Optimal. Leaf size=21 \[ -x^2-\log \left (-4+\log \left (\frac {4 x}{\log ^2(3+x)}\right )\right ) \]
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Rubi [A]
time = 0.98, antiderivative size = 23, normalized size of antiderivative = 1.10, number of steps
used = 4, number of rules used = 3, integrand size = 90, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.033, Rules used = {6820, 6874,
6816} \begin {gather*} -x^2-\log \left (4-\log \left (\frac {4 x}{\log ^2(x+3)}\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 6816
Rule 6820
Rule 6874
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-2 x+(3+x) \log (3+x) \left (1-8 x^2+2 x^2 \log \left (\frac {4 x}{\log ^2(3+x)}\right )\right )}{x (3+x) \log (3+x) \left (4-\log \left (\frac {4 x}{\log ^2(3+x)}\right )\right )} \, dx\\ &=\int \left (-2 x+\frac {2 x-3 \log (3+x)-x \log (3+x)}{x (3+x) \log (3+x) \left (-4+\log \left (\frac {4 x}{\log ^2(3+x)}\right )\right )}\right ) \, dx\\ &=-x^2+\int \frac {2 x-3 \log (3+x)-x \log (3+x)}{x (3+x) \log (3+x) \left (-4+\log \left (\frac {4 x}{\log ^2(3+x)}\right )\right )} \, dx\\ &=-x^2-\log \left (4-\log \left (\frac {4 x}{\log ^2(3+x)}\right )\right )\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.15, size = 23, normalized size = 1.10 \begin {gather*} -x^2-\log \left (4-\log \left (\frac {4 x}{\log ^2(3+x)}\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
3.
time = 0.39, size = 183, normalized size = 8.71
method | result | size |
risch | \(-x^{2}-\ln \left (\ln \left (\ln \left (3+x \right )\right )+\frac {i \left (\pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (\frac {i}{\ln \left (3+x \right )^{2}}\right ) \mathrm {csgn}\left (\frac {i x}{\ln \left (3+x \right )^{2}}\right )-\pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (\frac {i x}{\ln \left (3+x \right )^{2}}\right )^{2}-\pi \,\mathrm {csgn}\left (\frac {i}{\ln \left (3+x \right )^{2}}\right ) \mathrm {csgn}\left (\frac {i x}{\ln \left (3+x \right )^{2}}\right )^{2}-\pi \mathrm {csgn}\left (i \ln \left (3+x \right )\right )^{2} \mathrm {csgn}\left (i \ln \left (3+x \right )^{2}\right )+2 \pi \,\mathrm {csgn}\left (i \ln \left (3+x \right )\right ) \mathrm {csgn}\left (i \ln \left (3+x \right )^{2}\right )^{2}-\pi \mathrm {csgn}\left (i \ln \left (3+x \right )^{2}\right )^{3}+\pi \mathrm {csgn}\left (\frac {i x}{\ln \left (3+x \right )^{2}}\right )^{3}+4 i \ln \left (2\right )+2 i \ln \left (x \right )-8 i\right )}{4}\right )\) | \(183\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.50, size = 24, normalized size = 1.14 \begin {gather*} -x^{2} - \log \left (-\log \left (2\right ) - \frac {1}{2} \, \log \left (x\right ) + \log \left (\log \left (x + 3\right )\right ) + 2\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.39, size = 21, normalized size = 1.00 \begin {gather*} -x^{2} - \log \left (\log \left (\frac {4 \, x}{\log \left (x + 3\right )^{2}}\right ) - 4\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.20, size = 19, normalized size = 0.90 \begin {gather*} - x^{2} - \log {\left (\log {\left (\frac {4 x}{\log {\left (x + 3 \right )}^{2}} \right )} - 4 \right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.49, size = 24, normalized size = 1.14 \begin {gather*} -x^{2} - \log \left (\log \left (\log \left (x + 3\right )^{2}\right ) - \log \left (4 \, x\right ) + 4\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 5.98, size = 21, normalized size = 1.00 \begin {gather*} -\ln \left (\ln \left (\frac {4\,x}{{\ln \left (x+3\right )}^2}\right )-4\right )-x^2 \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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