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 = 0.46, 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 = {1607, 6857,
464, 213, 2341, 2505} \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 213
Rule 464
Rule 1607
Rule 2341
Rule 2505
Rule 6857
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 = 0.04, 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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Maple [A]
time = 0.44, size = 41, normalized size = 1.52
method | result | size |
risch | \(\frac {\ln \left (4 x^{2}-\frac {52}{3}\right )}{4 x}-\frac {\ln \left (x \right )-2}{x}\) | \(24\) |
default | \(-\frac {\ln \left (3\right )}{4 x}+\frac {\ln \left (3 x^{2}-13\right )}{4 x}+\frac {\ln \left (2\right )}{2 x}-\frac {\ln \left (x \right )}{x}+\frac {2}{x}\) | \(41\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.54, size = 33, normalized size = 1.22 \begin {gather*} -\frac {\log \left (3\right ) - 2 \, \log \left (2\right ) - \log \left (3 \, x^{2} - 13\right ) + 4 \, \log \left (x\right ) + 4}{4 \, x} + \frac {3}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.41, size = 19, normalized size = 0.70 \begin {gather*} \frac {\log \left (4 \, x^{2} - \frac {52}{3}\right ) - 4 \, \log \left (x\right ) + 8}{4 \, x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.14, size = 20, normalized size = 0.74 \begin {gather*} - \frac {\log {\left (x \right )}}{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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Giac [A]
time = 0.43, size = 30, normalized size = 1.11 \begin {gather*} -\frac {\log \left (3\right ) - 8}{4 \, x} + \frac {\log \left (12 \, x^{2} - 52\right )}{4 \, x} - \frac {\log \left (x\right )}{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 \left (x\right )+8}{4\,x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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