Optimal. Leaf size=13 \[ \frac {3 \left (1+\frac {2}{x}\right )}{\log (x)} \]
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Rubi [A]
time = 0.17, antiderivative size = 16, normalized size of antiderivative = 1.23, number of steps
used = 13, number of rules used = 9, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.529, Rules used = {6873, 12,
6874, 2395, 2343, 2346, 2209, 2339, 30} \begin {gather*} \frac {6}{x \log (x)}+\frac {3}{\log (x)} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 30
Rule 2209
Rule 2339
Rule 2343
Rule 2346
Rule 2395
Rule 6873
Rule 6874
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {3 (-2-x-2 \log (x))}{x^2 \log ^2(x)} \, dx\\ &=3 \int \frac {-2-x-2 \log (x)}{x^2 \log ^2(x)} \, dx\\ &=3 \int \left (\frac {-2-x}{x^2 \log ^2(x)}-\frac {2}{x^2 \log (x)}\right ) \, dx\\ &=3 \int \frac {-2-x}{x^2 \log ^2(x)} \, dx-6 \int \frac {1}{x^2 \log (x)} \, dx\\ &=3 \int \left (-\frac {2}{x^2 \log ^2(x)}-\frac {1}{x \log ^2(x)}\right ) \, dx-6 \text {Subst}\left (\int \frac {e^{-x}}{x} \, dx,x,\log (x)\right )\\ &=-6 \text {Ei}(-\log (x))-3 \int \frac {1}{x \log ^2(x)} \, dx-6 \int \frac {1}{x^2 \log ^2(x)} \, dx\\ &=-6 \text {Ei}(-\log (x))+\frac {6}{x \log (x)}-3 \text {Subst}\left (\int \frac {1}{x^2} \, dx,x,\log (x)\right )+6 \int \frac {1}{x^2 \log (x)} \, dx\\ &=-6 \text {Ei}(-\log (x))+\frac {3}{\log (x)}+\frac {6}{x \log (x)}+6 \text {Subst}\left (\int \frac {e^{-x}}{x} \, dx,x,\log (x)\right )\\ &=\frac {3}{\log (x)}+\frac {6}{x \log (x)}\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.05, size = 14, normalized size = 1.08 \begin {gather*} -\frac {3 (-2-x)}{x \log (x)} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.03, size = 17, normalized size = 1.31
method | result | size |
risch | \(\frac {6+3 x}{x \ln \left (x \right )}\) | \(13\) |
norman | \(\frac {6+3 x}{x \ln \left (x \right )}\) | \(14\) |
default | \(\frac {3}{\ln \left (x \right )}+\frac {6}{x \ln \left (x \right )}\) | \(17\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [C] Result contains higher order function than in optimal. Order 4 vs. order
3.
time = 0.43, size = 20, normalized size = 1.54 \begin {gather*} \frac {3}{\log \left (x\right )} - 6 \, {\rm Ei}\left (-\log \left (x\right )\right ) + 6 \, \Gamma \left (-1, \log \left (x\right )\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.31, size = 12, normalized size = 0.92 \begin {gather*} \frac {3 \, {\left (x + 2\right )}}{x \log \left (x\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.03, size = 8, normalized size = 0.62 \begin {gather*} \frac {3 x + 6}{x \log {\left (x \right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.42, size = 12, normalized size = 0.92 \begin {gather*} \frac {3 \, {\left (x + 2\right )}}{x \log \left (x\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.34, size = 12, normalized size = 0.92 \begin {gather*} \frac {3\,\left (x+2\right )}{x\,\ln \left (x\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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