Optimal. Leaf size=128 \[ -\frac {1}{2} \left (1-x^3\right )^{2/3}+\frac {1}{5} \left (1-x^3\right )^{5/3}-\frac {1}{8} \left (1-x^3\right )^{8/3}-\frac {\tan ^{-1}\left (\frac {1+2^{2/3} \sqrt [3]{1-x^3}}{\sqrt {3}}\right )}{\sqrt [3]{2} \sqrt {3}}+\frac {\log \left (1+x^3\right )}{6 \sqrt [3]{2}}-\frac {\log \left (\sqrt [3]{2}-\sqrt [3]{1-x^3}\right )}{2 \sqrt [3]{2}} \]
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Rubi [A]
time = 0.06, antiderivative size = 128, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 6, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {457, 90, 57,
631, 210, 31} \begin {gather*} -\frac {\text {ArcTan}\left (\frac {2^{2/3} \sqrt [3]{1-x^3}+1}{\sqrt {3}}\right )}{\sqrt [3]{2} \sqrt {3}}-\frac {1}{8} \left (1-x^3\right )^{8/3}+\frac {1}{5} \left (1-x^3\right )^{5/3}-\frac {1}{2} \left (1-x^3\right )^{2/3}+\frac {\log \left (x^3+1\right )}{6 \sqrt [3]{2}}-\frac {\log \left (\sqrt [3]{2}-\sqrt [3]{1-x^3}\right )}{2 \sqrt [3]{2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 31
Rule 57
Rule 90
Rule 210
Rule 457
Rule 631
Rubi steps
\begin {align*} \int \frac {x^{11}}{\sqrt [3]{1-x^3} \left (1+x^3\right )} \, dx &=\frac {1}{3} \text {Subst}\left (\int \frac {x^3}{\sqrt [3]{1-x} (1+x)} \, dx,x,x^3\right )\\ &=\frac {1}{3} \text {Subst}\left (\int \left (\frac {1}{\sqrt [3]{1-x}}-(1-x)^{2/3}+(1-x)^{5/3}-\frac {1}{\sqrt [3]{1-x} (1+x)}\right ) \, dx,x,x^3\right )\\ &=-\frac {1}{2} \left (1-x^3\right )^{2/3}+\frac {1}{5} \left (1-x^3\right )^{5/3}-\frac {1}{8} \left (1-x^3\right )^{8/3}-\frac {1}{3} \text {Subst}\left (\int \frac {1}{\sqrt [3]{1-x} (1+x)} \, dx,x,x^3\right )\\ &=-\frac {1}{2} \left (1-x^3\right )^{2/3}+\frac {1}{5} \left (1-x^3\right )^{5/3}-\frac {1}{8} \left (1-x^3\right )^{8/3}+\frac {\log \left (1+x^3\right )}{6 \sqrt [3]{2}}-\frac {1}{2} \text {Subst}\left (\int \frac {1}{2^{2/3}+\sqrt [3]{2} x+x^2} \, dx,x,\sqrt [3]{1-x^3}\right )+\frac {\text {Subst}\left (\int \frac {1}{\sqrt [3]{2}-x} \, dx,x,\sqrt [3]{1-x^3}\right )}{2 \sqrt [3]{2}}\\ &=-\frac {1}{2} \left (1-x^3\right )^{2/3}+\frac {1}{5} \left (1-x^3\right )^{5/3}-\frac {1}{8} \left (1-x^3\right )^{8/3}+\frac {\log \left (1+x^3\right )}{6 \sqrt [3]{2}}-\frac {\log \left (\sqrt [3]{2}-\sqrt [3]{1-x^3}\right )}{2 \sqrt [3]{2}}+\frac {\text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+2^{2/3} \sqrt [3]{1-x^3}\right )}{\sqrt [3]{2}}\\ &=-\frac {1}{2} \left (1-x^3\right )^{2/3}+\frac {1}{5} \left (1-x^3\right )^{5/3}-\frac {1}{8} \left (1-x^3\right )^{8/3}-\frac {\tan ^{-1}\left (\frac {1+2^{2/3} \sqrt [3]{1-x^3}}{\sqrt {3}}\right )}{\sqrt [3]{2} \sqrt {3}}+\frac {\log \left (1+x^3\right )}{6 \sqrt [3]{2}}-\frac {\log \left (\sqrt [3]{2}-\sqrt [3]{1-x^3}\right )}{2 \sqrt [3]{2}}\\ \end {align*}
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Mathematica [A]
time = 0.12, size = 139, normalized size = 1.09 \begin {gather*} \frac {1}{120} \left (-3 \left (1-x^3\right )^{2/3} \left (17-2 x^3+5 x^6\right )-20\ 2^{2/3} \sqrt {3} \tan ^{-1}\left (\frac {1+2^{2/3} \sqrt [3]{1-x^3}}{\sqrt {3}}\right )-20\ 2^{2/3} \log \left (-2+2^{2/3} \sqrt [3]{1-x^3}\right )+10\ 2^{2/3} \log \left (2+2^{2/3} \sqrt [3]{1-x^3}+\sqrt [3]{2} \left (1-x^3\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 = 5.07, size = 497, normalized size = 3.88
method | result | size |
trager | \(\text {Expression too large to display}\) | \(497\) |
