Optimal. Leaf size=538 \[ -\frac{\sqrt{2} \left (1-\sqrt [3]{1-x^2}\right ) \sqrt{\frac{\left (1-x^2\right )^{2/3}+\sqrt [3]{1-x^2}+1}{\left (-\sqrt [3]{1-x^2}-\sqrt{3}+1\right )^2}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{-\sqrt [3]{1-x^2}+\sqrt{3}+1}{-\sqrt [3]{1-x^2}-\sqrt{3}+1}\right ),4 \sqrt{3}-7\right )}{3 \sqrt [4]{3} \sqrt{-\frac{1-\sqrt [3]{1-x^2}}{\left (-\sqrt [3]{1-x^2}-\sqrt{3}+1\right )^2}} x}+\frac{x}{3 \left (-\sqrt [3]{1-x^2}-\sqrt{3}+1\right )}-\frac{\left (1-x^2\right )^{2/3}}{3 x}-\frac{\tan ^{-1}\left (\frac{\sqrt{3} \left (1-\sqrt [3]{2} \sqrt [3]{1-x^2}\right )}{x}\right )}{6\ 2^{2/3} \sqrt{3}}-\frac{\tanh ^{-1}\left (\frac{x}{\sqrt [3]{2} \sqrt [3]{1-x^2}+1}\right )}{6\ 2^{2/3}}+\frac{\sqrt{2+\sqrt{3}} \left (1-\sqrt [3]{1-x^2}\right ) \sqrt{\frac{\left (1-x^2\right )^{2/3}+\sqrt [3]{1-x^2}+1}{\left (-\sqrt [3]{1-x^2}-\sqrt{3}+1\right )^2}} E\left (\sin ^{-1}\left (\frac{-\sqrt [3]{1-x^2}+\sqrt{3}+1}{-\sqrt [3]{1-x^2}-\sqrt{3}+1}\right )|-7+4 \sqrt{3}\right )}{2\ 3^{3/4} \sqrt{-\frac{1-\sqrt [3]{1-x^2}}{\left (-\sqrt [3]{1-x^2}-\sqrt{3}+1\right )^2}} x}-\frac{\tan ^{-1}\left (\frac{\sqrt{3}}{x}\right )}{6\ 2^{2/3} \sqrt{3}}+\frac{\tanh ^{-1}(x)}{18\ 2^{2/3}} \]
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Rubi [A] time = 0.23598, antiderivative size = 538, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.318, Rules used = {480, 530, 235, 304, 219, 1879, 393} \[ \frac{x}{3 \left (-\sqrt [3]{1-x^2}-\sqrt{3}+1\right )}-\frac{\left (1-x^2\right )^{2/3}}{3 x}-\frac{\tan ^{-1}\left (\frac{\sqrt{3} \left (1-\sqrt [3]{2} \sqrt [3]{1-x^2}\right )}{x}\right )}{6\ 2^{2/3} \sqrt{3}}-\frac{\tanh ^{-1}\left (\frac{x}{\sqrt [3]{2} \sqrt [3]{1-x^2}+1}\right )}{6\ 2^{2/3}}-\frac{\sqrt{2} \left (1-\sqrt [3]{1-x^2}\right ) \sqrt{\frac{\left (1-x^2\right )^{2/3}+\sqrt [3]{1-x^2}+1}{\left (-\sqrt [3]{1-x^2}-\sqrt{3}+1\right )^2}} F\left (\sin ^{-1}\left (\frac{-\sqrt [3]{1-x^2}+\sqrt{3}+1}{-\sqrt [3]{1-x^2}-\sqrt{3}+1}\right )|-7+4 \sqrt{3}\right )}{3 \sqrt [4]{3} \sqrt{-\frac{1-\sqrt [3]{1-x^2}}{\left (-\sqrt [3]{1-x^2}-\sqrt{3}+1\right )^2}} x}+\frac{\sqrt{2+\sqrt{3}} \left (1-\sqrt [3]{1-x^2}\right ) \sqrt{\frac{\left (1-x^2\right )^{2/3}+\sqrt [3]{1-x^2}+1}{\left (-\sqrt [3]{1-x^2}-\sqrt{3}+1\right )^2}} E\left (\sin ^{-1}\left (\frac{-\sqrt [3]{1-x^2}+\sqrt{3}+1}{-\sqrt [3]{1-x^2}-\sqrt{3}+1}\right )|-7+4 \sqrt{3}\right )}{2\ 3^{3/4} \sqrt{-\frac{1-\sqrt [3]{1-x^2}}{\left (-\sqrt [3]{1-x^2}-\sqrt{3}+1\right )^2}} x}-\frac{\tan ^{-1}\left (\frac{\sqrt{3}}{x}\right )}{6\ 2^{2/3} \sqrt{3}}+\frac{\tanh ^{-1}(x)}{18\ 2^{2/3}} \]
Antiderivative was successfully verified.
