Optimal. Leaf size=16 \[ 3+x^2+\frac {16 (-2+x)}{9 \log (x)} \]
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Rubi [A] time = 0.27, antiderivative size = 21, normalized size of antiderivative = 1.31, number of steps used = 12, number of rules used = 8, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {12, 6741, 6742, 2353, 2297, 2298, 2302, 30} \begin {gather*} x^2+\frac {16 x}{9 \log (x)}-\frac {32}{9 \log (x)} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 30
Rule 2297
Rule 2298
Rule 2302
Rule 2353
Rule 6741
Rule 6742
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\frac {1}{9} \int \frac {32-16 x+16 x \log (x)+18 x^2 \log ^2(x)}{x \log ^2(x)} \, dx\\ &=\frac {1}{9} \int \frac {2 \left (16-8 x+8 x \log (x)+9 x^2 \log ^2(x)\right )}{x \log ^2(x)} \, dx\\ &=\frac {2}{9} \int \frac {16-8 x+8 x \log (x)+9 x^2 \log ^2(x)}{x \log ^2(x)} \, dx\\ &=\frac {2}{9} \int \left (9 x-\frac {8 (-2+x)}{x \log ^2(x)}+\frac {8}{\log (x)}\right ) \, dx\\ &=x^2-\frac {16}{9} \int \frac {-2+x}{x \log ^2(x)} \, dx+\frac {16}{9} \int \frac {1}{\log (x)} \, dx\\ &=x^2+\frac {16 \text {li}(x)}{9}-\frac {16}{9} \int \left (\frac {1}{\log ^2(x)}-\frac {2}{x \log ^2(x)}\right ) \, dx\\ &=x^2+\frac {16 \text {li}(x)}{9}-\frac {16}{9} \int \frac {1}{\log ^2(x)} \, dx+\frac {32}{9} \int \frac {1}{x \log ^2(x)} \, dx\\ &=x^2+\frac {16 x}{9 \log (x)}+\frac {16 \text {li}(x)}{9}-\frac {16}{9} \int \frac {1}{\log (x)} \, dx+\frac {32}{9} \operatorname {Subst}\left (\int \frac {1}{x^2} \, dx,x,\log (x)\right )\\ &=x^2-\frac {32}{9 \log (x)}+\frac {16 x}{9 \log (x)}\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.01, size = 21, normalized size = 1.31 \begin {gather*} x^2-\frac {32}{9 \log (x)}+\frac {16 x}{9 \log (x)} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.58, size = 18, normalized size = 1.12 \begin {gather*} \frac {9 \, x^{2} \log \relax (x) + 16 \, x - 32}{9 \, \log \relax (x)} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 2.20, size = 13, normalized size = 0.81 \begin {gather*} x^{2} + \frac {16 \, {\left (x - 2\right )}}{9 \, \log \relax (x)} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 14, normalized size = 0.88
method | result | size |
risch | \(x^{2}+\frac {\frac {16 x}{9}-\frac {32}{9}}{\ln \relax (x )}\) | \(14\) |
norman | \(\frac {-\frac {32}{9}+x^{2} \ln \relax (x )+\frac {16 x}{9}}{\ln \relax (x )}\) | \(17\) |
default | \(x^{2}+\frac {16 x}{9 \ln \relax (x )}-\frac {32}{9 \ln \relax (x )}\) | \(18\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [C] time = 0.55, size = 23, normalized size = 1.44 \begin {gather*} x^{2} - \frac {32}{9 \, \log \relax (x)} + \frac {16}{9} \, {\rm Ei}\left (\log \relax (x)\right ) - \frac {16}{9} \, \Gamma \left (-1, -\log \relax (x)\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.15, size = 14, normalized size = 0.88 \begin {gather*} x^2+\frac {\frac {16\,x}{9}-\frac {32}{9}}{\ln \relax (x)} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.09, size = 12, normalized size = 0.75 \begin {gather*} x^{2} + \frac {16 x - 32}{9 \log {\relax (x )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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