Optimal. Leaf size=101 \[ -\frac {\left (1-x^2\right )^{2/3}}{8 \left (x^2+3\right )}-\frac {\log \left (x^2+3\right )}{48\ 2^{2/3}}+\frac {\log \left (2^{2/3}-\sqrt [3]{1-x^2}\right )}{16\ 2^{2/3}}+\frac {\tan ^{-1}\left (\frac {\sqrt [3]{2-2 x^2}+1}{\sqrt {3}}\right )}{8\ 2^{2/3} \sqrt {3}} \]
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Rubi [A] time = 0.06, antiderivative size = 101, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {444, 51, 55, 617, 204, 31} \[ -\frac {\left (1-x^2\right )^{2/3}}{8 \left (x^2+3\right )}-\frac {\log \left (x^2+3\right )}{48\ 2^{2/3}}+\frac {\log \left (2^{2/3}-\sqrt [3]{1-x^2}\right )}{16\ 2^{2/3}}+\frac {\tan ^{-1}\left (\frac {\sqrt [3]{2-2 x^2}+1}{\sqrt {3}}\right )}{8\ 2^{2/3} \sqrt {3}} \]
Antiderivative was successfully verified.
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Rule 31
Rule 51
Rule 55
Rule 204
Rule 444
Rule 617
Rubi steps
\begin {align*} \int \frac {x}{\sqrt [3]{1-x^2} \left (3+x^2\right )^2} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{\sqrt [3]{1-x} (3+x)^2} \, dx,x,x^2\right )\\ &=-\frac {\left (1-x^2\right )^{2/3}}{8 \left (3+x^2\right )}+\frac {1}{24} \operatorname {Subst}\left (\int \frac {1}{\sqrt [3]{1-x} (3+x)} \, dx,x,x^2\right )\\ &=-\frac {\left (1-x^2\right )^{2/3}}{8 \left (3+x^2\right )}-\frac {\log \left (3+x^2\right )}{48\ 2^{2/3}}+\frac {1}{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 {\operatorname {Subst}\left (\int \frac {1}{2^{2/3}-x} \, dx,x,\sqrt [3]{1-x^2}\right )}{16\ 2^{2/3}}\\ &=-\frac {\left (1-x^2\right )^{2/3}}{8 \left (3+x^2\right )}-\frac {\log \left (3+x^2\right )}{48\ 2^{2/3}}+\frac {\log \left (2^{2/3}-\sqrt [3]{1-x^2}\right )}{16\ 2^{2/3}}-\frac {\operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+\sqrt [3]{2-2 x^2}\right )}{8\ 2^{2/3}}\\ &=-\frac {\left (1-x^2\right )^{2/3}}{8 \left (3+x^2\right )}+\frac {\tan ^{-1}\left (\frac {1+\sqrt [3]{2-2 x^2}}{\sqrt {3}}\right )}{8\ 2^{2/3} \sqrt {3}}-\frac {\log \left (3+x^2\right )}{48\ 2^{2/3}}+\frac {\log \left (2^{2/3}-\sqrt [3]{1-x^2}\right )}{16\ 2^{2/3}}\\ \end {align*}
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Mathematica [C] time = 0.01, size = 34, normalized size = 0.34 \[ -\frac {3}{64} \left (1-x^2\right )^{2/3} \, _2F_1\left (\frac {2}{3},2;\frac {5}{3};\frac {1}{4} \left (1-x^2\right )\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 1.10, size = 125, normalized size = 1.24 \[ \frac {4 \cdot 4^{\frac {1}{6}} \sqrt {3} {\left (x^{2} + 3\right )} \arctan \left (\frac {1}{6} \cdot 4^{\frac {1}{6}} {\left (4^{\frac {1}{3}} \sqrt {3} + 2 \, \sqrt {3} {\left (-x^{2} + 1\right )}^{\frac {1}{3}}\right )}\right ) - 4^{\frac {2}{3}} {\left (x^{2} + 3\right )} \log \left (4^{\frac {2}{3}} + 4^{\frac {1}{3}} {\left (-x^{2} + 1\right )}^{\frac {1}{3}} + {\left (-x^{2} + 1\right )}^{\frac {2}{3}}\right ) + 2 \cdot 4^{\frac {2}{3}} {\left (x^{2} + 3\right )} \log \left (-4^{\frac {1}{3}} + {\left (-x^{2} + 1\right )}^{\frac {1}{3}}\right ) - 24 \, {\left (-x^{2} + 1\right )}^{\frac {2}{3}}}{192 \, {\left (x^{2} + 3\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.39, size = 104, normalized size = 1.03 \[ \frac {1}{96} \cdot 4^{\frac {2}{3}} \sqrt {3} \arctan \left (\frac {1}{12} \cdot 4^{\frac {2}{3}} \sqrt {3} {\left (4^{\frac {1}{3}} + 2 \, {\left (-x^{2} + 1\right )}^{\frac {1}{3}}\right )}\right ) - \frac {1}{192} \cdot 4^{\frac {2}{3}} \log \left (4^{\frac {2}{3}} + 4^{\frac {1}{3}} {\left (-x^{2} + 1\right )}^{\frac {1}{3}} + {\left (-x^{2} + 1\right )}^{\frac {2}{3}}\right ) + \frac {1}{96} \cdot 4^{\frac {2}{3}} \log \left (4^{\frac {1}{3}} - {\left (-x^{2} + 1\right )}^{\frac {1}{3}}\right ) - \frac {{\left (-x^{2} + 1\right )}^{\frac {2}{3}}}{8 \, {\left (x^{2} + 3\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 4.95, size = 657, normalized size = 6.50 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.03, size = 104, normalized size = 1.03 \[ \frac {1}{96} \cdot 4^{\frac {2}{3}} \sqrt {3} \arctan \left (\frac {1}{12} \cdot 4^{\frac {2}{3}} \sqrt {3} {\left (4^{\frac {1}{3}} + 2 \, {\left (-x^{2} + 1\right )}^{\frac {1}{3}}\right )}\right ) - \frac {1}{192} \cdot 4^{\frac {2}{3}} \log \left (4^{\frac {2}{3}} + 4^{\frac {1}{3}} {\left (-x^{2} + 1\right )}^{\frac {1}{3}} + {\left (-x^{2} + 1\right )}^{\frac {2}{3}}\right ) + \frac {1}{96} \cdot 4^{\frac {2}{3}} \log \left (-4^{\frac {1}{3}} + {\left (-x^{2} + 1\right )}^{\frac {1}{3}}\right ) - \frac {{\left (-x^{2} + 1\right )}^{\frac {2}{3}}}{8 \, {\left (x^{2} + 3\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.86, size = 126, normalized size = 1.25 \[ \frac {2^{1/3}\,\ln \left (\frac {{\left (1-x^2\right )}^{1/3}}{64}-\frac {2^{2/3}}{64}\right )}{48}-\frac {{\left (1-x^2\right )}^{2/3}}{8\,\left (x^2+3\right )}+\frac {2^{1/3}\,\ln \left (\frac {{\left (1-x^2\right )}^{1/3}}{64}-\frac {2^{2/3}\,{\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}^2}{256}\right )\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}{96}-\frac {2^{1/3}\,\ln \left (\frac {{\left (1-x^2\right )}^{1/3}}{64}-\frac {2^{2/3}\,{\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}^2}{256}\right )\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}{96} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x}{\sqrt [3]{- \left (x - 1\right ) \left (x + 1\right )} \left (x^{2} + 3\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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