Optimal. Leaf size=25 \[ \frac {x \left (-x+\log \left (\frac {5}{\log (4) \log (2 x)}\right )\right )}{192 e^6} \]
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Rubi [A]
time = 0.04, antiderivative size = 32, normalized size of antiderivative = 1.28, number of steps
used = 6, number of rules used = 4, integrand size = 40, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {12, 6820, 2335,
2600} \begin {gather*} \frac {x \log \left (\frac {5}{\log (4) \log (2 x)}\right )}{192 e^6}-\frac {x^2}{192 e^6} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 2335
Rule 2600
Rule 6820
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\frac {\int \frac {-1-2 x \log (2 x)+\log (2 x) \log \left (\frac {5}{\log (4) \log (2 x)}\right )}{\log (2 x)} \, dx}{192 e^6}\\ &=\frac {\int \left (-2 x-\frac {1}{\log (2 x)}+\log \left (\frac {5}{\log (4) \log (2 x)}\right )\right ) \, dx}{192 e^6}\\ &=-\frac {x^2}{192 e^6}-\frac {\int \frac {1}{\log (2 x)} \, dx}{192 e^6}+\frac {\int \log \left (\frac {5}{\log (4) \log (2 x)}\right ) \, dx}{192 e^6}\\ &=-\frac {x^2}{192 e^6}+\frac {x \log \left (\frac {5}{\log (4) \log (2 x)}\right )}{192 e^6}-\frac {\text {li}(2 x)}{384 e^6}+\frac {\int \frac {1}{\log (2 x)} \, dx}{192 e^6}\\ &=-\frac {x^2}{192 e^6}+\frac {x \log \left (\frac {5}{\log (4) \log (2 x)}\right )}{192 e^6}\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.03, size = 28, normalized size = 1.12 \begin {gather*} \frac {-x^2+x \log \left (\frac {5}{\log (4) \log (2 x)}\right )}{192 e^6} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 4 vs. order
3.
time = 0.58, size = 60, normalized size = 2.40
method | result | size |
norman | \(-\frac {{\mathrm e}^{-6} x^{2}}{192}+\frac {x \,{\mathrm e}^{-6} \ln \left (\frac {5}{2 \ln \left (2\right ) \ln \left (2 x \right )}\right )}{192}\) | \(31\) |
risch | \(-\frac {{\mathrm e}^{-6} x \ln \left (\ln \left (2 x \right )\right )}{192}-\frac {{\mathrm e}^{-6} x \left (2 \ln \left (2\right )-2 \ln \left (5\right )+2 \ln \left (\ln \left (2\right )\right )+2 x \right )}{384}\) | \(34\) |
default | \(\frac {{\mathrm e}^{-6} \left (\frac {\expIntegral \left (1, -\ln \left (2\right )-\ln \left (x \right )\right )}{2}-x \ln \left (2\right )-x^{2}-x \ln \left (\ln \left (2\right )\right )+x \ln \left (5\right )+\ln \left (\frac {1}{\ln \left (2 x \right )}\right ) x -\frac {\expIntegral \left (1, -\ln \left (2 x \right )\right )}{2}\right )}{192}\) | \(60\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 24, normalized size = 0.96 \begin {gather*} -\frac {1}{192} \, {\left (x^{2} - x \log \left (\frac {5}{2 \, \log \left (2\right ) \log \left (2 \, x\right )}\right )\right )} e^{\left (-6\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.35, size = 24, normalized size = 0.96 \begin {gather*} -\frac {1}{192} \, {\left (x^{2} - x \log \left (\frac {5}{2 \, \log \left (2\right ) \log \left (2 \, x\right )}\right )\right )} e^{\left (-6\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.11, size = 27, normalized size = 1.08 \begin {gather*} - \frac {x^{2}}{192 e^{6}} + \frac {x \log {\left (\frac {5}{2 \log {\left (2 \right )} \log {\left (2 x \right )}} \right )}}{192 e^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.40, size = 29, normalized size = 1.16 \begin {gather*} -\frac {1}{192} \, {\left (x^{2} - x \log \left (5\right ) + x \log \left (2\right ) + x \log \left (\log \left (2\right )\right ) + x \log \left (\log \left (2 \, x\right )\right )\right )} e^{\left (-6\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.03, size = 22, normalized size = 0.88 \begin {gather*} -\frac {x\,{\mathrm {e}}^{-6}\,\left (x-\ln \left (\frac {5}{2\,\ln \left (2\,x\right )\,\ln \left (2\right )}\right )\right )}{192} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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