Optimal. Leaf size=124 \[ \frac {1}{2 x^2 \sqrt [3]{1-x^3}}-\frac {\left (1-x^3\right )^{2/3}}{x^2}+\frac {\tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{2} x}{\sqrt [3]{1-x^3}}}{\sqrt {3}}\right )}{2 \sqrt [3]{2} \sqrt {3}}+\frac {\log \left (1+x^3\right )}{12 \sqrt [3]{2}}-\frac {\log \left (-\sqrt [3]{2} x-\sqrt [3]{1-x^3}\right )}{4 \sqrt [3]{2}} \]
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Rubi [A]
time = 0.05, antiderivative size = 124, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {483, 597, 12,
384} \begin {gather*} \frac {\text {ArcTan}\left (\frac {1-\frac {2 \sqrt [3]{2} x}{\sqrt [3]{1-x^3}}}{\sqrt {3}}\right )}{2 \sqrt [3]{2} \sqrt {3}}+\frac {\log \left (x^3+1\right )}{12 \sqrt [3]{2}}-\frac {\log \left (-\sqrt [3]{1-x^3}-\sqrt [3]{2} x\right )}{4 \sqrt [3]{2}}-\frac {\left (1-x^3\right )^{2/3}}{x^2}+\frac {1}{2 x^2 \sqrt [3]{1-x^3}} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 384
Rule 483
Rule 597
Rubi steps
\begin {align*} \int \frac {1}{x^3 \left (1-x^3\right )^{4/3} \left (1+x^3\right )} \, dx &=-\frac {14+56 x^3-91 x^6-42 x^9+63 x^{12}-7 \left (1-x^3\right )^2 \left (2+12 x^3+9 x^6\right ) \, _2F_1\left (\frac {1}{3},1;\frac {4}{3};-\frac {2 x^3}{1-x^3}\right )-30 x^6 \, _2F_1\left (2,\frac {7}{3};\frac {10}{3};-\frac {2 x^3}{1-x^3}\right )-84 x^9 \, _2F_1\left (2,\frac {7}{3};\frac {10}{3};-\frac {2 x^3}{1-x^3}\right )-54 x^{12} \, _2F_1\left (2,\frac {7}{3};\frac {10}{3};-\frac {2 x^3}{1-x^3}\right )-18 x^6 \left (1+x^3\right )^2 \, _3F_2\left (2,2,\frac {7}{3};1,\frac {10}{3};-\frac {2 x^3}{1-x^3}\right )}{14 x^5 \left (1-x^3\right )^{7/3}}\\ \end {align*}
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Mathematica [A]
time = 0.33, size = 148, normalized size = 1.19 \begin {gather*} \frac {1}{24} \left (\frac {12 \left (-1+2 x^3\right )}{x^2 \sqrt [3]{1-x^3}}+2\ 2^{2/3} \sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} x}{x-2^{2/3} \sqrt [3]{1-x^3}}\right )-2\ 2^{2/3} \log \left (2 x+2^{2/3} \sqrt [3]{1-x^3}\right )+2^{2/3} \log \left (-2 x^2+2^{2/3} x \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 = 15.23, size = 777, normalized size = 6.27
method | result | size |
risch | \(\text {Expression too large to display}\) | \(777\) |
trager | \(\text {Expression too large to display}\) | \(1170\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 340 vs.
\(2 (95) = 190\).
time = 9.24, size = 340, normalized size = 2.74 \begin {gather*} -\frac {2 \, \sqrt {6} 2^{\frac {1}{6}} \left (-1\right )^{\frac {1}{3}} {\left (x^{5} - x^{2}\right )} \arctan \left (\frac {2^{\frac {1}{6}} {\left (6 \, \sqrt {6} 2^{\frac {2}{3}} \left (-1\right )^{\frac {2}{3}} {\left (5 \, x^{7} + 4 \, x^{4} - x\right )} {\left (-x^{3} + 1\right )}^{\frac {2}{3}} - 12 \, \sqrt {6} \left (-1\right )^{\frac {1}{3}} {\left (19 \, x^{8} - 16 \, x^{5} + x^{2}\right )} {\left (-x^{3} + 1\right )}^{\frac {1}{3}} - \sqrt {6} 2^{\frac {1}{3}} {\left (71 \, x^{9} - 111 \, x^{6} + 33 \, x^{3} - 1\right )}\right )}}{6 \, {\left (109 \, x^{9} - 105 \, x^{6} + 3 \, x^{3} + 1\right )}}\right ) - 2 \cdot 2^{\frac {2}{3}} \left (-1\right )^{\frac {1}{3}} {\left (x^{5} - x^{2}\right )} \log \left (\frac {6 \cdot 2^{\frac {1}{3}} \left (-1\right )^{\frac {2}{3}} {\left (-x^{3} + 1\right )}^{\frac {1}{3}} x^{2} - 2^{\frac {2}{3}} \left (-1\right )^{\frac {1}{3}} {\left (x^{3} + 1\right )} + 6 \, {\left (-x^{3} + 1\right )}^{\frac {2}{3}} x}{x^{3} + 1}\right ) + 2^{\frac {2}{3}} \left (-1\right )^{\frac {1}{3}} {\left (x^{5} - x^{2}\right )} \log \left (-\frac {3 \cdot 2^{\frac {2}{3}} \left (-1\right )^{\frac {1}{3}} {\left (5 \, x^{4} - x\right )} {\left (-x^{3} + 1\right )}^{\frac {2}{3}} - 2^{\frac {1}{3}} \left (-1\right )^{\frac {2}{3}} {\left (19 \, x^{6} - 16 \, x^{3} + 1\right )} + 12 \, {\left (2 \, x^{5} - x^{2}\right )} {\left (-x^{3} + 1\right )}^{\frac {1}{3}}}{x^{6} + 2 \, x^{3} + 1}\right ) + 36 \, {\left (2 \, x^{3} - 1\right )} {\left (-x^{3} + 1\right )}^{\frac {2}{3}}}{72 \, {\left (x^{5} - x^{2}\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 {1}{x^{3} \left (- \left (x - 1\right ) \left (x^{2} + x + 1\right )\right )^{\frac {4}{3}} \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 [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {1}{x^3\,{\left (1-x^3\right )}^{4/3}\,\left (x^3+1\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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