Optimal. Leaf size=14 \[ \frac {(2+x) \log ^2(x)}{96 x^2} \]
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Rubi [B] time = 0.17, antiderivative size = 61, normalized size of antiderivative = 4.36, number of steps used = 13, number of rules used = 8, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {12, 14, 37, 2334, 43, 2353, 2305, 2304} \begin {gather*} \frac {\log ^2(x)}{48 x^2}-\frac {(x+2)^2 \log (x)}{192 x^2}+\frac {\log (x)}{48 x^2}+\frac {\log ^2(x)}{96 x}+\frac {\log (x)}{48 x}+\frac {\log (x)}{192} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 14
Rule 37
Rule 43
Rule 2304
Rule 2305
Rule 2334
Rule 2353
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\frac {1}{96} \int \frac {(4+2 x) \log (x)+(-4-x) \log ^2(x)}{x^3} \, dx\\ &=\frac {1}{96} \int \left (\frac {2 (2+x) \log (x)}{x^3}-\frac {(4+x) \log ^2(x)}{x^3}\right ) \, dx\\ &=-\left (\frac {1}{96} \int \frac {(4+x) \log ^2(x)}{x^3} \, dx\right )+\frac {1}{48} \int \frac {(2+x) \log (x)}{x^3} \, dx\\ &=-\frac {(2+x)^2 \log (x)}{192 x^2}-\frac {1}{96} \int \left (\frac {4 \log ^2(x)}{x^3}+\frac {\log ^2(x)}{x^2}\right ) \, dx-\frac {1}{48} \int -\frac {(2+x)^2}{4 x^3} \, dx\\ &=-\frac {(2+x)^2 \log (x)}{192 x^2}+\frac {1}{192} \int \frac {(2+x)^2}{x^3} \, dx-\frac {1}{96} \int \frac {\log ^2(x)}{x^2} \, dx-\frac {1}{24} \int \frac {\log ^2(x)}{x^3} \, dx\\ &=-\frac {(2+x)^2 \log (x)}{192 x^2}+\frac {\log ^2(x)}{48 x^2}+\frac {\log ^2(x)}{96 x}+\frac {1}{192} \int \left (\frac {4}{x^3}+\frac {4}{x^2}+\frac {1}{x}\right ) \, dx-\frac {1}{48} \int \frac {\log (x)}{x^2} \, dx-\frac {1}{24} \int \frac {\log (x)}{x^3} \, dx\\ &=\frac {\log (x)}{192}+\frac {\log (x)}{48 x^2}+\frac {\log (x)}{48 x}-\frac {(2+x)^2 \log (x)}{192 x^2}+\frac {\log ^2(x)}{48 x^2}+\frac {\log ^2(x)}{96 x}\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.04, size = 22, normalized size = 1.57 \begin {gather*} \frac {1}{96} \left (\frac {2 \log ^2(x)}{x^2}+\frac {\log ^2(x)}{x}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.64, size = 12, normalized size = 0.86 \begin {gather*} \frac {{\left (x + 2\right )} \log \relax (x)^{2}}{96 \, x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.23, size = 12, normalized size = 0.86 \begin {gather*} \frac {{\left (x + 2\right )} \log \relax (x)^{2}}{96 \, x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 13, normalized size = 0.93
method | result | size |
risch | \(\frac {\ln \relax (x )^{2} \left (2+x \right )}{96 x^{2}}\) | \(13\) |
norman | \(\frac {\frac {\ln \relax (x )^{2}}{48}+\frac {x \ln \relax (x )^{2}}{96}}{x^{2}}\) | \(19\) |
default | \(\frac {\ln \relax (x )^{2}}{96 x}+\frac {\ln \relax (x )^{2}}{48 x^{2}}\) | \(20\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.37, size = 57, normalized size = 4.07 \begin {gather*} \frac {\log \relax (x)^{2} + 2 \, \log \relax (x) + 2}{96 \, x} - \frac {\log \relax (x)}{48 \, x} + \frac {2 \, \log \relax (x)^{2} + 2 \, \log \relax (x) + 1}{96 \, x^{2}} - \frac {1}{48 \, x} - \frac {\log \relax (x)}{48 \, x^{2}} - \frac {1}{96 \, x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.13, size = 12, normalized size = 0.86 \begin {gather*} \frac {{\ln \relax (x)}^2\,\left (x+2\right )}{96\,x^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.13, size = 12, normalized size = 0.86 \begin {gather*} \frac {\left (x + 2\right ) \log {\relax (x )}^{2}}{96 x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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