Integrand size = 21, antiderivative size = 74 \[ \int \frac {1}{\sqrt [3]{1-x^2} \left (9-x^2\right )} \, dx=\frac {\arctan \left (\frac {\sqrt {3} \left (1-\sqrt [3]{1-x^2}\right )}{x}\right )}{4 \sqrt {3}}+\frac {1}{12} \text {arctanh}\left (\frac {x}{3}\right )-\frac {1}{12} \text {arctanh}\left (\frac {\left (1-\sqrt [3]{1-x^2}\right )^2}{3 x}\right ) \]
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Time = 0.01 (sec) , antiderivative size = 74, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.048, Rules used = {404} \[ \int \frac {1}{\sqrt [3]{1-x^2} \left (9-x^2\right )} \, dx=\frac {\arctan \left (\frac {\sqrt {3} \left (1-\sqrt [3]{1-x^2}\right )}{x}\right )}{4 \sqrt {3}}-\frac {1}{12} \text {arctanh}\left (\frac {\left (1-\sqrt [3]{1-x^2}\right )^2}{3 x}\right )+\frac {1}{12} \text {arctanh}\left (\frac {x}{3}\right ) \]
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Rule 404
Rubi steps \begin{align*} \text {integral}& = \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*}
Result contains higher order function than in optimal. Order 6 vs. order 3 in optimal.
Time = 8.55 (sec) , antiderivative size = 125, normalized size of antiderivative = 1.69 \[ \int \frac {1}{\sqrt [3]{1-x^2} \left (9-x^2\right )} \, dx=\frac {\sqrt [3]{\frac {-1+x}{-3+x}} \sqrt [3]{\frac {1+x}{-3+x}} \operatorname {AppellF1}\left (\frac {2}{3},\frac {1}{3},\frac {1}{3},\frac {5}{3},-\frac {4}{-3+x},-\frac {2}{-3+x}\right )-\sqrt [3]{\frac {-1+x}{3+x}} \sqrt [3]{\frac {1+x}{3+x}} \operatorname {AppellF1}\left (\frac {2}{3},\frac {1}{3},\frac {1}{3},\frac {5}{3},\frac {2}{3+x},\frac {4}{3+x}\right )}{4 \sqrt [3]{1-x^2}} \]
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Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 2.63 (sec) , antiderivative size = 539, normalized size of antiderivative = 7.28
method | result | size |
trager | \(-\frac {\ln \left (\frac {288 \left (-x^{2}+1\right )^{\frac {1}{3}} \operatorname {RootOf}\left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right )^{2} x +36 \operatorname {RootOf}\left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right ) \left (-x^{2}+1\right )^{\frac {1}{3}} x -576 \operatorname {RootOf}\left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right )^{2} x -6 \operatorname {RootOf}\left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right ) x^{2}+3 \left (-x^{2}+1\right )^{\frac {2}{3}}-36 \operatorname {RootOf}\left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right ) \left (-x^{2}+1\right )^{\frac {1}{3}}+x \left (-x^{2}+1\right )^{\frac {1}{3}}-24 \operatorname {RootOf}\left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right ) x -3 \left (-x^{2}+1\right )^{\frac {1}{3}}-18 \operatorname {RootOf}\left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right )}{\left (-3+x \right ) \left (3+x \right )}\right )}{12}-\ln \left (\frac {288 \left (-x^{2}+1\right )^{\frac {1}{3}} \operatorname {RootOf}\left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right )^{2} x +36 \operatorname {RootOf}\left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right ) \left (-x^{2}+1\right )^{\frac {1}{3}} x -576 \operatorname {RootOf}\left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right )^{2} x -6 \operatorname {RootOf}\left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right ) x^{2}+3 \left (-x^{2}+1\right )^{\frac {2}{3}}-36 \operatorname {RootOf}\left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right ) \left (-x^{2}+1\right )^{\frac {1}{3}}+x \left (-x^{2}+1\right )^{\frac {1}{3}}-24 \operatorname {RootOf}\left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right ) x -3 \left (-x^{2}+1\right )^{\frac {1}{3}}-18 \operatorname {RootOf}\left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right )}{\left (-3+x \right ) \left (3+x \right )}\right ) \operatorname {RootOf}\left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right )+\operatorname {RootOf}\left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right ) \ln \left (\frac {576 \left (-x^{2}+1\right )^{\frac {1}{3}} \operatorname {RootOf}\left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right )^{2} x +24 \operatorname {RootOf}\left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right ) \left (-x^{2}+1\right )^{\frac {1}{3}} x -1152 \operatorname {RootOf}\left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right )^{2} x +12 \operatorname {RootOf}\left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right ) x^{2}+6 \left (-x^{2}+1\right )^{\frac {2}{3}}+72 \operatorname {RootOf}\left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right ) \left (-x^{2}+1\right )^{\frac {1}{3}}-144 \operatorname {RootOf}\left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right ) x +x^{2}+36 \operatorname {RootOf}\left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right )-4 x +3}{\left (-3+x \right ) \left (3+x \right )}\right )\) | \(539\) |
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Leaf count of result is larger than twice the leaf count of optimal. 269 vs. \(2 (53) = 106\).
