Optimal. Leaf size=27 \[ \frac {2-x-\log (x)+\frac {1}{4} \log \left (4 \left (-\frac {13}{3}+x^2\right )\right )}{x} \]
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Rubi [A] time = 0.42, antiderivative size = 30, normalized size of antiderivative = 1.11, number of steps used = 10, number of rules used = 6, integrand size = 51, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {1593, 6725, 453, 207, 2304, 2455} \begin {gather*} \frac {\log \left (4 x^2-\frac {52}{3}\right )}{4 x}+\frac {2}{x}-\frac {\log (x)}{x} \end {gather*}
Antiderivative was successfully verified.
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Rule 207
Rule 453
Rule 1593
Rule 2304
Rule 2455
Rule 6725
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {156-30 x^2+\left (-52+12 x^2\right ) \log (x)+\left (13-3 x^2\right ) \log \left (\frac {1}{3} \left (-52+12 x^2\right )\right )}{x^2 \left (-52+12 x^2\right )} \, dx\\ &=\int \left (\frac {78-15 x^2-26 \log (x)+6 x^2 \log (x)}{2 x^2 \left (-13+3 x^2\right )}-\frac {\log \left (-\frac {52}{3}+4 x^2\right )}{4 x^2}\right ) \, dx\\ &=-\left (\frac {1}{4} \int \frac {\log \left (-\frac {52}{3}+4 x^2\right )}{x^2} \, dx\right )+\frac {1}{2} \int \frac {78-15 x^2-26 \log (x)+6 x^2 \log (x)}{x^2 \left (-13+3 x^2\right )} \, dx\\ &=\frac {\log \left (-\frac {52}{3}+4 x^2\right )}{4 x}+\frac {1}{2} \int \left (-\frac {3 \left (-26+5 x^2\right )}{x^2 \left (-13+3 x^2\right )}+\frac {2 \log (x)}{x^2}\right ) \, dx-2 \int \frac {1}{-\frac {52}{3}+4 x^2} \, dx\\ &=\frac {1}{2} \sqrt {\frac {3}{13}} \tanh ^{-1}\left (\sqrt {\frac {3}{13}} x\right )+\frac {\log \left (-\frac {52}{3}+4 x^2\right )}{4 x}-\frac {3}{2} \int \frac {-26+5 x^2}{x^2 \left (-13+3 x^2\right )} \, dx+\int \frac {\log (x)}{x^2} \, dx\\ &=\frac {2}{x}+\frac {1}{2} \sqrt {\frac {3}{13}} \tanh ^{-1}\left (\sqrt {\frac {3}{13}} x\right )-\frac {\log (x)}{x}+\frac {\log \left (-\frac {52}{3}+4 x^2\right )}{4 x}+\frac {3}{2} \int \frac {1}{-13+3 x^2} \, dx\\ &=\frac {2}{x}-\frac {\log (x)}{x}+\frac {\log \left (-\frac {52}{3}+4 x^2\right )}{4 x}\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.05, size = 31, normalized size = 1.15 \begin {gather*} \frac {1}{4} \left (\frac {8}{x}-\frac {4 \log (x)}{x}+\frac {\log \left (-\frac {52}{3}+4 x^2\right )}{x}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.53, size = 19, normalized size = 0.70 \begin {gather*} \frac {\log \left (4 \, x^{2} - \frac {52}{3}\right ) - 4 \, \log \relax (x) + 8}{4 \, x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.24, size = 30, normalized size = 1.11 \begin {gather*} -\frac {\log \relax (3) - 8}{4 \, x} + \frac {\log \left (12 \, x^{2} - 52\right )}{4 \, x} - \frac {\log \relax (x)}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.10, size = 24, normalized size = 0.89
method | result | size |
risch | \(\frac {\ln \left (4 x^{2}-\frac {52}{3}\right )}{4 x}-\frac {\ln \relax (x )-2}{x}\) | \(24\) |
default | \(-\frac {\ln \relax (3)}{4 x}+\frac {\ln \left (3 x^{2}-13\right )}{4 x}+\frac {\ln \relax (2)}{2 x}-\frac {\ln \relax (x )}{x}+\frac {2}{x}\) | \(41\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.48, size = 33, normalized size = 1.22 \begin {gather*} -\frac {\log \relax (3) - 2 \, \log \relax (2) - \log \left (3 \, x^{2} - 13\right ) + 4 \, \log \relax (x) + 4}{4 \, x} + \frac {3}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.37, size = 19, normalized size = 0.70 \begin {gather*} \frac {\ln \left (4\,x^2-\frac {52}{3}\right )-4\,\ln \relax (x)+8}{4\,x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.36, size = 20, normalized size = 0.74 \begin {gather*} - \frac {\log {\relax (x )}}{x} + \frac {\log {\left (4 x^{2} - \frac {52}{3} \right )}}{4 x} + \frac {2}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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