Optimal. Leaf size=30 \[ \log \left (9 e^{-2 x+2 x^2} x^4 \log ^2(5) \log ^2\left (\frac {\log (x)}{x^2}\right )\right ) \]
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Rubi [A]
time = 0.17, antiderivative size = 23, normalized size of antiderivative = 0.77, number of steps
used = 5, number of rules used = 3, integrand size = 43, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.070, Rules used = {6874, 14, 6816}
\begin {gather*} 2 x^2+2 \log \left (\log \left (\frac {\log (x)}{x^2}\right )\right )-2 x+4 \log (x) \end {gather*}
Antiderivative was successfully verified.
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Rule 14
Rule 6816
Rule 6874
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \left (\frac {2 \left (2-x+2 x^2\right )}{x}-\frac {2 (-1+2 \log (x))}{x \log (x) \log \left (\frac {\log (x)}{x^2}\right )}\right ) \, dx\\ &=2 \int \frac {2-x+2 x^2}{x} \, dx-2 \int \frac {-1+2 \log (x)}{x \log (x) \log \left (\frac {\log (x)}{x^2}\right )} \, dx\\ &=2 \log \left (\log \left (\frac {\log (x)}{x^2}\right )\right )+2 \int \left (-1+\frac {2}{x}+2 x\right ) \, dx\\ &=-2 x+2 x^2+4 \log (x)+2 \log \left (\log \left (\frac {\log (x)}{x^2}\right )\right )\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.06, size = 23, normalized size = 0.77 \begin {gather*} -2 x+2 x^2+4 \log (x)+2 \log \left (\log \left (\frac {\log (x)}{x^2}\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.96, size = 24, normalized size = 0.80
method | result | size |
norman | \(2 x^{2}-2 x +4 \ln \left (x \right )+2 \ln \left (\ln \left (\frac {\ln \left (x \right )}{x^{2}}\right )\right )\) | \(24\) |
risch | \(2 x^{2}-2 x +4 \ln \left (x \right )+2 \ln \left (\ln \left (\ln \left (x \right )\right )-\frac {i \left (\pi \,\mathrm {csgn}\left (i \ln \left (x \right )\right ) \mathrm {csgn}\left (\frac {i}{x^{2}}\right ) \mathrm {csgn}\left (\frac {i \ln \left (x \right )}{x^{2}}\right )-\pi \,\mathrm {csgn}\left (i \ln \left (x \right )\right ) \mathrm {csgn}\left (\frac {i \ln \left (x \right )}{x^{2}}\right )^{2}-\pi \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i x^{2}\right )+2 \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{2}\right )^{2}-\pi \mathrm {csgn}\left (i x^{2}\right )^{3}-\pi \,\mathrm {csgn}\left (\frac {i}{x^{2}}\right ) \mathrm {csgn}\left (\frac {i \ln \left (x \right )}{x^{2}}\right )^{2}+\pi \mathrm {csgn}\left (\frac {i \ln \left (x \right )}{x^{2}}\right )^{3}-4 i \ln \left (x \right )\right )}{2}\right )\) | \(154\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 24, normalized size = 0.80 \begin {gather*} 2 \, x^{2} - 2 \, x + 4 \, \log \left (x\right ) + 2 \, \log \left (-2 \, \log \left (x\right ) + \log \left (\log \left (x\right )\right )\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.35, size = 23, normalized size = 0.77 \begin {gather*} 2 \, x^{2} - 2 \, x + 4 \, \log \left (x\right ) + 2 \, \log \left (\log \left (\frac {\log \left (x\right )}{x^{2}}\right )\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.08, size = 24, normalized size = 0.80 \begin {gather*} 2 x^{2} - 2 x + 4 \log {\left (x \right )} + 2 \log {\left (\log {\left (\frac {\log {\left (x \right )}}{x^{2}} \right )} \right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.40, size = 26, normalized size = 0.87 \begin {gather*} 2 \, x^{2} - 2 \, x + 4 \, \log \left (x\right ) + 2 \, \log \left (2 \, \log \left (x\right ) - \log \left (\log \left (x\right )\right )\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.18, size = 23, normalized size = 0.77 \begin {gather*} 4\,\ln \left (x\right )-2\,x+2\,\ln \left (\ln \left (\frac {\ln \left (x\right )}{x^2}\right )\right )+2\,x^2 \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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