Optimal. Leaf size=23 \[ \text{LogIntegral}(x)-\frac{1}{2} \log (x-\log (x))+\frac{1}{2} \log (x+\log (x)) \]
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Rubi [A] time = 0.247992, antiderivative size = 23, normalized size of antiderivative = 1., 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 6741
Rule 6742
Rule 6684
Rule 2298
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*}
Mathematica [A] time = 0.0901356, size = 23, normalized size = 1. \[ \text{LogIntegral}(x)-\frac{1}{2} \log (x-\log (x))+\frac{1}{2} \log (x+\log (x)) \]
Antiderivative was successfully verified.
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Maple [F] time = 0.017, size = 0, normalized size = 0. \begin{align*} \int{\frac{-{x}^{2}-\ln \left ( x \right ) +2\, \left ( \ln \left ( x \right ) \right ) ^{2}}{-{x}^{2}\ln \left ( x \right ) + \left ( \ln \left ( x \right ) \right ) ^{3}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\log \left (x\right )}\,{d x} + \frac{1}{2} \, \log \left (x + \log \left (x\right )\right ) - \frac{1}{2} \, \log \left (-x + \log \left (x\right )\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.35184, size = 84, normalized size = 3.65 \begin{align*} \frac{1}{2} \, \log \left (x + \log \left (x\right )\right ) - \frac{1}{2} \, \log \left (-x + \log \left (x\right )\right ) + \logintegral \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.549804, size = 19, normalized size = 0.83 \begin{align*} - \frac{\log{\left (- x + \log{\left (x \right )} \right )}}{2} + \frac{\log{\left (x + \log{\left (x \right )} \right )}}{2} + \operatorname{li}{\left (x \right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.09277, size = 32, normalized size = 1.39 \begin{align*}{\rm Ei}\left (\log \left (x\right )\right ) - \frac{1}{2} \, \log \left (x - \log \left (x\right )\right ) + \frac{1}{2} \, \log \left (-x - \log \left (x\right )\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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