Optimal. Leaf size=136 \[ \frac{\log \left (x^2+3\right )}{12\ 2^{2/3}}+\frac{1}{4} \log \left (1-\sqrt [3]{1-x^2}\right )-\frac{\log \left (2^{2/3}-\sqrt [3]{1-x^2}\right )}{4\ 2^{2/3}}-\frac{\tan ^{-1}\left (\frac{\sqrt [3]{2-2 x^2}+1}{\sqrt{3}}\right )}{2\ 2^{2/3} \sqrt{3}}+\frac{\tan ^{-1}\left (\frac{2 \sqrt [3]{1-x^2}+1}{\sqrt{3}}\right )}{2 \sqrt{3}}-\frac{\log (x)}{6} \]
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Rubi [A] time = 0.0990665, antiderivative size = 136, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 7, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.318, Rules used = {446, 86, 55, 618, 204, 31, 617} \[ \frac{\log \left (x^2+3\right )}{12\ 2^{2/3}}+\frac{1}{4} \log \left (1-\sqrt [3]{1-x^2}\right )-\frac{\log \left (2^{2/3}-\sqrt [3]{1-x^2}\right )}{4\ 2^{2/3}}-\frac{\tan ^{-1}\left (\frac{\sqrt [3]{2-2 x^2}+1}{\sqrt{3}}\right )}{2\ 2^{2/3} \sqrt{3}}+\frac{\tan ^{-1}\left (\frac{2 \sqrt [3]{1-x^2}+1}{\sqrt{3}}\right )}{2 \sqrt{3}}-\frac{\log (x)}{6} \]
Antiderivative was successfully verified.
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Rule 446
Rule 86
Rule 55
Rule 618
Rule 204
Rule 31
Rule 617
Rubi steps
\begin{align*} \int \frac{1}{x \sqrt [3]{1-x^2} \left (3+x^2\right )} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{\sqrt [3]{1-x} x (3+x)} \, dx,x,x^2\right )\\ &=\frac{1}{6} \operatorname{Subst}\left (\int \frac{1}{\sqrt [3]{1-x} x} \, dx,x,x^2\right )-\frac{1}{6} \operatorname{Subst}\left (\int \frac{1}{\sqrt [3]{1-x} (3+x)} \, dx,x,x^2\right )\\ &=-\frac{\log (x)}{6}+\frac{\log \left (3+x^2\right )}{12\ 2^{2/3}}-\frac{1}{4} \operatorname{Subst}\left (\int \frac{1}{1-x} \, dx,x,\sqrt [3]{1-x^2}\right )+\frac{1}{4} \operatorname{Subst}\left (\int \frac{1}{1+x+x^2} \, dx,x,\sqrt [3]{1-x^2}\right )-\frac{1}{4} \operatorname{Subst}\left (\int \frac{1}{2 \sqrt [3]{2}+2^{2/3} x+x^2} \, dx,x,\sqrt [3]{1-x^2}\right )+\frac{\operatorname{Subst}\left (\int \frac{1}{2^{2/3}-x} \, dx,x,\sqrt [3]{1-x^2}\right )}{4\ 2^{2/3}}\\ &=-\frac{\log (x)}{6}+\frac{\log \left (3+x^2\right )}{12\ 2^{2/3}}+\frac{1}{4} \log \left (1-\sqrt [3]{1-x^2}\right )-\frac{\log \left (2^{2/3}-\sqrt [3]{1-x^2}\right )}{4\ 2^{2/3}}-\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,1+2 \sqrt [3]{1-x^2}\right )+\frac{\operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,1+\sqrt [3]{2-2 x^2}\right )}{2\ 2^{2/3}}\\ &=-\frac{\tan ^{-1}\left (\frac{1+\sqrt [3]{2-2 x^2}}{\sqrt{3}}\right )}{2\ 2^{2/3} \sqrt{3}}+\frac{\tan ^{-1}\left (\frac{1+2 \sqrt [3]{1-x^2}}{\sqrt{3}}\right )}{2 \sqrt{3}}-\frac{\log (x)}{6}+\frac{\log \left (3+x^2\right )}{12\ 2^{2/3}}+\frac{1}{4} \log \left (1-\sqrt [3]{1-x^2}\right )-\frac{\log \left (2^{2/3}-\sqrt [3]{1-x^2}\right )}{4\ 2^{2/3}}\\ \end{align*}
Mathematica [A] time = 0.0359399, size = 127, normalized size = 0.93 \[ \frac{1}{24} \left (\sqrt [3]{2} \log \left (x^2+3\right )+6 \log \left (1-\sqrt [3]{1-x^2}\right )-3 \sqrt [3]{2} \log \left (2^{2/3}-\sqrt [3]{1-x^2}\right )-2 \sqrt [3]{2} \sqrt{3} \tan ^{-1}\left (\frac{\sqrt [3]{2-2 x^2}+1}{\sqrt{3}}\right )+4 \sqrt{3} \tan ^{-1}\left (\frac{2 \sqrt [3]{1-x^2}+1}{\sqrt{3}}\right )-4 \log (x)\right ) \]
Antiderivative was successfully verified.
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Maple [F] time = 0.039, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{x \left ({x}^{2}+3 \right ) }{\frac{1}{\sqrt [3]{-{x}^{2}+1}}}}\, 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}{{\left (x^{2} + 3\right )}{\left (-x^{2} + 1\right )}^{\frac{1}{3}} x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.5695, size = 591, normalized size = 4.35 \begin{align*} -\frac{1}{12} \cdot 4^{\frac{1}{6}} \sqrt{3} \left (-1\right )^{\frac{1}{3}} \arctan \left (\frac{1}{6} \cdot 4^{\frac{1}{6}}{\left (2 \, \sqrt{3} \left (-1\right )^{\frac{1}{3}}{\left (-x^{2} + 1\right )}^{\frac{1}{3}} - 4^{\frac{1}{3}} \sqrt{3}\right )}\right ) - \frac{1}{48} \cdot 4^{\frac{2}{3}} \left (-1\right )^{\frac{1}{3}} \log \left (4^{\frac{1}{3}} \left (-1\right )^{\frac{2}{3}}{\left (-x^{2} + 1\right )}^{\frac{1}{3}} - 4^{\frac{2}{3}} \left (-1\right )^{\frac{1}{3}} +{\left (-x^{2} + 1\right )}^{\frac{2}{3}}\right ) + \frac{1}{24} \cdot 4^{\frac{2}{3}} \left (-1\right )^{\frac{1}{3}} \log \left (-4^{\frac{1}{3}} \left (-1\right )^{\frac{2}{3}} +{\left (-x^{2} + 1\right )}^{\frac{1}{3}}\right ) + \frac{1}{6} \, \sqrt{3} \arctan \left (\frac{2}{3} \, \sqrt{3}{\left (-x^{2} + 1\right )}^{\frac{1}{3}} + \frac{1}{3} \, \sqrt{3}\right ) - \frac{1}{12} \, \log \left ({\left (-x^{2} + 1\right )}^{\frac{2}{3}} +{\left (-x^{2} + 1\right )}^{\frac{1}{3}} + 1\right ) + \frac{1}{6} \, \log \left ({\left (-x^{2} + 1\right )}^{\frac{1}{3}} - 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{x \sqrt [3]{- \left (x - 1\right ) \left (x + 1\right )} \left (x^{2} + 3\right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: NotImplementedError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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