\int \frac {1}{x} \, dx,x,\log (x)\right )\\ &=\log (x)-\log \left (1-e^2+3 x\right )+\log (\log (x))\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.04, size = 19, normalized size = 1.12 \begin {gather*} \log (x)-\log \left (1-e^2+3 x\right )+\log (\log (x)) \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(62\) vs.
\(2(16)=32\).
time = 0.62, size = 63, normalized size = 3.71
method | result | size |
norman | \(\ln \left (x \right )-\ln \left ({\mathrm e}^{2}-1-3 x \right )+\ln \left (\ln \left (x \right )\right )\) | \(17\) |
risch | \(-\ln \left (1+3 x -{\mathrm e}^{2}\right )+\ln \left (x \right )+\ln \left (\ln \left (x \right )\right )\) | \(19\) |
default | \(\ln \left (\ln \left (x \right )\right )-\frac {{\mathrm e}^{2} \ln \left (1+3 x -{\mathrm e}^{2}\right )}{{\mathrm e}^{2}-1}+\frac {{\mathrm e}^{2} \ln \left (x \right )}{{\mathrm e}^{2}-1}+\frac {\ln \left (1+3 x -{\mathrm e}^{2}\right )}{{\mathrm e}^{2}-1}-\frac {\ln \left (x \right )}{{\mathrm e}^{2}-1}\) | \(63\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 18, normalized size = 1.06 \begin {gather*} -\log \left (3 \, x - e^{2} + 1\right ) + \log \left (x\right ) + \log \left (\log \left (x\right )\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.46, size = 18, normalized size = 1.06 \begin {gather*} -\log \left (3 \, x - e^{2} + 1\right ) + \log \left (x\right ) + \log \left (\log \left (x\right )\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 141 vs.
\(2 (14) = 28\).
time = 0.19, size = 141, normalized size = 8.29 \begin {gather*} \left (1 - e^{2}\right ) \left (\frac {\log {\left (x - \frac {e^{4}}{6 \left (-1 + e\right ) \left (1 + e\right )} - \frac {e^{2}}{6} - \frac {1}{6 \left (-1 + e\right ) \left (1 + e\right )} + \frac {1}{6} + \frac {e^{2}}{3 \left (-1 + e\right ) \left (1 + e\right )} \right )}}{\left (-1 + e\right ) \left (1 + e\right )} - \frac {\log {\left (x - \frac {e^{2}}{6} - \frac {e^{2}}{3 \left (-1 + e\right ) \left (1 + e\right )} + \frac {1}{6 \left (-1 + e\right ) \left (1 + e\right )} + \frac {1}{6} + \frac {e^{4}}{6 \left (-1 + e\right ) \left (1 + e\right )} \right )}}{\left (-1 + e\right ) \left (1 + e\right )}\right ) + \log {\left (\log {\left (x \right )} \right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.39, size = 18, normalized size = 1.06 \begin {gather*} -\log \left (3 \, x - e^{2} + 1\right ) + \log \left (x\right ) + \log \left (\log \left (x\right )\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 4.56, size = 18, normalized size = 1.06 \begin {gather*} \ln \left (\ln \left (x\right )\right )-\ln \left (3\,x-{\mathrm {e}}^2+1\right )+\ln \left (x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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