Optimal. Leaf size=101 \[ \frac{3 \left (1-x^2\right )^{2/3}}{8 \left (x^2+3\right )}-\frac{3 \log \left (x^2+3\right )}{16\ 2^{2/3}}+\frac{9 \log \left (2^{2/3}-\sqrt [3]{1-x^2}\right )}{16\ 2^{2/3}}+\frac{3 \sqrt{3} \tan ^{-1}\left (\frac{\sqrt [3]{2-2 x^2}+1}{\sqrt{3}}\right )}{8\ 2^{2/3}} \]
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Rubi [A] time = 0.0659721, antiderivative size = 101, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273, Rules used = {446, 78, 55, 617, 204, 31} \[ \frac{3 \left (1-x^2\right )^{2/3}}{8 \left (x^2+3\right )}-\frac{3 \log \left (x^2+3\right )}{16\ 2^{2/3}}+\frac{9 \log \left (2^{2/3}-\sqrt [3]{1-x^2}\right )}{16\ 2^{2/3}}+\frac{3 \sqrt{3} \tan ^{-1}\left (\frac{\sqrt [3]{2-2 x^2}+1}{\sqrt{3}}\right )}{8\ 2^{2/3}} \]
Antiderivative was successfully verified.
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Rule 446
Rule 78
Rule 55
Rule 617
Rule 204
Rule 31
Rubi steps
\begin{align*} \int \frac{x^3}{\sqrt [3]{1-x^2} \left (3+x^2\right )^2} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{x}{\sqrt [3]{1-x} (3+x)^2} \, dx,x,x^2\right )\\ &=\frac{3 \left (1-x^2\right )^{2/3}}{8 \left (3+x^2\right )}+\frac{3}{8} \operatorname{Subst}\left (\int \frac{1}{\sqrt [3]{1-x} (3+x)} \, dx,x,x^2\right )\\ &=\frac{3 \left (1-x^2\right )^{2/3}}{8 \left (3+x^2\right )}-\frac{3 \log \left (3+x^2\right )}{16\ 2^{2/3}}+\frac{9}{16} \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{9 \operatorname{Subst}\left (\int \frac{1}{2^{2/3}-x} \, dx,x,\sqrt [3]{1-x^2}\right )}{16\ 2^{2/3}}\\ &=\frac{3 \left (1-x^2\right )^{2/3}}{8 \left (3+x^2\right )}-\frac{3 \log \left (3+x^2\right )}{16\ 2^{2/3}}+\frac{9 \log \left (2^{2/3}-\sqrt [3]{1-x^2}\right )}{16\ 2^{2/3}}-\frac{9 \operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,1+\sqrt [3]{2-2 x^2}\right )}{8\ 2^{2/3}}\\ &=\frac{3 \left (1-x^2\right )^{2/3}}{8 \left (3+x^2\right )}+\frac{3 \sqrt{3} \tan ^{-1}\left (\frac{1+\sqrt [3]{2-2 x^2}}{\sqrt{3}}\right )}{8\ 2^{2/3}}-\frac{3 \log \left (3+x^2\right )}{16\ 2^{2/3}}+\frac{9 \log \left (2^{2/3}-\sqrt [3]{1-x^2}\right )}{16\ 2^{2/3}}\\ \end{align*}
Mathematica [A] time = 0.0906556, size = 97, normalized size = 0.96 \[ \frac{3}{32} \left (\frac{4 \left (1-x^2\right )^{2/3}}{x^2+3}-\sqrt [3]{2} \log \left (x^2+3\right )+3 \sqrt [3]{2} \log \left (2^{2/3}-\sqrt [3]{1-x^2}\right )+2 \sqrt [3]{2} \sqrt{3} \tan ^{-1}\left (\frac{\sqrt [3]{2-2 x^2}+1}{\sqrt{3}}\right )\right ) \]
Antiderivative was successfully verified.
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Maple [F] time = 0.042, size = 0, normalized size = 0. \begin{align*} \int{\frac{{x}^{3}}{ \left ({x}^{2}+3 \right ) ^{2}}{\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 [A] time = 1.50316, size = 140, normalized size = 1.39 \begin{align*} \frac{3}{32} \cdot 4^{\frac{2}{3}} \sqrt{3} \arctan \left (\frac{1}{12} \cdot 4^{\frac{2}{3}} \sqrt{3}{\left (4^{\frac{1}{3}} + 2 \,{\left (-x^{2} + 1\right )}^{\frac{1}{3}}\right )}\right ) - \frac{3}{64} \cdot 4^{\frac{2}{3}} \log \left (4^{\frac{2}{3}} + 4^{\frac{1}{3}}{\left (-x^{2} + 1\right )}^{\frac{1}{3}} +{\left (-x^{2} + 1\right )}^{\frac{2}{3}}\right ) + \frac{3}{32} \cdot 4^{\frac{2}{3}} \log \left (-4^{\frac{1}{3}} +{\left (-x^{2} + 1\right )}^{\frac{1}{3}}\right ) + \frac{3 \,{\left (-x^{2} + 1\right )}^{\frac{2}{3}}}{8 \,{\left (x^{2} + 3\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.53422, size = 351, normalized size = 3.48 \begin{align*} \frac{3 \,{\left (4 \cdot 4^{\frac{1}{6}} \sqrt{3}{\left (x^{2} + 3\right )} \arctan \left (\frac{1}{6} \cdot 4^{\frac{1}{6}} \sqrt{3}{\left (4^{\frac{1}{3}} + 2 \,{\left (-x^{2} + 1\right )}^{\frac{1}{3}}\right )}\right ) - 4^{\frac{2}{3}}{\left (x^{2} + 3\right )} \log \left (4^{\frac{2}{3}} + 4^{\frac{1}{3}}{\left (-x^{2} + 1\right )}^{\frac{1}{3}} +{\left (-x^{2} + 1\right )}^{\frac{2}{3}}\right ) + 2 \cdot 4^{\frac{2}{3}}{\left (x^{2} + 3\right )} \log \left (-4^{\frac{1}{3}} +{\left (-x^{2} + 1\right )}^{\frac{1}{3}}\right ) + 8 \,{\left (-x^{2} + 1\right )}^{\frac{2}{3}}\right )}}{64 \,{\left (x^{2} + 3\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \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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