risch | \(\text {Expression too large to display}\) | \(790\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.59, size = 119, normalized size = 0.93 \begin {gather*} -\frac {1}{8} \, {\left (-x^{3} + 1\right )}^{\frac {8}{3}} - \frac {1}{6} \, \sqrt {3} 2^{\frac {2}{3}} \arctan \left (\frac {1}{6} \, \sqrt {3} 2^{\frac {2}{3}} {\left (2^{\frac {1}{3}} + 2 \, {\left (-x^{3} + 1\right )}^{\frac {1}{3}}\right )}\right ) + \frac {1}{5} \, {\left (-x^{3} + 1\right )}^{\frac {5}{3}} + \frac {1}{12} \cdot 2^{\frac {2}{3}} \log \left (2^{\frac {2}{3}} + 2^{\frac {1}{3}} {\left (-x^{3} + 1\right )}^{\frac {1}{3}} + {\left (-x^{3} + 1\right )}^{\frac {2}{3}}\right ) - \frac {1}{6} \cdot 2^{\frac {2}{3}} \log \left (-2^{\frac {1}{3}} + {\left (-x^{3} + 1\right )}^{\frac {1}{3}}\right ) - \frac {1}{2} \, {\left (-x^{3} + 1\right )}^{\frac {2}{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.38, size = 137, normalized size = 1.07 \begin {gather*} -\frac {1}{6} \, \sqrt {6} 2^{\frac {1}{6}} \left (-1\right )^{\frac {1}{3}} \arctan \left (\frac {1}{6} \cdot 2^{\frac {1}{6}} {\left (2 \, \sqrt {6} \left (-1\right )^{\frac {1}{3}} {\left (-x^{3} + 1\right )}^{\frac {1}{3}} - \sqrt {6} 2^{\frac {1}{3}}\right )}\right ) - \frac {1}{12} \cdot 2^{\frac {2}{3}} \left (-1\right )^{\frac {1}{3}} \log \left (2^{\frac {1}{3}} \left (-1\right )^{\frac {2}{3}} {\left (-x^{3} + 1\right )}^{\frac {1}{3}} - 2^{\frac {2}{3}} \left (-1\right )^{\frac {1}{3}} + {\left (-x^{3} + 1\right )}^{\frac {2}{3}}\right ) + \frac {1}{6} \cdot 2^{\frac {2}{3}} \left (-1\right )^{\frac {1}{3}} \log \left (-2^{\frac {1}{3}} \left (-1\right )^{\frac {2}{3}} + {\left (-x^{3} + 1\right )}^{\frac {1}{3}}\right ) - \frac {1}{40} \, {\left (5 \, x^{6} - 2 \, x^{3} + 17\right )} {\left (-x^{3} + 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^{11}}{\sqrt [3]{- \left (x - 1\right ) \left (x^{2} + x + 1\right )} \left (x + 1\right ) \left (x^{2} - x + 1\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.68, size = 127, normalized size = 0.99 \begin {gather*} -\frac {1}{8} \, {\left (x^{3} - 1\right )}^{2} {\left (-x^{3} + 1\right )}^{\frac {2}{3}} - \frac {1}{6} \, \sqrt {3} 2^{\frac {2}{3}} \arctan \left (\frac {1}{6} \, \sqrt {3} 2^{\frac {2}{3}} {\left (2^{\frac {1}{3}} + 2 \, {\left (-x^{3} + 1\right )}^{\frac {1}{3}}\right )}\right ) + \frac {1}{5} \, {\left (-x^{3} + 1\right )}^{\frac {5}{3}} + \frac {1}{12} \cdot 2^{\frac {2}{3}} \log \left (2^{\frac {2}{3}} + 2^{\frac {1}{3}} {\left (-x^{3} + 1\right )}^{\frac {1}{3}} + {\left (-x^{3} + 1\right )}^{\frac {2}{3}}\right ) - \frac {1}{6} \cdot 2^{\frac {2}{3}} \log \left ({\left | -2^{\frac {1}{3}} + {\left (-x^{3} + 1\right )}^{\frac {1}{3}} \right |}\right ) - \frac {1}{2} \, {\left (-x^{3} + 1\right )}^{\frac {2}{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 4.73, size = 133, normalized size = 1.04 \begin {gather*} \frac {{\left (1-x^3\right )}^{5/3}}{5}-\frac {{\left (1-x^3\right )}^{2/3}}{2}-\frac {2^{2/3}\,\ln \left ({\left (1-x^3\right )}^{1/3}-2^{1/3}\right )}{6}-\frac {{\left (1-x^3\right )}^{8/3}}{8}-\frac {2^{2/3}\,\ln \left ({\left (1-x^3\right )}^{1/3}-\frac {2^{1/3}\,{\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}^2}{4}\right )\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}{12}+\frac {2^{2/3}\,\ln \left ({\left (1-x^3\right )}^{1/3}-\frac {2^{1/3}\,{\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}^2}{4}\right )\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}{12} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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