Optimal. Leaf size=16 \[ \frac {3}{4} \log \left (1-\left (x^2\right )^{2/3}\right ) \]
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Rubi [A] time = 0.06, antiderivative size = 16, normalized size of antiderivative = 1.00, 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 260
Rule 1593
Rule 6715
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*}
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Mathematica [A] time = 0.03, size = 16, normalized size = 1.00 \[ \frac {3}{4} \log \left (1-\left (x^2\right )^{2/3}\right ) \]
Antiderivative was successfully verified.
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fricas [B] time = 0.45, size = 32, normalized size = 2.00 \[ -3 \, \log \left (\frac {{\left (x^{2}\right )}^{\frac {1}{3}}}{x}\right ) + \frac {3}{4} \, \log \left (-\frac {x^{2} - {\left (x^{2}\right )}^{\frac {1}{3}}}{x^{2}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.47, size = 16, normalized size = 1.00 \[ \frac {3}{4} \, \log \left ({\left | \left (x \mathrm {sgn}\relax (x)\right )^{\frac {1}{3}} x \mathrm {sgn}\relax (x) - 1 \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.05, size = 70, normalized size = 4.38 \[ \frac {\ln \left (\left (x^{2}\right )^{\frac {1}{3}}-1\right )}{2}+\frac {\ln \left (\left (x^{2}\right )^{\frac {1}{3}}+1\right )}{2}-\frac {\ln \left (\left (x^{2}\right )^{\frac {2}{3}}-\left (x^{2}\right )^{\frac {1}{3}}+1\right )}{4}-\frac {\ln \left (\left (x^{2}\right )^{\frac {2}{3}}+\left (x^{2}\right )^{\frac {1}{3}}+1\right )}{4}+\frac {\ln \left (x^{2}-1\right )}{4}+\frac {\ln \left (x^{2}+1\right )}{4} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.64, size = 21, normalized size = 1.31 \[ \frac {3}{4} \, \log \left ({\left (x^{2}\right )}^{\frac {1}{3}} + 1\right ) + \frac {3}{4} \, \log \left ({\left (x^{2}\right )}^{\frac {1}{3}} - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.34, size = 10, normalized size = 0.62 \[ \frac {3\,\ln \left ({\left (x^2\right )}^{2/3}-1\right )}{4} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.22, size = 19, normalized size = 1.19 \[ - \frac {\log {\relax (x )}}{2} + \frac {3 \log {\left (x^{2} - \sqrt [3]{x^{2}} \right )}}{4} \]
Verification of antiderivative is not currently implemented for this CAS.
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