Time = 0.54 (sec) , antiderivative size = 269, normalized size of antiderivative = 3.64 \[ \int \frac {1}{\sqrt [3]{1-x^2} \left (9-x^2\right )} \, dx=-\frac {1}{36} \, \sqrt {3} \arctan \left (\frac {36 \, \sqrt {3} {\left (x^{4} - 32 \, x^{3} - 42 \, x^{2} + 9\right )} {\left (-x^{2} + 1\right )}^{\frac {2}{3}} + 12 \, \sqrt {3} {\left (x^{5} + 27 \, x^{4} - 210 \, x^{3} - 54 \, x^{2} + 81 \, x + 27\right )} {\left (-x^{2} + 1\right )}^{\frac {1}{3}} + \sqrt {3} {\left (x^{6} - 108 \, x^{5} - 567 \, x^{4} + 1080 \, x^{3} + 459 \, x^{2} - 972 \, x - 405\right )}}{3 \, {\left (x^{6} + 108 \, x^{5} - 1647 \, x^{4} - 1080 \, x^{3} + 891 \, x^{2} + 972 \, x + 243\right )}}\right ) - \frac {1}{72} \, \log \left (\frac {x^{3} + 33 \, x^{2} + 18 \, {\left (-x^{2} + 1\right )}^{\frac {2}{3}} {\left (x + 1\right )} - 6 \, {\left (x^{2} + 6 \, x - 3\right )} {\left (-x^{2} + 1\right )}^{\frac {1}{3}} - 9 \, x - 9}{x^{3} + 9 \, x^{2} + 27 \, x + 27}\right ) + \frac {1}{36} \, \log \left (-\frac {x^{3} - 33 \, x^{2} + 18 \, {\left (-x^{2} + 1\right )}^{\frac {2}{3}} {\left (x - 1\right )} + 6 \, {\left (x^{2} - 6 \, x - 3\right )} {\left (-x^{2} + 1\right )}^{\frac {1}{3}} - 9 \, x + 9}{x^{3} + 9 \, x^{2} + 27 \, x + 27}\right ) \]
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\[ \int \frac {1}{\sqrt [3]{1-x^2} \left (9-x^2\right )} \, dx=- \int \frac {1}{x^{2} \sqrt [3]{1 - x^{2}} - 9 \sqrt [3]{1 - x^{2}}}\, dx \]
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\[ \int \frac {1}{\sqrt [3]{1-x^2} \left (9-x^2\right )} \, dx=\int { -\frac {1}{{\left (x^{2} - 9\right )} {\left (-x^{2} + 1\right )}^{\frac {1}{3}}} \,d x } \]
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\[ \int \frac {1}{\sqrt [3]{1-x^2} \left (9-x^2\right )} \, dx=\int { -\frac {1}{{\left (x^{2} - 9\right )} {\left (-x^{2} + 1\right )}^{\frac {1}{3}}} \,d x } \]
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Timed out. \[ \int \frac {1}{\sqrt [3]{1-x^2} \left (9-x^2\right )} \, dx=-\int \frac {1}{{\left (1-x^2\right )}^{1/3}\,\left (x^2-9\right )} \,d x \]
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