Optimal. Leaf size=25 \[ 7+2 x+\frac {x^2}{8}-\frac {x+\frac {x}{\log (x)}}{x} \]
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Rubi [A]
time = 0.05, antiderivative size = 17, normalized size of antiderivative = 0.68, number of steps
used = 5, number of rules used = 4, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {12, 6820, 2339,
30} \begin {gather*} \frac {x^2}{8}+2 x-\frac {1}{\log (x)} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 30
Rule 2339
Rule 6820
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\frac {1}{4} \int \frac {4+\left (8 x+x^2\right ) \log ^2(x)}{x \log ^2(x)} \, dx\\ &=\frac {1}{4} \int \left (8+x+\frac {4}{x \log ^2(x)}\right ) \, dx\\ &=2 x+\frac {x^2}{8}+\int \frac {1}{x \log ^2(x)} \, dx\\ &=2 x+\frac {x^2}{8}+\text {Subst}\left (\int \frac {1}{x^2} \, dx,x,\log (x)\right )\\ &=2 x+\frac {x^2}{8}-\frac {1}{\log (x)}\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.00, size = 17, normalized size = 0.68 \begin {gather*} 2 x+\frac {x^2}{8}-\frac {1}{\log (x)} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.03, size = 16, normalized size = 0.64
method | result | size |
default | \(\frac {x^{2}}{8}+2 x -\frac {1}{\ln \left (x \right )}\) | \(16\) |
risch | \(\frac {x^{2}}{8}+2 x -\frac {1}{\ln \left (x \right )}\) | \(16\) |
norman | \(\frac {-1+2 x \ln \left (x \right )+\frac {x^{2} \ln \left (x \right )}{8}}{\ln \left (x \right )}\) | \(20\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.26, size = 15, normalized size = 0.60 \begin {gather*} \frac {1}{8} \, x^{2} + 2 \, x - \frac {1}{\log \left (x\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.37, size = 18, normalized size = 0.72 \begin {gather*} \frac {{\left (x^{2} + 16 \, x\right )} \log \left (x\right ) - 8}{8 \, \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 = 12, normalized size = 0.48 \begin {gather*} \frac {x^{2}}{8} + 2 x - \frac {1}{\log {\left (x \right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.39, size = 15, normalized size = 0.60 \begin {gather*} \frac {1}{8} \, x^{2} + 2 \, x - \frac {1}{\log \left (x\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.42, size = 15, normalized size = 0.60 \begin {gather*} 2\,x-\frac {1}{\ln \left (x\right )}+\frac {x^2}{8} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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