Optimal. Leaf size=25 \[ \frac {5}{3} \left (15+\frac {\log ^2(x)}{x^2}-\log \left (\log \left (\frac {x^2}{16}\right )\right )\right ) \]
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Rubi [A]
time = 0.12, antiderivative size = 38, normalized size of antiderivative = 1.52, number of steps
used = 10, number of rules used = 8, integrand size = 43, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.186, Rules used = {12, 6820, 14,
2341, 2413, 2340, 2339, 29} \begin {gather*} -\frac {5 (1-\log (x)) \log (x)}{3 x^2}+\frac {5 \log (x)}{3 x^2}-\frac {5}{3} \log \left (\log \left (\frac {x^2}{16}\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 14
Rule 29
Rule 2339
Rule 2340
Rule 2341
Rule 2413
Rule 6820
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\frac {1}{3} \int \frac {-10 x^2+\left (10 \log (x)-10 \log ^2(x)\right ) \log \left (\frac {x^2}{16}\right )}{x^3 \log \left (\frac {x^2}{16}\right )} \, dx\\ &=\frac {1}{3} \int \frac {10 \left (\log (x)-\log ^2(x)-\frac {x^2}{\log \left (\frac {x^2}{16}\right )}\right )}{x^3} \, dx\\ &=\frac {10}{3} \int \frac {\log (x)-\log ^2(x)-\frac {x^2}{\log \left (\frac {x^2}{16}\right )}}{x^3} \, dx\\ &=\frac {10}{3} \int \left (-\frac {(-1+\log (x)) \log (x)}{x^3}-\frac {1}{x \log \left (\frac {x^2}{16}\right )}\right ) \, dx\\ &=-\left (\frac {10}{3} \int \frac {(-1+\log (x)) \log (x)}{x^3} \, dx\right )-\frac {10}{3} \int \frac {1}{x \log \left (\frac {x^2}{16}\right )} \, dx\\ &=\frac {5 \log (x)}{6 x^2}-\frac {5 (1-\log (x)) \log (x)}{3 x^2}-\frac {5}{3} \text {Subst}\left (\int \frac {1}{x} \, dx,x,\log \left (\frac {x^2}{16}\right )\right )+\frac {10}{3} \int \frac {1-2 \log (x)}{4 x^3} \, dx\\ &=\frac {5 \log (x)}{6 x^2}-\frac {5 (1-\log (x)) \log (x)}{3 x^2}-\frac {5}{3} \log \left (\log \left (\frac {x^2}{16}\right )\right )+\frac {5}{6} \int \frac {1-2 \log (x)}{x^3} \, dx\\ &=\frac {5 \log (x)}{3 x^2}-\frac {5 (1-\log (x)) \log (x)}{3 x^2}-\frac {5}{3} \log \left (\log \left (\frac {x^2}{16}\right )\right )\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.02, size = 25, normalized size = 1.00 \begin {gather*} \frac {5 \log ^2(x)}{3 x^2}-\frac {5}{3} \log \left (\log \left (\frac {x^2}{16}\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.28, size = 23, normalized size = 0.92
method | result | size |
default | \(\frac {5 \ln \left (x \right )^{2}}{3 x^{2}}-\frac {5 \ln \left (-4 \ln \left (2\right )+\ln \left (x^{2}\right )\right )}{3}\) | \(23\) |
risch | \(\frac {5 \ln \left (x \right )^{2}}{3 x^{2}}-\frac {5 \ln \left (\ln \left (x \right )-\frac {i \left (\pi \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i x^{2}\right )-2 \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{2}\right )^{2}+\pi \mathrm {csgn}\left (i x^{2}\right )^{3}-8 i \ln \left (2\right )\right )}{4}\right )}{3}\) | \(70\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 39, normalized size = 1.56 \begin {gather*} \frac {5 \, {\left (2 \, \log \left (x\right )^{2} + 2 \, \log \left (x\right ) + 1\right )}}{6 \, x^{2}} - \frac {5 \, \log \left (x\right )}{3 \, x^{2}} - \frac {5}{6 \, x^{2}} - \frac {5}{3} \, \log \left (\log \left (\frac {1}{16} \, x^{2}\right )\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.37, size = 26, normalized size = 1.04 \begin {gather*} -\frac {5 \, {\left (x^{2} \log \left (-4 \, \log \left (2\right ) + 2 \, \log \left (x\right )\right ) - \log \left (x\right )^{2}\right )}}{3 \, x^{2}} \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 = 0.96 \begin {gather*} - \frac {5 \log {\left (\log {\left (x \right )} - 2 \log {\left (2 \right )} \right )}}{3} + \frac {5 \log {\left (x \right )}^{2}}{3 x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.39, size = 22, normalized size = 0.88 \begin {gather*} \frac {5 \, \log \left (x\right )^{2}}{3 \, x^{2}} - \frac {5}{3} \, \log \left (-4 \, \log \left (2\right ) + 2 \, \log \left (x\right )\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 4.39, size = 19, normalized size = 0.76 \begin {gather*} \frac {5\,{\ln \left (x\right )}^2}{3\,x^2}-\frac {5\,\ln \left (\ln \left (\frac {x^2}{16}\right )\right )}{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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