Optimal. Leaf size=127 \[ \frac {2}{5} \left (1-x^3\right )^{5/3}-\frac {1}{4} \left (1-x^3\right )^{8/3}+\frac {1}{11} \left (1-x^3\right )^{11/3}+\frac {\tan ^{-1}\left (\frac {1+2^{2/3} \sqrt [3]{1-x^3}}{\sqrt {3}}\right )}{\sqrt [3]{2} \sqrt {3}}-\frac {\log \left (1+x^3\right )}{6 \sqrt [3]{2}}+\frac {\log \left (\sqrt [3]{2}-\sqrt [3]{1-x^3}\right )}{2 \sqrt [3]{2}} \]
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Rubi [A]
time = 0.06, antiderivative size = 127, 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}{11} \left (1-x^3\right )^{11/3}-\frac {1}{4} \left (1-x^3\right )^{8/3}+\frac {2}{5} \left (1-x^3\right )^{5/3}-\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^{14}}{\sqrt [3]{1-x^3} \left (1+x^3\right )} \, dx &=\frac {1}{3} \text {Subst}\left (\int \frac {x^4}{\sqrt [3]{1-x} (1+x)} \, dx,x,x^3\right )\\ &=\frac {1}{3} \text {Subst}\left (\int \left (-2 (1-x)^{2/3}+2 (1-x)^{5/3}-(1-x)^{8/3}+\frac {1}{\sqrt [3]{1-x} (1+x)}\right ) \, dx,x,x^3\right )\\ &=\frac {2}{5} \left (1-x^3\right )^{5/3}-\frac {1}{4} \left (1-x^3\right )^{8/3}+\frac {1}{11} \left (1-x^3\right )^{11/3}+\frac {1}{3} \text {Subst}\left (\int \frac {1}{\sqrt [3]{1-x} (1+x)} \, dx,x,x^3\right )\\ &=\frac {2}{5} \left (1-x^3\right )^{5/3}-\frac {1}{4} \left (1-x^3\right )^{8/3}+\frac {1}{11} \left (1-x^3\right )^{11/3}-\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 {2}{5} \left (1-x^3\right )^{5/3}-\frac {1}{4} \left (1-x^3\right )^{8/3}+\frac {1}{11} \left (1-x^3\right )^{11/3}-\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 {2}{5} \left (1-x^3\right )^{5/3}-\frac {1}{4} \left (1-x^3\right )^{8/3}+\frac {1}{11} \left (1-x^3\right )^{11/3}+\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 = 0.13, size = 145, normalized size = 1.14 \begin {gather*} \frac {1}{220} \left (1-x^3\right )^{2/3} \left (53-38 x^3+5 x^6-20 x^9\right )+\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 (-2+2^{2/3} \sqrt [3]{1-x^3}\right )}{3 \sqrt [3]{2}}-\frac {\log \left (2+2^{2/3} \sqrt [3]{1-x^3}+\sqrt [3]{2} \left (1-x^3\right )^{2/3}\right )}{6 \sqrt [3]{2}} \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 = 5.02, size = 509, normalized size = 4.01
method | result | size |
risch | \(\text {Expression too large to display}\) | \(509\) |
trager | \(\text {Expression too large to display}\) | \(787\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.66, size = 119, normalized size = 0.94 \begin {gather*} \frac {1}{11} \, {\left (-x^{3} + 1\right )}^{\frac {11}{3}} - \frac {1}{4} \, {\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 {2}{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 ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.55, size = 118, normalized size = 0.93 \begin {gather*} -\frac {1}{220} \, {\left (20 \, x^{9} - 5 \, x^{6} + 38 \, x^{3} - 53\right )} {\left (-x^{3} + 1\right )}^{\frac {2}{3}} + \frac {1}{6} \, \sqrt {6} 2^{\frac {1}{6}} \arctan \left (\frac {1}{6} \cdot 2^{\frac {1}{6}} {\left (\sqrt {6} 2^{\frac {1}{3}} + 2 \, \sqrt {6} {\left (-x^{3} + 1\right )}^{\frac {1}{3}}\right )}\right ) - \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 ) \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^{14}}{\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 = 0.88, size = 134, normalized size = 1.06 \begin {gather*} -\frac {1}{11} \, {\left (x^{3} - 1\right )}^{3} {\left (-x^{3} + 1\right )}^{\frac {2}{3}} - \frac {1}{4} \, {\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 {2}{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 ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 5.03, size = 133, normalized size = 1.05 \begin {gather*} \frac {2^{2/3}\,\ln \left ({\left (1-x^3\right )}^{1/3}-2^{1/3}\right )}{6}+\frac {2\,{\left (1-x^3\right )}^{5/3}}{5}-\frac {{\left (1-x^3\right )}^{8/3}}{4}+\frac {{\left (1-x^3\right )}^{11/3}}{11}+\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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