Optimal. Leaf size=23 \[ \text {li}(x)-\frac {1}{2} \log (x-\log (x))+\frac {1}{2} \log (x+\log (x)) \]
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Rubi [A] time = 0.25, antiderivative size = 23, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.129, Rules used = {6741, 6742, 6684, 2298} \[ \text {LogIntegral}(x)-\frac {1}{2} \log (x-\log (x))+\frac {1}{2} \log (x+\log (x)) \]
Antiderivative was successfully verified.
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Rule 2298
Rule 6684
Rule 6741
Rule 6742
Rubi steps
\begin {align*} \int \frac {-x^2-\log (x)+2 \log ^2(x)}{-x^2 \log (x)+\log ^3(x)} \, dx &=\int \frac {x^2+\log (x)-2 \log ^2(x)}{\log (x) \left (x^2-\log ^2(x)\right )} \, dx\\ &=\int \left (\frac {1-x}{2 x (x-\log (x))}+\frac {1}{\log (x)}+\frac {1+x}{2 x (x+\log (x))}\right ) \, dx\\ &=\frac {1}{2} \int \frac {1-x}{x (x-\log (x))} \, dx+\frac {1}{2} \int \frac {1+x}{x (x+\log (x))} \, dx+\int \frac {1}{\log (x)} \, dx\\ &=-\frac {1}{2} \log (x-\log (x))+\frac {1}{2} \log (x+\log (x))+\text {li}(x)\\ \end {align*}
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Mathematica [F] time = 0.29, size = 0, normalized size = 0.00 \[ \int \frac {-x^2-\log (x)+2 \log ^2(x)}{-x^2 \log (x)+\log ^3(x)} \, dx \]
Verification is Not applicable to the result.
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fricas [A] time = 0.40, size = 19, normalized size = 0.83 \[ \frac {1}{2} \, \log \left (x + \log \relax (x)\right ) - \frac {1}{2} \, \log \left (-x + \log \relax (x)\right ) + \operatorname {log\_integral}\relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.15, size = 24, normalized size = 1.04 \[ {\rm Ei}\left (\log \relax (x)\right ) - \frac {1}{2} \, \log \left (x - \log \relax (x)\right ) + \frac {1}{2} \, \log \left (-x - \log \relax (x)\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.02, size = 0, normalized size = 0.00 \[ \int \frac {-x^{2}+2 \ln \relax (x )^{2}-\ln \relax (x )}{-x^{2} \ln \relax (x )+\ln \relax (x )^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\log \relax (x)}\,{d x} + \frac {1}{2} \, \log \left (x + \log \relax (x)\right ) - \frac {1}{2} \, \log \left (-x + \log \relax (x)\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.27, size = 19, normalized size = 0.83 \[ \frac {\ln \left (x+\ln \relax (x)\right )}{2}-\frac {\ln \left (x-\ln \relax (x)\right )}{2}+\mathrm {logint}\relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.57, size = 19, normalized size = 0.83 \[ - \frac {\log {\left (- x + \log {\relax (x )} \right )}}{2} + \frac {\log {\left (x + \log {\relax (x )} \right )}}{2} + \operatorname {li}{\relax (x )} \]
Verification of antiderivative is not currently implemented for this CAS.
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