Optimal. Leaf size=14 \[ \frac {1}{3} \log ^2((-13-x) x) \]
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Rubi [B] Leaf count is larger than twice the leaf count of optimal. \(75\) vs. \(2(14)=28\).
time = 0.30, antiderivative size = 75, normalized size of antiderivative = 5.36, number of steps
used = 18, number of rules used = 10, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.370, Rules used = {1607, 2608,
2604, 2404, 2338, 2354, 2438, 2465, 2439, 2437} \begin {gather*} \frac {2}{3} \log \left (-x^2-13 x\right ) \log (x)+\frac {2}{3} \log (x+13) \log \left (-x^2-13 x\right )-\frac {1}{3} \log ^2(x)-\frac {1}{3} \log ^2(x+13)-\frac {2}{3} \log \left (\frac {x}{13}+1\right ) \log (x)-\frac {2}{3} \log (13) \log (x) \end {gather*}
Antiderivative was successfully verified.
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Rule 1607
Rule 2338
Rule 2354
Rule 2404
Rule 2437
Rule 2438
Rule 2439
Rule 2465
Rule 2604
Rule 2608
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {(26+4 x) \log \left (-13 x-x^2\right )}{x (39+3 x)} \, dx\\ &=\int \left (\frac {2 \log \left (-13 x-x^2\right )}{3 x}+\frac {2 \log \left (-13 x-x^2\right )}{3 (13+x)}\right ) \, dx\\ &=\frac {2}{3} \int \frac {\log \left (-13 x-x^2\right )}{x} \, dx+\frac {2}{3} \int \frac {\log \left (-13 x-x^2\right )}{13+x} \, dx\\ &=\frac {2}{3} \log (x) \log \left (-13 x-x^2\right )+\frac {2}{3} \log (13+x) \log \left (-13 x-x^2\right )-\frac {2}{3} \int \frac {(-13-2 x) \log (x)}{-13 x-x^2} \, dx-\frac {2}{3} \int \frac {(-13-2 x) \log (13+x)}{-13 x-x^2} \, dx\\ &=\frac {2}{3} \log (x) \log \left (-13 x-x^2\right )+\frac {2}{3} \log (13+x) \log \left (-13 x-x^2\right )-\frac {2}{3} \int \frac {(-13-2 x) \log (x)}{(-13-x) x} \, dx-\frac {2}{3} \int \frac {(-13-2 x) \log (13+x)}{(-13-x) x} \, dx\\ &=\frac {2}{3} \log (x) \log \left (-13 x-x^2\right )+\frac {2}{3} \log (13+x) \log \left (-13 x-x^2\right )-\frac {2}{3} \int \left (\frac {\log (x)}{x}+\frac {\log (x)}{13+x}\right ) \, dx-\frac {2}{3} \int \left (\frac {\log (13+x)}{x}+\frac {\log (13+x)}{13+x}\right ) \, dx\\ &=\frac {2}{3} \log (x) \log \left (-13 x-x^2\right )+\frac {2}{3} \log (13+x) \log \left (-13 x-x^2\right )-\frac {2}{3} \int \frac {\log (x)}{x} \, dx-\frac {2}{3} \int \frac {\log (x)}{13+x} \, dx-\frac {2}{3} \int \frac {\log (13+x)}{x} \, dx-\frac {2}{3} \int \frac {\log (13+x)}{13+x} \, dx\\ &=-\frac {2}{3} \log (13) \log (x)-\frac {2}{3} \log \left (1+\frac {x}{13}\right ) \log (x)-\frac {\log ^2(x)}{3}+\frac {2}{3} \log (x) \log \left (-13 x-x^2\right )+\frac {2}{3} \log (13+x) \log \left (-13 x-x^2\right )-\frac {2}{3} \text {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,13+x\right )\\ &=-\frac {2}{3} \log (13) \log (x)-\frac {2}{3} \log \left (1+\frac {x}{13}\right ) \log (x)-\frac {\log ^2(x)}{3}-\frac {1}{3} \log ^2(13+x)+\frac {2}{3} \log (x) \log \left (-13 x-x^2\right )+\frac {2}{3} \log (13+x) \log \left (-13 x-x^2\right )\\ \end {aligned} \end {gather*}
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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(63\) vs. \(2(14)=28\).
time = 0.02, size = 63, normalized size = 4.50 \begin {gather*} \frac {2}{3} \left (-\log (13) \log (x)-\frac {\log ^2(x)}{2}-\log (x) \log \left (\frac {13+x}{13}\right )-\frac {1}{2} \log ^2(13+x)+\log (x) \log (-x (13+x))+\log (13+x) \log (-x (13+x))\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(74\) vs.
\(2(12)=24\).
time = 1.10, size = 75, normalized size = 5.36
method | result | size |
norman | \(\frac {\ln \left (-x^{2}-13 x \right )^{2}}{3}\) | \(15\) |
risch | \(\frac {\ln \left (-x^{2}-13 x \right )^{2}}{3}\) | \(15\) |
default | \(\frac {2 \ln \left (x \right ) \ln \left (-x^{2}-13 x \right )}{3}-\frac {\ln \left (x \right )^{2}}{3}-\frac {2 \ln \left (x \right ) \ln \left (\frac {x}{13}+1\right )}{3}+\frac {2 \ln \left (x +13\right ) \ln \left (-x^{2}-13 x \right )}{3}-\frac {2 \left (\ln \left (x +13\right )-\ln \left (\frac {x}{13}+1\right )\right ) \ln \left (-\frac {x}{13}\right )}{3}-\frac {\ln \left (x +13\right )^{2}}{3}\) | \(75\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 27 vs.
\(2 (11) = 22\).
time = 0.27, size = 27, normalized size = 1.93 \begin {gather*} \frac {1}{3} \, \log \left (x\right )^{2} + \frac {2}{3} \, \log \left (x\right ) \log \left (-x - 13\right ) + \frac {1}{3} \, \log \left (-x - 13\right )^{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.37, size = 14, normalized size = 1.00 \begin {gather*} \frac {1}{3} \, \log \left (-x^{2} - 13 \, x\right )^{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.05, size = 12, normalized size = 0.86 \begin {gather*} \frac {\log {\left (- x^{2} - 13 x \right )}^{2}}{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 6.71, size = 11, normalized size = 0.79 \begin {gather*} \frac {{\ln \left (-x\,\left (x+13\right )\right )}^2}{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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