Optimal. Leaf size=28 \[ \frac {1}{4} \log ^2\left (4+\frac {3}{4 x \log \left (-2+e^5\right ) \log (x)}\right ) \]
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Rubi [A]
time = 1.43, antiderivative size = 28, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 7, integrand size = 63, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {6873, 12, 6874,
2339, 29, 6820, 6818} \begin {gather*} \frac {1}{4} \log ^2\left (\frac {3}{4 x \log \left (e^5-2\right ) \log (x)}+4\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 29
Rule 2339
Rule 6818
Rule 6820
Rule 6873
Rule 6874
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {3 (-1-\log (x)) \log \left (\frac {3+16 x \log \left (-2+e^5\right ) \log (x)}{4 x \log \left (-2+e^5\right ) \log (x)}\right )}{2 x \log (x) \left (3+16 x \log \left (-2+e^5\right ) \log (x)\right )} \, dx\\ &=\frac {3}{2} \int \frac {(-1-\log (x)) \log \left (\frac {3+16 x \log \left (-2+e^5\right ) \log (x)}{4 x \log \left (-2+e^5\right ) \log (x)}\right )}{x \log (x) \left (3+16 x \log \left (-2+e^5\right ) \log (x)\right )} \, dx\\ &=\frac {3}{2} \int \frac {(-1-\log (x)) \log \left (4+\frac {3}{4 x \log \left (-2+e^5\right ) \log (x)}\right )}{x \log (x) \left (3+16 x \log \left (-2+e^5\right ) \log (x)\right )} \, dx\\ &=\frac {1}{4} \log ^2\left (4+\frac {3}{4 x \log \left (-2+e^5\right ) \log (x)}\right )\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.04, size = 28, normalized size = 1.00 \begin {gather*} \frac {1}{4} \log ^2\left (4+\frac {3}{4 x \log \left (-2+e^5\right ) \log (x)}\right ) \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. \(96\) vs.
\(2(23)=46\).
time = 11.71, size = 97, normalized size = 3.46
method | result | size |
default | \(\frac {\ln \left (x \right ) \ln \left (\ln \left ({\mathrm e}^{5}-2\right )\right )}{2}+\frac {\ln \left (\ln \left ({\mathrm e}^{5}-2\right )\right ) \ln \left (\ln \left (x \right )\right )}{2}-\frac {\ln \left (\ln \left ({\mathrm e}^{5}-2\right )\right ) \ln \left (16 x \ln \left (x \right ) \ln \left ({\mathrm e}^{5}-2\right )+3\right )}{2}+\ln \left (2\right ) \ln \left (x \right )+\ln \left (2\right ) \ln \left (\ln \left (x \right )\right )-\ln \left (2\right ) \ln \left (16 x \ln \left (x \right ) \ln \left ({\mathrm e}^{5}-2\right )+3\right )+\frac {\ln \left (\frac {16 x \ln \left (x \right ) \ln \left ({\mathrm e}^{5}-2\right )+3}{x \ln \left (x \right )}\right )^{2}}{4}\) | \(97\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.42, size = 33, normalized size = 1.18 \begin {gather*} \frac {1}{4} \, \log \left (\frac {16 \, x \log \left (x\right ) \log \left (e^{5} - 2\right ) + 3}{4 \, x \log \left (x\right ) \log \left (e^{5} - 2\right )}\right )^{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.13, size = 32, normalized size = 1.14 \begin {gather*} \frac {\log {\left (\frac {4 x \log {\left (x \right )} \log {\left (-2 + e^{5} \right )} + \frac {3}{4}}{x \log {\left (x \right )} \log {\left (-2 + e^{5} \right )}} \right )}^{2}}{4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 7.54, size = 32, normalized size = 1.14 \begin {gather*} \frac {{\ln \left (\frac {4\,x\,\ln \left ({\mathrm {e}}^5-2\right )\,\ln \left (x\right )+\frac {3}{4}}{x\,\ln \left ({\mathrm {e}}^5-2\right )\,\ln \left (x\right )}\right )}^2}{4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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