Optimal. Leaf size=73 \[ -\frac{1}{2} \sqrt{3} \tanh ^{-1}\left (\frac{\sqrt{3} \sqrt [3]{x}}{x^{2/3}+1}\right )-\frac{1}{2} \tan ^{-1}\left (\sqrt{3}-2 \sqrt [3]{x}\right )+\frac{1}{2} \tan ^{-1}\left (2 \sqrt [3]{x}+\sqrt{3}\right )+\tan ^{-1}\left (\sqrt [3]{x}\right ) \]
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Rubi [A] time = 0.257018, antiderivative size = 100, normalized size of antiderivative = 1.37, number of steps used = 11, number of rules used = 7, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.538, Rules used = {329, 295, 634, 618, 204, 628, 203} \[ \frac{1}{4} \sqrt{3} \log \left (x^{2/3}-\sqrt{3} \sqrt [3]{x}+1\right )-\frac{1}{4} \sqrt{3} \log \left (x^{2/3}+\sqrt{3} \sqrt [3]{x}+1\right )-\frac{1}{2} \tan ^{-1}\left (\sqrt{3}-2 \sqrt [3]{x}\right )+\frac{1}{2} \tan ^{-1}\left (2 \sqrt [3]{x}+\sqrt{3}\right )+\tan ^{-1}\left (\sqrt [3]{x}\right ) \]
Antiderivative was successfully verified.
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Rule 329
Rule 295
Rule 634
Rule 618
Rule 204
Rule 628
Rule 203
Rubi steps
\begin{align*} \int \frac{x^{2/3}}{1+x^2} \, dx &=3 \operatorname{Subst}\left (\int \frac{x^4}{1+x^6} \, dx,x,\sqrt [3]{x}\right )\\ &=\operatorname{Subst}\left (\int \frac{1}{1+x^2} \, dx,x,\sqrt [3]{x}\right )+\operatorname{Subst}\left (\int \frac{-\frac{1}{2}+\frac{\sqrt{3} x}{2}}{1-\sqrt{3} x+x^2} \, dx,x,\sqrt [3]{x}\right )+\operatorname{Subst}\left (\int \frac{-\frac{1}{2}-\frac{\sqrt{3} x}{2}}{1+\sqrt{3} x+x^2} \, dx,x,\sqrt [3]{x}\right )\\ &=\tan ^{-1}\left (\sqrt [3]{x}\right )+\frac{1}{4} \operatorname{Subst}\left (\int \frac{1}{1-\sqrt{3} x+x^2} \, dx,x,\sqrt [3]{x}\right )+\frac{1}{4} \operatorname{Subst}\left (\int \frac{1}{1+\sqrt{3} x+x^2} \, dx,x,\sqrt [3]{x}\right )+\frac{1}{4} \sqrt{3} \operatorname{Subst}\left (\int \frac{-\sqrt{3}+2 x}{1-\sqrt{3} x+x^2} \, dx,x,\sqrt [3]{x}\right )-\frac{1}{4} \sqrt{3} \operatorname{Subst}\left (\int \frac{\sqrt{3}+2 x}{1+\sqrt{3} x+x^2} \, dx,x,\sqrt [3]{x}\right )\\ &=\tan ^{-1}\left (\sqrt [3]{x}\right )+\frac{1}{4} \sqrt{3} \log \left (1-\sqrt{3} \sqrt [3]{x}+x^{2/3}\right )-\frac{1}{4} \sqrt{3} \log \left (1+\sqrt{3} \sqrt [3]{x}+x^{2/3}\right )-\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,-\sqrt{3}+2 \sqrt [3]{x}\right )-\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,\sqrt{3}+2 \sqrt [3]{x}\right )\\ &=-\frac{1}{2} \tan ^{-1}\left (\sqrt{3}-2 \sqrt [3]{x}\right )+\frac{1}{2} \tan ^{-1}\left (\sqrt{3}+2 \sqrt [3]{x}\right )+\tan ^{-1}\left (\sqrt [3]{x}\right )+\frac{1}{4} \sqrt{3} \log \left (1-\sqrt{3} \sqrt [3]{x}+x^{2/3}\right )-\frac{1}{4} \sqrt{3} \log \left (1+\sqrt{3} \sqrt [3]{x}+x^{2/3}\right )\\ \end{align*}