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Rule 480
Rule 530
Rule 235
Rule 304
Rule 219
Rule 1879
Rule 393
Rubi steps
\begin{align*} \int \frac{1}{x^2 \sqrt [3]{1-x^2} \left (3+x^2\right )} \, dx &=-\frac{\left (1-x^2\right )^{2/3}}{3 x}+\frac{1}{3} \int \frac{-2-\frac{x^2}{3}}{\sqrt [3]{1-x^2} \left (3+x^2\right )} \, dx\\ &=-\frac{\left (1-x^2\right )^{2/3}}{3 x}-\frac{1}{9} \int \frac{1}{\sqrt [3]{1-x^2}} \, dx-\frac{1}{3} \int \frac{1}{\sqrt [3]{1-x^2} \left (3+x^2\right )} \, dx\\ &=-\frac{\left (1-x^2\right )^{2/3}}{3 x}-\frac{\tan ^{-1}\left (\frac{\sqrt{3}}{x}\right )}{6\ 2^{2/3} \sqrt{3}}-\frac{\tan ^{-1}\left (\frac{\sqrt{3} \left (1-\sqrt [3]{2} \sqrt [3]{1-x^2}\right )}{x}\right )}{6\ 2^{2/3} \sqrt{3}}+\frac{\tanh ^{-1}(x)}{18\ 2^{2/3}}-\frac{\tanh ^{-1}\left (\frac{x}{1+\sqrt [3]{2} \sqrt [3]{1-x^2}}\right )}{6\ 2^{2/3}}+\frac{\sqrt{-x^2} \operatorname{Subst}\left (\int \frac{x}{\sqrt{-1+x^3}} \, dx,x,\sqrt [3]{1-x^2}\right )}{6 x}\\ &=-\frac{\left (1-x^2\right )^{2/3}}{3 x}-\frac{\tan ^{-1}\left (\frac{\sqrt{3}}{x}\right )}{6\ 2^{2/3} \sqrt{3}}-\frac{\tan ^{-1}\left (\frac{\sqrt{3} \left (1-\sqrt [3]{2} \sqrt [3]{1-x^2}\right )}{x}\right )}{6\ 2^{2/3} \sqrt{3}}+\frac{\tanh ^{-1}(x)}{18\ 2^{2/3}}-\frac{\tanh ^{-1}\left (\frac{x}{1+\sqrt [3]{2} \sqrt [3]{1-x^2}}\right )}{6\ 2^{2/3}}-\frac{\sqrt{-x^2} \operatorname{Subst}\left (\int \frac{1+\sqrt{3}-x}{\sqrt{-1+x^3}} \, dx,x,\sqrt [3]{1-x^2}\right )}{6 x}+\frac{\left (\sqrt{\frac{1}{2} \left (2+\sqrt{3}\right )} \sqrt{-x^2}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{-1+x^3}} \, dx,x,\sqrt [3]{1-x^2}\right )}{3 x}\\ &=-\frac{\left (1-x^2\right )^{2/3}}{3 x}+\frac{x}{3 \left (1-\sqrt{3}-\sqrt [3]{1-x^2}\right )}-\frac{\tan ^{-1}\left (\frac{\sqrt{3}}{x}\right )}{6\ 2^{2/3} \sqrt{3}}-\frac{\tan ^{-1}\left (\frac{\sqrt{3} \left (1-\sqrt [3]{2} \sqrt [3]{1-x^2}\right )}{x}\right )}{6\ 2^{2/3} \sqrt{3}}+\frac{\tanh ^{-1}(x)}{18\ 2^{2/3}}-\frac{\tanh ^{-1}\left (\frac{x}{1+\sqrt [3]{2} \sqrt [3]{1-x^2}}\right )}{6\ 2^{2/3}}+\frac{\sqrt{2+\sqrt{3}} \left (1-\sqrt [3]{1-x^2}\right ) \sqrt{\frac{1+\sqrt [3]{1-x^2}+\left (1-x^2\right )^{2/3}}{\left (1-\sqrt{3}-\sqrt [3]{1-x^2}\right )^2}} E\left (\sin ^{-1}\left (\frac{1+\sqrt{3}-\sqrt [3]{1-x^2}}{1-\sqrt{3}-\sqrt [3]{1-x^2}}\right )|-7+4 \sqrt{3}\right )}{2\ 3^{3/4} x \sqrt{-\frac{1-\sqrt [3]{1-x^2}}{\left (1-\sqrt{3}-\sqrt [3]{1-x^2}\right )^2}}}-\frac{\sqrt{2} \left (1-\sqrt [3]{1-x^2}\right ) \sqrt{\frac{1+\sqrt [3]{1-x^2}+\left (1-x^2\right )^{2/3}}{\left (1-\sqrt{3}-\sqrt [3]{1-x^2}\right )^2}} F\left (\sin ^{-1}\left (\frac{1+\sqrt{3}-\sqrt [3]{1-x^2}}{1-\sqrt{3}-\sqrt [3]{1-x^2}}\right )|-7+4 \sqrt{3}\right )}{3 \sqrt [4]{3} x \sqrt{-\frac{1-\sqrt [3]{1-x^2}}{\left (1-\sqrt{3}-\sqrt [3]{1-x^2}\right )^2}}}\\ \end{align*}
Mathematica [C] time = 0.0942054, size = 161, normalized size = 0.3 \[ \frac{\frac{18 x^2 F_1\left (\frac{1}{2};\frac{1}{3},1;\frac{3}{2};x^2,-\frac{x^2}{3}\right )}{\left (x^2+3\right ) \left (2 x^2 \left (F_1\left (\frac{3}{2};\frac{1}{3},2;\frac{5}{2};x^2,-\frac{x^2}{3}\right )-F_1\left (\frac{3}{2};\frac{4}{3},1;\frac{5}{2};x^2,-\frac{x^2}{3}\right )\right )-9 F_1\left (\frac{1}{2};\frac{1}{3},1;\frac{3}{2};x^2,-\frac{x^2}{3}\right )\right )}+x^2-1}{3 x \sqrt [3]{1-x^2}}-\frac{1}{81} x^3 F_1\left (\frac{3}{2};\frac{1}{3},1;\frac{5}{2};x^2,-\frac{x^2}{3}\right ) \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.046, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{2} \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^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-\frac{{\left (-x^{2} + 1\right )}^{\frac{2}{3}}}{x^{6} + 2 \, x^{4} - 3 \, x^{2}}, x\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^{2} \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] 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^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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