Optimal. Leaf size=18 \[ \frac {1}{5} x (-5+2 x) (7+x-\log (\log (x))) \]
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Rubi [A] time = 0.19, antiderivative size = 33, normalized size of antiderivative = 1.83, number of steps used = 17, number of rules used = 8, integrand size = 37, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.216, Rules used = {12, 6742, 2330, 2298, 2309, 2178, 2520, 2522} \begin {gather*} \frac {2 x^3}{5}+\frac {9 x^2}{5}-\frac {2}{5} x^2 \log (\log (x))-7 x+x \log (\log (x)) \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 2178
Rule 2298
Rule 2309
Rule 2330
Rule 2520
Rule 2522
Rule 6742
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\frac {1}{5} \int \frac {5-2 x+\left (-35+18 x+6 x^2\right ) \log (x)+(5-4 x) \log (x) \log (\log (x))}{\log (x)} \, dx\\ &=\frac {1}{5} \int \left (\frac {5-2 x-35 \log (x)+18 x \log (x)+6 x^2 \log (x)}{\log (x)}-(-5+4 x) \log (\log (x))\right ) \, dx\\ &=\frac {1}{5} \int \frac {5-2 x-35 \log (x)+18 x \log (x)+6 x^2 \log (x)}{\log (x)} \, dx-\frac {1}{5} \int (-5+4 x) \log (\log (x)) \, dx\\ &=\frac {1}{5} \int \left (-35+18 x+6 x^2+\frac {5-2 x}{\log (x)}\right ) \, dx-\frac {1}{5} \int (-5 \log (\log (x))+4 x \log (\log (x))) \, dx\\ &=-7 x+\frac {9 x^2}{5}+\frac {2 x^3}{5}+\frac {1}{5} \int \frac {5-2 x}{\log (x)} \, dx-\frac {4}{5} \int x \log (\log (x)) \, dx+\int \log (\log (x)) \, dx\\ &=-7 x+\frac {9 x^2}{5}+\frac {2 x^3}{5}+x \log (\log (x))-\frac {2}{5} x^2 \log (\log (x))+\frac {1}{5} \int \left (\frac {5}{\log (x)}-\frac {2 x}{\log (x)}\right ) \, dx+\frac {2}{5} \int \frac {x}{\log (x)} \, dx-\int \frac {1}{\log (x)} \, dx\\ &=-7 x+\frac {9 x^2}{5}+\frac {2 x^3}{5}+x \log (\log (x))-\frac {2}{5} x^2 \log (\log (x))-\text {li}(x)-\frac {2}{5} \int \frac {x}{\log (x)} \, dx+\frac {2}{5} \operatorname {Subst}\left (\int \frac {e^{2 x}}{x} \, dx,x,\log (x)\right )+\int \frac {1}{\log (x)} \, dx\\ &=-7 x+\frac {9 x^2}{5}+\frac {2 x^3}{5}+\frac {2}{5} \text {Ei}(2 \log (x))+x \log (\log (x))-\frac {2}{5} x^2 \log (\log (x))-\frac {2}{5} \operatorname {Subst}\left (\int \frac {e^{2 x}}{x} \, dx,x,\log (x)\right )\\ &=-7 x+\frac {9 x^2}{5}+\frac {2 x^3}{5}+x \log (\log (x))-\frac {2}{5} x^2 \log (\log (x))\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.09, size = 18, normalized size = 1.00 \begin {gather*} \frac {1}{5} x (-5+2 x) (7+x-\log (\log (x))) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.85, size = 28, normalized size = 1.56 \begin {gather*} \frac {2}{5} \, x^{3} + \frac {9}{5} \, x^{2} - \frac {1}{5} \, {\left (2 \, x^{2} - 5 \, x\right )} \log \left (\log \relax (x)\right ) - 7 \, x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.19, size = 27, normalized size = 1.50 \begin {gather*} \frac {2}{5} \, x^{3} - \frac {2}{5} \, x^{2} \log \left (\log \relax (x)\right ) + \frac {9}{5} \, x^{2} + x \log \left (\log \relax (x)\right ) - 7 \, x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 28, normalized size = 1.56
method | result | size |
norman | \(x \ln \left (\ln \relax (x )\right )-7 x +\frac {9 x^{2}}{5}+\frac {2 x^{3}}{5}-\frac {2 x^{2} \ln \left (\ln \relax (x )\right )}{5}\) | \(28\) |
risch | \(\frac {\left (-2 x^{2}+5 x \right ) \ln \left (\ln \relax (x )\right )}{5}+\frac {2 x^{3}}{5}+\frac {9 x^{2}}{5}-7 x\) | \(29\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.41, size = 27, normalized size = 1.50 \begin {gather*} \frac {2}{5} \, x^{3} - \frac {2}{5} \, x^{2} \log \left (\log \relax (x)\right ) + \frac {9}{5} \, x^{2} + x \log \left (\log \relax (x)\right ) - 7 \, x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.65, size = 16, normalized size = 0.89 \begin {gather*} \frac {x\,\left (2\,x-5\right )\,\left (x-\ln \left (\ln \relax (x)\right )+7\right )}{5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.35, size = 29, normalized size = 1.61 \begin {gather*} \frac {2 x^{3}}{5} + \frac {9 x^{2}}{5} - 7 x + \left (- \frac {2 x^{2}}{5} + x\right ) \log {\left (\log {\relax (x )} \right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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