Mathematica [C] time = 0.0047313, size = 22, normalized size = 0.3 \[ \frac{3}{5} x^{5/3} \, _2F_1\left (\frac{5}{6},1;\frac{11}{6};-x^2\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.024, size = 69, normalized size = 1. \begin{align*} \arctan \left ( \sqrt [3]{x} \right ) +{\frac{1}{2}\arctan \left ( 2\,\sqrt [3]{x}-\sqrt{3} \right ) }+{\frac{1}{2}\arctan \left ( 2\,\sqrt [3]{x}+\sqrt{3} \right ) }+{\frac{\sqrt{3}}{4}\ln \left ( 1+{x}^{{\frac{2}{3}}}-\sqrt [3]{x}\sqrt{3} \right ) }-{\frac{\sqrt{3}}{4}\ln \left ( 1+{x}^{{\frac{2}{3}}}+\sqrt [3]{x}\sqrt{3} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 2.03088, size = 92, normalized size = 1.26 \begin{align*} -\frac{1}{4} \, \sqrt{3} \log \left (\sqrt{3} x^{\frac{1}{3}} + x^{\frac{2}{3}} + 1\right ) + \frac{1}{4} \, \sqrt{3} \log \left (-\sqrt{3} x^{\frac{1}{3}} + x^{\frac{2}{3}} + 1\right ) + \frac{1}{2} \, \arctan \left (\sqrt{3} + 2 \, x^{\frac{1}{3}}\right ) + \frac{1}{2} \, \arctan \left (-\sqrt{3} + 2 \, x^{\frac{1}{3}}\right ) + \arctan \left (x^{\frac{1}{3}}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.744, size = 381, normalized size = 5.22 \begin{align*} -\frac{1}{4} \, \sqrt{3} \log \left (16 \, \sqrt{3} x^{\frac{1}{3}} + 16 \, x^{\frac{2}{3}} + 16\right ) + \frac{1}{4} \, \sqrt{3} \log \left (-16 \, \sqrt{3} x^{\frac{1}{3}} + 16 \, x^{\frac{2}{3}} + 16\right ) - \arctan \left (\sqrt{3} + \frac{1}{2} \, \sqrt{-16 \, \sqrt{3} x^{\frac{1}{3}} + 16 \, x^{\frac{2}{3}} + 16} - 2 \, x^{\frac{1}{3}}\right ) - \arctan \left (-\sqrt{3} + 2 \, \sqrt{\sqrt{3} x^{\frac{1}{3}} + x^{\frac{2}{3}} + 1} - 2 \, x^{\frac{1}{3}}\right ) + \arctan \left (x^{\frac{1}{3}}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 2.34158, size = 94, normalized size = 1.29 \begin{align*} \frac{\sqrt{3} \log{\left (4 x^{\frac{2}{3}} - 4 \sqrt{3} \sqrt [3]{x} + 4 \right )}}{4} - \frac{\sqrt{3} \log{\left (4 x^{\frac{2}{3}} + 4 \sqrt{3} \sqrt [3]{x} + 4 \right )}}{4} + \operatorname{atan}{\left (\sqrt [3]{x} \right )} + \frac{\operatorname{atan}{\left (2 \sqrt [3]{x} - \sqrt{3} \right )}}{2} + \frac{\operatorname{atan}{\left (2 \sqrt [3]{x} + \sqrt{3} \right )}}{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.50239, size = 92, normalized size = 1.26 \begin{align*} -\frac{1}{4} \, \sqrt{3} \log \left (\sqrt{3} x^{\frac{1}{3}} + x^{\frac{2}{3}} + 1\right ) + \frac{1}{4} \, \sqrt{3} \log \left (-\sqrt{3} x^{\frac{1}{3}} + x^{\frac{2}{3}} + 1\right ) + \frac{1}{2} \, \arctan \left (\sqrt{3} + 2 \, x^{\frac{1}{3}}\right ) + \frac{1}{2} \, \arctan \left (-\sqrt{3} + 2 \, x^{\frac{1}{3}}\right ) + \arctan \left (x^{\frac{1}{3}}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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