Optimal. Leaf size=16 \[ \frac{3}{4} \log \left (1-\left (x^2\right )^{2/3}\right ) \]
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Rubi [A] time = 0.061193, antiderivative size = 16, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.176, Rules used = {6715, 1593, 260} \[ \frac{3}{4} \log \left (1-\left (x^2\right )^{2/3}\right ) \]
Antiderivative was successfully verified.
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Rule 6715
Rule 1593
Rule 260
Rubi steps
\begin{align*} \int \frac{x}{x^2-\sqrt [3]{x^2}} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{-\sqrt [3]{x}+x} \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{\left (-1+x^{2/3}\right ) \sqrt [3]{x}} \, dx,x,x^2\right )\\ &=\frac{3}{4} \log \left (1-\left (x^2\right )^{2/3}\right )\\ \end{align*}
Mathematica [A] time = 0.0267944, size = 16, normalized size = 1. \[ \frac{3}{4} \log \left (1-\left (x^2\right )^{2/3}\right ) \]
Antiderivative was successfully verified.
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Maple [B] time = 0.033, size = 70, normalized size = 4.4 \begin{align*}{\frac{\ln \left ({x}^{2}-1 \right ) }{4}}+{\frac{\ln \left ({x}^{2}+1 \right ) }{4}}+{\frac{1}{2}\ln \left ( \sqrt [3]{{x}^{2}}-1 \right ) }-{\frac{1}{4}\ln \left ( \left ({x}^{2} \right ) ^{{\frac{2}{3}}}+\sqrt [3]{{x}^{2}}+1 \right ) }-{\frac{1}{4}\ln \left ( \left ({x}^{2} \right ) ^{{\frac{2}{3}}}-\sqrt [3]{{x}^{2}}+1 \right ) }+{\frac{1}{2}\ln \left ( \sqrt [3]{{x}^{2}}+1 \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.12492, size = 28, normalized size = 1.75 \begin{align*} \frac{3}{4} \, \log \left ({\left (x^{2}\right )}^{\frac{1}{3}} + 1\right ) + \frac{3}{4} \, \log \left ({\left (x^{2}\right )}^{\frac{1}{3}} - 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.64885, size = 80, normalized size = 5. \begin{align*} -3 \, \log \left (\frac{{\left (x^{2}\right )}^{\frac{1}{3}}}{x}\right ) + \frac{3}{4} \, \log \left (-\frac{x^{2} -{\left (x^{2}\right )}^{\frac{1}{3}}}{x^{2}}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.204456, size = 19, normalized size = 1.19 \begin{align*} - \frac{\log{\left (x \right )}}{2} + \frac{3 \log{\left (x^{2} - \sqrt [3]{x^{2}} \right )}}{4} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.14409, size = 22, normalized size = 1.38 \begin{align*} \frac{3}{4} \, \log \left ({\left | \left (x \mathrm{sgn}\left (x\right )\right )^{\frac{1}{3}} x \mathrm{sgn}\left (x\right ) - 1 \right |}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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