Optimal. Leaf size=19 \[ \frac {1}{9} \left (x+\log ^2\left (\left (-16-\frac {4 x}{5}\right ) \log (5)\right )\right ) \]
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Rubi [A]
time = 0.07, antiderivative size = 22, normalized size of antiderivative = 1.16, number of steps
used = 5, number of rules used = 4, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {6874, 2437, 12,
2338} \begin {gather*} \frac {x}{9}+\frac {1}{9} \log ^2\left (-\frac {4}{5} (x+20) \log (5)\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 2338
Rule 2437
Rule 6874
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \left (\frac {1}{9}+\frac {2 \log \left (-16 \log (5)-\frac {4}{5} x \log (5)\right )}{9 (20+x)}\right ) \, dx\\ &=\frac {x}{9}+\frac {2}{9} \int \frac {\log \left (-16 \log (5)-\frac {4}{5} x \log (5)\right )}{20+x} \, dx\\ &=\frac {x}{9}-\frac {5 \text {Subst}\left (\int -\frac {4 \log (5) \log (x)}{5 x} \, dx,x,-16 \log (5)-\frac {4}{5} x \log (5)\right )}{18 \log (5)}\\ &=\frac {x}{9}+\frac {2}{9} \text {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,-16 \log (5)-\frac {4}{5} x \log (5)\right )\\ &=\frac {x}{9}+\frac {1}{9} \log ^2\left (-\frac {4}{5} (20+x) \log (5)\right )\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.03, size = 19, normalized size = 1.00 \begin {gather*} \frac {1}{9} \left (20+x+\log ^2\left (-\frac {4}{5} (20+x) \log (5)\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.04, size = 34, normalized size = 1.79
method | result | size |
norman | \(\frac {x}{9}+\frac {\ln \left (\frac {\left (-4 x -80\right ) \ln \left (5\right )}{5}\right )^{2}}{9}\) | \(19\) |
risch | \(\frac {x}{9}+\frac {\ln \left (\frac {\left (-4 x -80\right ) \ln \left (5\right )}{5}\right )^{2}}{9}\) | \(19\) |
derivativedivides | \(-\frac {-4 \ln \left (5\right ) \ln \left (-\frac {4 x \ln \left (5\right )}{5}-16 \ln \left (5\right )\right )^{2}-4 x \ln \left (5\right )-80 \ln \left (5\right )}{36 \ln \left (5\right )}\) | \(34\) |
default | \(-\frac {-4 \ln \left (5\right ) \ln \left (-\frac {4 x \ln \left (5\right )}{5}-16 \ln \left (5\right )\right )^{2}-4 x \ln \left (5\right )-80 \ln \left (5\right )}{36 \ln \left (5\right )}\) | \(34\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [C] Result contains complex when optimal does not.
time = 0.48, size = 33, normalized size = 1.74 \begin {gather*} -\frac {2}{9} \, {\left (-i \, \pi + \log \left (5\right ) - 2 \, \log \left (2\right ) - \log \left (\log \left (5\right )\right )\right )} \log \left (x + 20\right ) + \frac {1}{9} \, \log \left (x + 20\right )^{2} + \frac {1}{9} \, x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.38, size = 16, normalized size = 0.84 \begin {gather*} \frac {1}{9} \, \log \left (-\frac {4}{5} \, {\left (x + 20\right )} \log \left (5\right )\right )^{2} + \frac {1}{9} \, x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.04, size = 19, normalized size = 1.00 \begin {gather*} \frac {x}{9} + \frac {\log {\left (\left (- \frac {4 x}{5} - 16\right ) \log {\left (5 \right )} \right )}^{2}}{9} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.43, size = 27, normalized size = 1.42 \begin {gather*} \frac {1}{9} \, \log \left (-4 \, x \log \left (5\right ) - 80 \, \log \left (5\right )\right )^{2} - \frac {2}{9} \, \log \left (5\right ) \log \left (x + 20\right ) + \frac {1}{9} \, x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.24, size = 18, normalized size = 0.95 \begin {gather*} \frac {{\ln \left (-\frac {\ln \left (5\right )\,\left (4\,x+80\right )}{5}\right )}^2}{9}+\frac {x}{9} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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