Optimal. Leaf size=74 \[ \frac {\tan ^{-1}\left (\frac {\sqrt {3} \left (1-\sqrt [3]{1-x^2}\right )}{x}\right )}{4 \sqrt {3}}-\frac {1}{12} \tanh ^{-1}\left (\frac {\left (1-\sqrt [3]{1-x^2}\right )^2}{3 x}\right )+\frac {1}{12} \tanh ^{-1}\left (\frac {x}{3}\right ) \]
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Rubi [A] time = 0.01, antiderivative size = 74, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.048, Rules used = {395} \begin {gather*} \frac {\tan ^{-1}\left (\frac {\sqrt {3} \left (1-\sqrt [3]{1-x^2}\right )}{x}\right )}{4 \sqrt {3}}-\frac {1}{12} \tanh ^{-1}\left (\frac {\left (1-\sqrt [3]{1-x^2}\right )^2}{3 x}\right )+\frac {1}{12} \tanh ^{-1}\left (\frac {x}{3}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 395
Rubi steps
\begin {align*} \int \frac {1}{\sqrt [3]{1-x^2} \left (9-x^2\right )} \, dx &=\frac {\tan ^{-1}\left (\frac {\sqrt {3} \left (1-\sqrt [3]{1-x^2}\right )}{x}\right )}{4 \sqrt {3}}+\frac {1}{12} \tanh ^{-1}\left (\frac {x}{3}\right )-\frac {1}{12} \tanh ^{-1}\left (\frac {\left (1-\sqrt [3]{1-x^2}\right )^2}{3 x}\right )\\ \end {align*}
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Mathematica [C] time = 0.05, size = 125, normalized size = 1.69 \begin {gather*} \frac {\sqrt [3]{\frac {x-1}{x-3}} \sqrt [3]{\frac {x+1}{x-3}} F_1\left (\frac {2}{3};\frac {1}{3},\frac {1}{3};\frac {5}{3};-\frac {4}{x-3},-\frac {2}{x-3}\right )-\sqrt [3]{\frac {x-1}{x+3}} \sqrt [3]{\frac {x+1}{x+3}} F_1\left (\frac {2}{3};\frac {1}{3},\frac {1}{3};\frac {5}{3};\frac {2}{x+3},\frac {4}{x+3}\right )}{4 \sqrt [3]{1-x^2}} \end {gather*}
Warning: Unable to verify antiderivative.
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IntegrateAlgebraic [F] time = 8.17, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\sqrt [3]{1-x^2} \left (9-x^2\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [B] time = 3.72, size = 269, normalized size = 3.64 \begin {gather*} -\frac {1}{36} \, \sqrt {3} \arctan \left (\frac {36 \, \sqrt {3} {\left (x^{4} - 32 \, x^{3} - 42 \, x^{2} + 9\right )} {\left (-x^{2} + 1\right )}^{\frac {2}{3}} + 12 \, \sqrt {3} {\left (x^{5} + 27 \, x^{4} - 210 \, x^{3} - 54 \, x^{2} + 81 \, x + 27\right )} {\left (-x^{2} + 1\right )}^{\frac {1}{3}} + \sqrt {3} {\left (x^{6} - 108 \, x^{5} - 567 \, x^{4} + 1080 \, x^{3} + 459 \, x^{2} - 972 \, x - 405\right )}}{3 \, {\left (x^{6} + 108 \, x^{5} - 1647 \, x^{4} - 1080 \, x^{3} + 891 \, x^{2} + 972 \, x + 243\right )}}\right ) - \frac {1}{72} \, \log \left (\frac {x^{3} + 33 \, x^{2} + 18 \, {\left (-x^{2} + 1\right )}^{\frac {2}{3}} {\left (x + 1\right )} - 6 \, {\left (x^{2} + 6 \, x - 3\right )} {\left (-x^{2} + 1\right )}^{\frac {1}{3}} - 9 \, x - 9}{x^{3} + 9 \, x^{2} + 27 \, x + 27}\right ) + \frac {1}{36} \, \log \left (-\frac {x^{3} - 33 \, x^{2} + 18 \, {\left (-x^{2} + 1\right )}^{\frac {2}{3}} {\left (x - 1\right )} + 6 \, {\left (x^{2} - 6 \, x - 3\right )} {\left (-x^{2} + 1\right )}^{\frac {1}{3}} - 9 \, x + 9}{x^{3} + 9 \, x^{2} + 27 \, x + 27}\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 {1}{{\left (x^{2} - 9\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 = 2.61, size = 539, normalized size = 7.28 \begin {gather*} -\RootOf \left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right ) \ln \left (\frac {-6 x^{2} \RootOf \left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right )+288 \left (-x^{2}+1\right )^{\frac {1}{3}} x \RootOf \left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right )^{2}-576 x \RootOf \left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right )^{2}+36 \left (-x^{2}+1\right )^{\frac {1}{3}} x \RootOf \left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right )-24 x \RootOf \left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right )+\left (-x^{2}+1\right )^{\frac {1}{3}} x -36 \left (-x^{2}+1\right )^{\frac {1}{3}} \RootOf \left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right )-18 \RootOf \left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right )+3 \left (-x^{2}+1\right )^{\frac {2}{3}}-3 \left (-x^{2}+1\right )^{\frac {1}{3}}}{\left (x -3\right ) \left (x +3\right )}\right )+\RootOf \left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right ) \ln \left (\frac {12 x^{2} \RootOf \left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right )+576 \left (-x^{2}+1\right )^{\frac {1}{3}} x \RootOf \left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right )^{2}-1152 x \RootOf \left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right )^{2}+x^{2}+24 \left (-x^{2}+1\right )^{\frac {1}{3}} x \RootOf \left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right )-144 x \RootOf \left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right )-4 x +72 \left (-x^{2}+1\right )^{\frac {1}{3}} \RootOf \left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right )+36 \RootOf \left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right )+6 \left (-x^{2}+1\right )^{\frac {2}{3}}+3}{\left (x -3\right ) \left (x +3\right )}\right )-\frac {\ln \left (\frac {-6 x^{2} \RootOf \left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right )+288 \left (-x^{2}+1\right )^{\frac {1}{3}} x \RootOf \left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right )^{2}-576 x \RootOf \left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right )^{2}+36 \left (-x^{2}+1\right )^{\frac {1}{3}} x \RootOf \left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right )-24 x \RootOf \left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right )+\left (-x^{2}+1\right )^{\frac {1}{3}} x -36 \left (-x^{2}+1\right )^{\frac {1}{3}} \RootOf \left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right )-18 \RootOf \left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right )+3 \left (-x^{2}+1\right )^{\frac {2}{3}}-3 \left (-x^{2}+1\right )^{\frac {1}{3}}}{\left (x -3\right ) \left (x +3\right )}\right )}{12} \end {gather*}
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 {1}{{\left (x^{2} - 9\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 {1}{{\left (1-x^2\right )}^{1/3}\,\left (x^2-9\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 {1}{x^{2} \sqrt [3]{1 - x^{2}} - 9 \sqrt [3]{1 - x^{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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