Optimal. Leaf size=20 \[ \log \left (8 \left (-1+\frac {25}{x^4 (4+x)^2 \log ^2(\log (4))}\right )\right ) \]
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Rubi [B] time = 0.11, antiderivative size = 48, normalized size of antiderivative = 2.40, number of steps used = 4, number of rules used = 2, integrand size = 42, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.048, Rules used = {2074, 1587} \begin {gather*} \log \left (x^3 (-\log (\log (4)))-4 x^2 \log (\log (4))+5\right )+\log \left (x^3 \log (\log (4))+4 x^2 \log (\log (4))+5\right )-4 \log (x)-2 \log (x+4) \end {gather*}
Antiderivative was successfully verified.
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Rule 1587
Rule 2074
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \left (-\frac {4}{x}-\frac {2}{4+x}+\frac {x (8+3 x) \log (\log (4))}{-5+4 x^2 \log (\log (4))+x^3 \log (\log (4))}+\frac {x (8+3 x) \log (\log (4))}{5+4 x^2 \log (\log (4))+x^3 \log (\log (4))}\right ) \, dx\\ &=-4 \log (x)-2 \log (4+x)+\log (\log (4)) \int \frac {x (8+3 x)}{-5+4 x^2 \log (\log (4))+x^3 \log (\log (4))} \, dx+\log (\log (4)) \int \frac {x (8+3 x)}{5+4 x^2 \log (\log (4))+x^3 \log (\log (4))} \, dx\\ &=-4 \log (x)-2 \log (4+x)+\log \left (5-4 x^2 \log (\log (4))-x^3 \log (\log (4))\right )+\log \left (5+4 x^2 \log (\log (4))+x^3 \log (\log (4))\right )\\ \end {aligned} \end {gather*}
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Mathematica [B] time = 0.03, size = 54, normalized size = 2.70 \begin {gather*} 50 \left (-\frac {2 \log (x)}{25}-\frac {1}{25} \log (4+x)+\frac {1}{50} \log \left (25-16 x^4 \log ^2(\log (4))-8 x^5 \log ^2(\log (4))-x^6 \log ^2(\log (4))\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.58, size = 36, normalized size = 1.80 \begin {gather*} \log \left ({\left (x^{6} + 8 \, x^{5} + 16 \, x^{4}\right )} \log \left (2 \, \log \relax (2)\right )^{2} - 25\right ) - 2 \, \log \left (x + 4\right ) - 4 \, \log \relax (x) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.16, size = 59, normalized size = 2.95 \begin {gather*} \log \left ({\left | x^{3} \log \left (2 \, \log \relax (2)\right ) + 4 \, x^{2} \log \left (2 \, \log \relax (2)\right ) + 5 \right |}\right ) + \log \left ({\left | x^{3} \log \left (2 \, \log \relax (2)\right ) + 4 \, x^{2} \log \left (2 \, \log \relax (2)\right ) - 5 \right |}\right ) - 2 \, \log \left ({\left | x + 4 \right |}\right ) - 4 \, \log \left ({\left | x \right |}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.30, size = 56, normalized size = 2.80
method | result | size |
default | \(-4 \ln \relax (x )+\ln \left (x^{3} \ln \left (2 \ln \relax (2)\right )+4 x^{2} \ln \left (2 \ln \relax (2)\right )-5\right )-2 \ln \left (4+x \right )+\ln \left (x^{3} \ln \left (2 \ln \relax (2)\right )+4 x^{2} \ln \left (2 \ln \relax (2)\right )+5\right )\) | \(56\) |
norman | \(-4 \ln \relax (x )+\ln \left (x^{3} \ln \left (2 \ln \relax (2)\right )+4 x^{2} \ln \left (2 \ln \relax (2)\right )-5\right )-2 \ln \left (4+x \right )+\ln \left (x^{3} \ln \left (2 \ln \relax (2)\right )+4 x^{2} \ln \left (2 \ln \relax (2)\right )+5\right )\) | \(56\) |
risch | \(-2 \ln \left (4+x \right )-4 \ln \left (-x \right )+\ln \left (\left (-\ln \relax (2)^{2}-2 \ln \relax (2) \ln \left (\ln \relax (2)\right )-\ln \left (\ln \relax (2)\right )^{2}\right ) x^{6}+\left (-8 \ln \relax (2)^{2}-16 \ln \relax (2) \ln \left (\ln \relax (2)\right )-8 \ln \left (\ln \relax (2)\right )^{2}\right ) x^{5}+\left (-16 \ln \relax (2)^{2}-32 \ln \relax (2) \ln \left (\ln \relax (2)\right )-16 \ln \left (\ln \relax (2)\right )^{2}\right ) x^{4}+25\right )\) | \(92\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.35, size = 55, normalized size = 2.75 \begin {gather*} \log \left (x^{3} \log \left (2 \, \log \relax (2)\right ) + 4 \, x^{2} \log \left (2 \, \log \relax (2)\right ) + 5\right ) + \log \left (x^{3} \log \left (2 \, \log \relax (2)\right ) + 4 \, x^{2} \log \left (2 \, \log \relax (2)\right ) - 5\right ) - 2 \, \log \left (x + 4\right ) - 4 \, \log \relax (x) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 5.83, size = 43, normalized size = 2.15 \begin {gather*} \ln \left ({\ln \left (\ln \relax (4)\right )}^2\,x^6+8\,{\ln \left (\ln \relax (4)\right )}^2\,x^5+16\,{\ln \left (\ln \relax (4)\right )}^2\,x^4-25\right )-2\,\ln \left (x+4\right )-4\,\ln \relax (x) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 15.88, size = 49, normalized size = 2.45 \begin {gather*} - 4 \log {\relax (x )} - 2 \log {\left (x + 4 \right )} + \log {\left (x^{6} + 8 x^{5} + 16 x^{4} - \frac {25}{2 \log {\relax (2 )} \log {\left (\log {\relax (2 )} \right )} + \log {\left (\log {\relax (2 )} \right )}^{2} + \log {\relax (2 )}^{2}} \right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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