Optimal. Leaf size=94 \[ -\frac {3}{4} \left (1-x^2\right )^{2/3}-\frac {3 \sqrt {3} \tan ^{-1}\left (\frac {1+\sqrt [3]{2-2 x^2}}{\sqrt {3}}\right )}{2\ 2^{2/3}}+\frac {3 \log \left (3+x^2\right )}{4\ 2^{2/3}}-\frac {9 \log \left (2^{2/3}-\sqrt [3]{1-x^2}\right )}{4\ 2^{2/3}} \]
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Rubi [A]
time = 0.05, antiderivative size = 94, normalized size of antiderivative = 1.00, 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 = {457, 81, 57,
631, 210, 31} \begin {gather*} -\frac {3 \sqrt {3} \text {ArcTan}\left (\frac {\sqrt [3]{2-2 x^2}+1}{\sqrt {3}}\right )}{2\ 2^{2/3}}-\frac {3}{4} \left (1-x^2\right )^{2/3}+\frac {3 \log \left (x^2+3\right )}{4\ 2^{2/3}}-\frac {9 \log \left (2^{2/3}-\sqrt [3]{1-x^2}\right )}{4\ 2^{2/3}} \end {gather*}
Antiderivative was successfully verified.
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Rule 31
Rule 57
Rule 81
Rule 210
Rule 457
Rule 631
Rubi steps
\begin {align*} \int \frac {x^3}{\sqrt [3]{1-x^2} \left (3+x^2\right )} \, dx &=\frac {1}{2} \text {Subst}\left (\int \frac {x}{\sqrt [3]{1-x} (3+x)} \, dx,x,x^2\right )\\ &=-\frac {3}{4} \left (1-x^2\right )^{2/3}-\frac {3}{2} \text {Subst}\left (\int \frac {1}{\sqrt [3]{1-x} (3+x)} \, dx,x,x^2\right )\\ &=-\frac {3}{4} \left (1-x^2\right )^{2/3}+\frac {3 \log \left (3+x^2\right )}{4\ 2^{2/3}}-\frac {9}{4} \text {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 \text {Subst}\left (\int \frac {1}{2^{2/3}-x} \, dx,x,\sqrt [3]{1-x^2}\right )}{4\ 2^{2/3}}\\ &=-\frac {3}{4} \left (1-x^2\right )^{2/3}+\frac {3 \log \left (3+x^2\right )}{4\ 2^{2/3}}-\frac {9 \log \left (2^{2/3}-\sqrt [3]{1-x^2}\right )}{4\ 2^{2/3}}+\frac {9 \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+\sqrt [3]{2-2 x^2}\right )}{2\ 2^{2/3}}\\ &=-\frac {3}{4} \left (1-x^2\right )^{2/3}-\frac {3 \sqrt {3} \tan ^{-1}\left (\frac {1+\sqrt [3]{2-2 x^2}}{\sqrt {3}}\right )}{2\ 2^{2/3}}+\frac {3 \log \left (3+x^2\right )}{4\ 2^{2/3}}-\frac {9 \log \left (2^{2/3}-\sqrt [3]{1-x^2}\right )}{4\ 2^{2/3}}\\ \end {align*}
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Mathematica [A]
time = 0.10, size = 105, normalized size = 1.12 \begin {gather*} -\frac {3}{8} \left (2 \left (1-x^2\right )^{2/3}+2 \sqrt [3]{2} \sqrt {3} \tan ^{-1}\left (\frac {1+\sqrt [3]{2-2 x^2}}{\sqrt {3}}\right )+2 \sqrt [3]{2} \log \left (-2+\sqrt [3]{2-2 x^2}\right )-\sqrt [3]{2} \log \left (4+2 \sqrt [3]{2-2 x^2}+\left (2-2 x^2\right )^{2/3}\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
3.
time = 7.61, size = 648, normalized size = 6.89
method | result | size |
risch | \(\text {Expression too large to display}\) | \(648\) |
trager | \(\text {Expression too large to display}\) | \(749\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.48, size = 97, normalized size = 1.03 \begin {gather*} -\frac {3}{8} \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}{16} \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}{8} \cdot 4^{\frac {2}{3}} \log \left (-4^{\frac {1}{3}} + {\left (-x^{2} + 1\right )}^{\frac {1}{3}}\right ) - \frac {3}{4} \, {\left (-x^{2} + 1\right )}^{\frac {2}{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.89, size = 122, normalized size = 1.30 \begin {gather*} -\frac {3}{4} \cdot 4^{\frac {1}{6}} \sqrt {3} \left (-1\right )^{\frac {1}{3}} \arctan \left (\frac {1}{6} \cdot 4^{\frac {1}{6}} \sqrt {3} {\left (2 \, \left (-1\right )^{\frac {1}{3}} {\left (-x^{2} + 1\right )}^{\frac {1}{3}} - 4^{\frac {1}{3}}\right )}\right ) - \frac {3}{16} \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 {3}{8} \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 {3}{4} \, {\left (-x^{2} + 1\right )}^{\frac {2}{3}} \end {gather*}
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 \begin {gather*} \int \frac {x^{3}}{\sqrt [3]{- \left (x - 1\right ) \left (x + 1\right )} \left (x^{2} + 3\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.59, size = 97, normalized size = 1.03 \begin {gather*} -\frac {3}{8} \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}{16} \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}{8} \cdot 4^{\frac {2}{3}} \log \left (4^{\frac {1}{3}} - {\left (-x^{2} + 1\right )}^{\frac {1}{3}}\right ) - \frac {3}{4} \, {\left (-x^{2} + 1\right )}^{\frac {2}{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.48, size = 117, normalized size = 1.24 \begin {gather*} -\frac {3\,2^{1/3}\,\ln \left (\frac {81\,{\left (1-x^2\right )}^{1/3}}{4}-\frac {81\,2^{2/3}}{4}\right )}{4}-\frac {3\,{\left (1-x^2\right )}^{2/3}}{4}-\frac {3\,2^{1/3}\,\ln \left (\frac {81\,{\left (1-x^2\right )}^{1/3}}{4}-\frac {81\,2^{2/3}\,{\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}^2}{16}\right )\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}{8}+\frac {3\,2^{1/3}\,\ln \left (\frac {81\,{\left (1-x^2\right )}^{1/3}}{4}-\frac {81\,2^{2/3}\,{\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}^2}{16}\right )\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}{8} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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