Optimal. Leaf size=23 \[ \frac {x \left (-x+x^3\right )}{\log (x)}+(x+10 \log (\log (2)))^2 \]
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Rubi [A]
time = 0.22, antiderivative size = 27, normalized size of antiderivative = 1.17, number of steps
used = 16, number of rules used = 5, integrand size = 40, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {6874, 2395,
2343, 2346, 2209} \begin {gather*} \frac {x^4}{\log (x)}-\frac {x^2}{\log (x)}+(x+10 \log (\log (2)))^2 \end {gather*}
Antiderivative was successfully verified.
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Rule 2209
Rule 2343
Rule 2346
Rule 2395
Rule 6874
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \left (\frac {x \left (1-x^2\right )}{\log ^2(x)}+\frac {2 x \left (-1+2 x^2\right )}{\log (x)}+2 (x+10 \log (\log (2)))\right ) \, dx\\ &=(x+10 \log (\log (2)))^2+2 \int \frac {x \left (-1+2 x^2\right )}{\log (x)} \, dx+\int \frac {x \left (1-x^2\right )}{\log ^2(x)} \, dx\\ &=(x+10 \log (\log (2)))^2+2 \int \left (-\frac {x}{\log (x)}+\frac {2 x^3}{\log (x)}\right ) \, dx+\int \left (\frac {x}{\log ^2(x)}-\frac {x^3}{\log ^2(x)}\right ) \, dx\\ &=(x+10 \log (\log (2)))^2-2 \int \frac {x}{\log (x)} \, dx+4 \int \frac {x^3}{\log (x)} \, dx+\int \frac {x}{\log ^2(x)} \, dx-\int \frac {x^3}{\log ^2(x)} \, dx\\ &=-\frac {x^2}{\log (x)}+\frac {x^4}{\log (x)}+(x+10 \log (\log (2)))^2+2 \int \frac {x}{\log (x)} \, dx-2 \text {Subst}\left (\int \frac {e^{2 x}}{x} \, dx,x,\log (x)\right )-4 \int \frac {x^3}{\log (x)} \, dx+4 \text {Subst}\left (\int \frac {e^{4 x}}{x} \, dx,x,\log (x)\right )\\ &=-2 \text {Ei}(2 \log (x))+4 \text {Ei}(4 \log (x))-\frac {x^2}{\log (x)}+\frac {x^4}{\log (x)}+(x+10 \log (\log (2)))^2+2 \text {Subst}\left (\int \frac {e^{2 x}}{x} \, dx,x,\log (x)\right )-4 \text {Subst}\left (\int \frac {e^{4 x}}{x} \, dx,x,\log (x)\right )\\ &=-\frac {x^2}{\log (x)}+\frac {x^4}{\log (x)}+(x+10 \log (\log (2)))^2\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.04, size = 27, normalized size = 1.17 \begin {gather*} x^2-\frac {x^2}{\log (x)}+\frac {x^4}{\log (x)}+20 x \log (\log (2)) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.61, size = 28, normalized size = 1.22
method | result | size |
risch | \(20 x \ln \left (\ln \left (2\right )\right )+x^{2}+\frac {x^{2} \left (x^{2}-1\right )}{\ln \left (x \right )}\) | \(24\) |
default | \(20 x \ln \left (\ln \left (2\right )\right )+x^{2}+\frac {x^{4}}{\ln \left (x \right )}-\frac {x^{2}}{\ln \left (x \right )}\) | \(28\) |
norman | \(\frac {x^{4}+x^{2} \ln \left (x \right )-x^{2}+20 x \ln \left (\ln \left (2\right )\right ) \ln \left (x \right )}{\ln \left (x \right )}\) | \(29\) |
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 = 0.31, size = 40, normalized size = 1.74 \begin {gather*} x^{2} + 20 \, x \log \left (\log \left (2\right )\right ) + 4 \, {\rm Ei}\left (4 \, \log \left (x\right )\right ) - 2 \, {\rm Ei}\left (2 \, \log \left (x\right )\right ) + 2 \, \Gamma \left (-1, -2 \, \log \left (x\right )\right ) - 4 \, \Gamma \left (-1, -4 \, \log \left (x\right )\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.38, size = 28, normalized size = 1.22 \begin {gather*} \frac {x^{4} + x^{2} \log \left (x\right ) + 20 \, x \log \left (x\right ) \log \left (\log \left (2\right )\right ) - x^{2}}{\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 = 20, normalized size = 0.87 \begin {gather*} x^{2} + 20 x \log {\left (\log {\left (2 \right )} \right )} + \frac {x^{4} - x^{2}}{\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 = 27, normalized size = 1.17 \begin {gather*} \frac {x^{4}}{\log \left (x\right )} + x^{2} + 20 \, x \log \left (\log \left (2\right )\right ) - \frac {x^{2}}{\log \left (x\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 7.09, size = 24, normalized size = 1.04 \begin {gather*} x\,\left (x+20\,\ln \left (\ln \left (2\right )\right )\right )-\frac {x\,\left (x-x^3\right )}{\ln \left (x\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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