Optimal. Leaf size=22 \[ -\frac {1}{x}+x-\frac {20 e^2 x}{3 \log (x \log (4))} \]
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Rubi [A] time = 0.24, antiderivative size = 22, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 51, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.078, Rules used = {12, 6688, 2297, 2298} \begin {gather*} x-\frac {1}{x}-\frac {20 e^2 x}{3 \log (x \log (4))} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 2297
Rule 2298
Rule 6688
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\frac {1}{3} \int \frac {20 e^2 x^2-20 e^2 x^2 \log (x \log (4))+\left (3+3 x^2\right ) \log ^2(x \log (4))}{x^2 \log ^2(x \log (4))} \, dx\\ &=\frac {1}{3} \int \left (3+\frac {3}{x^2}+\frac {20 e^2}{\log ^2(x \log (4))}-\frac {20 e^2}{\log (x \log (4))}\right ) \, dx\\ &=-\frac {1}{x}+x+\frac {1}{3} \left (20 e^2\right ) \int \frac {1}{\log ^2(x \log (4))} \, dx-\frac {1}{3} \left (20 e^2\right ) \int \frac {1}{\log (x \log (4))} \, dx\\ &=-\frac {1}{x}+x-\frac {20 e^2 x}{3 \log (x \log (4))}-\frac {20 e^2 \text {li}(x \log (4))}{3 \log (4)}+\frac {1}{3} \left (20 e^2\right ) \int \frac {1}{\log (x \log (4))} \, dx\\ &=-\frac {1}{x}+x-\frac {20 e^2 x}{3 \log (x \log (4))}\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.02, size = 22, normalized size = 1.00 \begin {gather*} -\frac {1}{x}+x-\frac {20 e^2 x}{3 \log (x \log (4))} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.66, size = 34, normalized size = 1.55 \begin {gather*} -\frac {20 \, x^{2} e^{2} - 3 \, {\left (x^{2} - 1\right )} \log \left (2 \, x \log \relax (2)\right )}{3 \, x \log \left (2 \, x \log \relax (2)\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.32, size = 52, normalized size = 2.36 \begin {gather*} -\frac {20 \, x^{2} e^{2} - 3 \, x^{2} \log \relax (2) - 3 \, x^{2} \log \left (x \log \relax (2)\right ) + 3 \, \log \relax (2) + 3 \, \log \left (x \log \relax (2)\right )}{3 \, {\left (x \log \relax (2) + x \log \left (x \log \relax (2)\right )\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.07, size = 24, normalized size = 1.09
method | result | size |
risch | \(\frac {x^{2}-1}{x}-\frac {20 x \,{\mathrm e}^{2}}{3 \ln \left (2 x \ln \relax (2)\right )}\) | \(24\) |
norman | \(\frac {x^{2} \ln \left (2 x \ln \relax (2)\right )-\frac {20 x^{2} {\mathrm e}^{2}}{3}-\ln \left (2 x \ln \relax (2)\right )}{x \ln \left (2 x \ln \relax (2)\right )}\) | \(39\) |
derivativedivides | \(\frac {2 \ln \relax (2) \left (\frac {5 \,{\mathrm e}^{2} \expIntegralEi \left (1, -\ln \left (2 x \ln \relax (2)\right )\right )}{\ln \relax (2)^{2}}-\frac {3}{2 x \ln \relax (2)}+\frac {3 x}{2 \ln \relax (2)}+\frac {5 \,{\mathrm e}^{2} \left (-\frac {2 x \ln \relax (2)}{\ln \left (2 x \ln \relax (2)\right )}-\expIntegralEi \left (1, -\ln \left (2 x \ln \relax (2)\right )\right )\right )}{\ln \relax (2)^{2}}\right )}{3}\) | \(74\) |
default | \(\frac {2 \ln \relax (2) \left (\frac {5 \,{\mathrm e}^{2} \expIntegralEi \left (1, -\ln \left (2 x \ln \relax (2)\right )\right )}{\ln \relax (2)^{2}}-\frac {3}{2 x \ln \relax (2)}+\frac {3 x}{2 \ln \relax (2)}+\frac {5 \,{\mathrm e}^{2} \left (-\frac {2 x \ln \relax (2)}{\ln \left (2 x \ln \relax (2)\right )}-\expIntegralEi \left (1, -\ln \left (2 x \ln \relax (2)\right )\right )\right )}{\ln \relax (2)^{2}}\right )}{3}\) | \(74\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [C] time = 0.61, size = 40, normalized size = 1.82 \begin {gather*} x - \frac {10 \, {\rm Ei}\left (\log \left (2 \, x \log \relax (2)\right )\right ) e^{2}}{3 \, \log \relax (2)} + \frac {10 \, e^{2} \Gamma \left (-1, -\log \left (2 \, x \log \relax (2)\right )\right )}{3 \, \log \relax (2)} - \frac {1}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.18, size = 20, normalized size = 0.91 \begin {gather*} x-\frac {1}{x}-\frac {20\,x\,{\mathrm {e}}^2}{3\,\ln \left (2\,x\,\ln \relax (2)\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.15, size = 20, normalized size = 0.91 \begin {gather*} x - \frac {20 x e^{2}}{3 \log {\left (2 x \log {\relax (2 )} \right )}} - \frac {1}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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