Optimal. Leaf size=168 \[ -\frac {\log \left (2 \sqrt [3]{1-x^2}+2^{2/3} x+2^{2/3}\right )}{2^{2/3}}+\frac {\log \left (-\sqrt [3]{2} x^2-2 \left (1-x^2\right )^{2/3}+\left (2^{2/3} x+2^{2/3}\right ) \sqrt [3]{1-x^2}-2 \sqrt [3]{2} x-\sqrt [3]{2}\right )}{2\ 2^{2/3}}-\frac {\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} \sqrt [3]{1-x^2}}{\sqrt [3]{1-x^2}-2^{2/3} x-2^{2/3}}\right )}{2^{2/3}} \]
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Rubi [A] time = 0.02, antiderivative size = 95, normalized size of antiderivative = 0.57, number of steps used = 1, number of rules used = 1, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used = {1008} \begin {gather*} \frac {\log \left (x^2+3\right )}{2\ 2^{2/3}}-\frac {3 \log \left ((x+1)^{2/3}+\sqrt [3]{2} \sqrt [3]{1-x}\right )}{2\ 2^{2/3}}+\frac {\sqrt {3} \tan ^{-1}\left (\frac {1}{\sqrt {3}}-\frac {2^{2/3} (x+1)^{2/3}}{\sqrt {3} \sqrt [3]{1-x}}\right )}{2^{2/3}} \end {gather*}
Antiderivative was successfully verified.
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Rule 1008
Rubi steps
\begin {align*} \int \frac {-3+x}{\sqrt [3]{1-x^2} \left (3+x^2\right )} \, dx &=\frac {\sqrt {3} \tan ^{-1}\left (\frac {1}{\sqrt {3}}-\frac {2^{2/3} (1+x)^{2/3}}{\sqrt {3} \sqrt [3]{1-x}}\right )}{2^{2/3}}+\frac {\log \left (3+x^2\right )}{2\ 2^{2/3}}-\frac {3 \log \left (\sqrt [3]{2} \sqrt [3]{1-x}+(1+x)^{2/3}\right )}{2\ 2^{2/3}}\\ \end {align*}
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Mathematica [C] time = 0.22, size = 143, normalized size = 0.85 \begin {gather*} \frac {1}{6} x^2 F_1\left (1;\frac {1}{3},1;2;x^2,-\frac {x^2}{3}\right )+\frac {27 x F_1\left (\frac {1}{2};\frac {1}{3},1;\frac {3}{2};x^2,-\frac {x^2}{3}\right )}{\sqrt [3]{1-x^2} \left (x^2+3\right ) \left (2 x^2 \left (F_1\left (\frac {3}{2};\frac {1}{3},2;\frac {5}{2};x^2,-\frac {x^2}{3}\right )-F_1\left (\frac {3}{2};\frac {4}{3},1;\frac {5}{2};x^2,-\frac {x^2}{3}\right )\right )-9 F_1\left (\frac {1}{2};\frac {1}{3},1;\frac {3}{2};x^2,-\frac {x^2}{3}\right )\right )} \end {gather*}
Warning: Unable to verify antiderivative.
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IntegrateAlgebraic [A] time = 0.25, size = 168, normalized size = 1.00 \begin {gather*} -\frac {\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} \sqrt [3]{1-x^2}}{-2^{2/3}-2^{2/3} x+\sqrt [3]{1-x^2}}\right )}{2^{2/3}}-\frac {\log \left (2^{2/3}+2^{2/3} x+2 \sqrt [3]{1-x^2}\right )}{2^{2/3}}+\frac {\log \left (-\sqrt [3]{2}-2 \sqrt [3]{2} x-\sqrt [3]{2} x^2+\left (2^{2/3}+2^{2/3} x\right ) \sqrt [3]{1-x^2}-2 \left (1-x^2\right )^{2/3}\right )}{2\ 2^{2/3}} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 6.50, size = 315, normalized size = 1.88 \begin {gather*} -\frac {1}{6} \cdot 4^{\frac {1}{6}} \sqrt {3} \left (-1\right )^{\frac {1}{3}} \arctan \left (\frac {4^{\frac {1}{6}} \sqrt {3} {\left (12 \cdot 4^{\frac {2}{3}} \left (-1\right )^{\frac {2}{3}} {\left (x^{4} + 3 \, x^{3} + 3 \, x^{2} + 9 \, x\right )} {\left (-x^{2} + 1\right )}^{\frac {2}{3}} - 12 \, \left (-1\right )^{\frac {1}{3}} {\left (x^{5} + 19 \, x^{4} + 42 \, x^{3} + 6 \, x^{2} - 27 \, x - 9\right )} {\left (-x^{2} + 1\right )}^{\frac {1}{3}} + 4^{\frac {1}{3}} {\left (x^{6} - 18 \, x^{5} - 117 \, x^{4} - 36 \, x^{3} + 207 \, x^{2} + 54 \, x - 27\right )}\right )}}{6 \, {\left (x^{6} + 54 \, x^{5} + 171 \, x^{4} + 108 \, x^{3} - 81 \, x^{2} - 162 \, x - 27\right )}}\right ) - \frac {1}{24} \cdot 4^{\frac {2}{3}} \left (-1\right )^{\frac {1}{3}} \log \left (-\frac {6 \cdot 4^{\frac {2}{3}} \left (-1\right )^{\frac {1}{3}} {\left (x^{2} + 3 \, x\right )} {\left (-x^{2} + 1\right )}^{\frac {2}{3}} - 4^{\frac {1}{3}} \left (-1\right )^{\frac {2}{3}} {\left (x^{4} + 18 \, x^{3} + 24 \, x^{2} - 18 \, x - 9\right )} + 6 \, {\left (x^{3} + 7 \, x^{2} + 3 \, x - 3\right )} {\left (-x^{2} + 1\right )}^{\frac {1}{3}}}{x^{4} + 6 \, x^{2} + 9}\right ) + \frac {1}{12} \cdot 4^{\frac {2}{3}} \left (-1\right )^{\frac {1}{3}} \log \left (\frac {6 \cdot 4^{\frac {1}{3}} \left (-1\right )^{\frac {2}{3}} {\left (-x^{2} + 1\right )}^{\frac {1}{3}} {\left (x + 1\right )} - 4^{\frac {2}{3}} \left (-1\right )^{\frac {1}{3}} {\left (x^{2} + 3\right )} + 12 \, {\left (-x^{2} + 1\right )}^{\frac {2}{3}}}{x^{2} + 3}\right ) \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*} \int \frac {x - 3}{{\left (x^{2} + 3\right )} {\left (-x^{2} + 1\right )}^{\frac {1}{3}}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 8.26, size = 1552, normalized size = 9.24
method | result | size |
trager | \(\text {Expression too large to display}\) | \(1552\) |
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*} \int \frac {x - 3}{{\left (x^{2} + 3\right )} {\left (-x^{2} + 1\right )}^{\frac {1}{3}}}\,{d x} \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 {x-3}{{\left (1-x^2\right )}^{1/3}\,\left (x^2+3\right )} \,d x \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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