Integrand size = 19, antiderivative size = 78 \[ \int \frac {1}{(3-x) \sqrt [3]{1-x^2}} \, dx=-\frac {1}{4} \sqrt {3} \arctan \left (\frac {1}{\sqrt {3}}-\frac {(1+x)^{2/3}}{\sqrt {3} \sqrt [3]{1-x}}\right )-\frac {1}{4} \log (3-x)+\frac {3}{8} \log \left (-\sqrt [3]{1-x}-\frac {1}{2} (1+x)^{2/3}\right ) \]
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Time = 0.01 (sec) , antiderivative size = 78, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {767, 124} \[ \int \frac {1}{(3-x) \sqrt [3]{1-x^2}} \, dx=-\frac {1}{4} \sqrt {3} \arctan \left (\frac {1}{\sqrt {3}}-\frac {(x+1)^{2/3}}{\sqrt {3} \sqrt [3]{1-x}}\right )-\frac {1}{4} \log (3-x)+\frac {3}{8} \log \left (-\frac {1}{2} (x+1)^{2/3}-\sqrt [3]{1-x}\right ) \]
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Rule 124
Rule 767
Rubi steps \begin{align*} \text {integral}& = \int \frac {1}{\sqrt [3]{1-x} (3-x) \sqrt [3]{1+x}} \, dx \\ & = -\frac {1}{4} \sqrt {3} \tan ^{-1}\left (\frac {1}{\sqrt {3}}-\frac {(1+x)^{2/3}}{\sqrt {3} \sqrt [3]{1-x}}\right )-\frac {1}{4} \log (3-x)+\frac {3}{8} \log \left (-\sqrt [3]{1-x}-\frac {1}{2} (1+x)^{2/3}\right ) \\ \end{align*}
Time = 0.19 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.37 \[ \int \frac {1}{(3-x) \sqrt [3]{1-x^2}} \, dx=\frac {1}{8} \left (-2 \sqrt {3} \arctan \left (\frac {\sqrt {3} \sqrt [3]{1-x^2}}{1+x-\sqrt [3]{1-x^2}}\right )+2 \log \left (1+x+2 \sqrt [3]{1-x^2}\right )-\log \left (1+2 x+x^2-2 (1+x) \sqrt [3]{1-x^2}+4 \left (1-x^2\right )^{2/3}\right )\right ) \]
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Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 2.19 (sec) , antiderivative size = 605, normalized size of antiderivative = 7.76
method | result | size |
trager | \(\frac {\ln \left (-\frac {448 \operatorname {RootOf}\left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right )^{2} x^{2}+864 \operatorname {RootOf}\left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right ) \left (-x^{2}+1\right )^{\frac {2}{3}}-516 \operatorname {RootOf}\left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right ) \left (-x^{2}+1\right )^{\frac {1}{3}} x +1344 \operatorname {RootOf}\left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right )^{2} x -20 \operatorname {RootOf}\left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right ) x^{2}-516 \left (-x^{2}+1\right )^{\frac {2}{3}}-516 \operatorname {RootOf}\left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right ) \left (-x^{2}+1\right )^{\frac {1}{3}}-474 x \left (-x^{2}+1\right )^{\frac {1}{3}}-1656 \operatorname {RootOf}\left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right ) x -3 x^{2}-474 \left (-x^{2}+1\right )^{\frac {1}{3}}+1596 \operatorname {RootOf}\left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right )+162 x -171}{\left (-3+x \right )^{2}}\right )}{4}-\frac {\ln \left (\frac {544 \operatorname {RootOf}\left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right )^{2} x^{2}+864 \operatorname {RootOf}\left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right ) \left (-x^{2}+1\right )^{\frac {2}{3}}+948 \operatorname {RootOf}\left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right ) \left (-x^{2}+1\right )^{\frac {1}{3}} x +1632 \operatorname {RootOf}\left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right )^{2} x +286 \operatorname {RootOf}\left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right ) x^{2}+948 \left (-x^{2}+1\right )^{\frac {2}{3}}+948 \operatorname {RootOf}\left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right ) \left (-x^{2}+1\right )^{\frac {1}{3}}+258 x \left (-x^{2}+1\right )^{\frac {1}{3}}+2796 \operatorname {RootOf}\left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right ) x +21 x^{2}+258 \left (-x^{2}+1\right )^{\frac {1}{3}}-1938 \operatorname {RootOf}\left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right )+234 x -171}{\left (-3+x \right )^{2}}\right )}{4}-\frac {\ln \left (\frac {544 \operatorname {RootOf}\left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right )^{2} x^{2}+864 \operatorname {RootOf}\left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right ) \left (-x^{2}+1\right )^{\frac {2}{3}}+948 \operatorname {RootOf}\left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right ) \left (-x^{2}+1\right )^{\frac {1}{3}} x +1632 \operatorname {RootOf}\left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right )^{2} x +286 \operatorname {RootOf}\left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right ) x^{2}+948 \left (-x^{2}+1\right )^{\frac {2}{3}}+948 \operatorname {RootOf}\left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right ) \left (-x^{2}+1\right )^{\frac {1}{3}}+258 x \left (-x^{2}+1\right )^{\frac {1}{3}}+2796 \operatorname {RootOf}\left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right ) x +21 x^{2}+258 \left (-x^{2}+1\right )^{\frac {1}{3}}-1938 \operatorname {RootOf}\left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right )+234 x -171}{\left (-3+x \right )^{2}}\right ) \operatorname {RootOf}\left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right )}{2}\) | \(605\) |
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Time = 0.45 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.45 \[ \int \frac {1}{(3-x) \sqrt [3]{1-x^2}} \, dx=-\frac {1}{4} \, \sqrt {3} \arctan \left (\frac {18031 \, \sqrt {3} {\left (-x^{2} + 1\right )}^{\frac {1}{3}} {\left (x + 1\right )} + \sqrt {3} {\left (5054 \, x^{2} - 8497 \, x + 23659\right )} + 57889 \, \sqrt {3} {\left (-x^{2} + 1\right )}^{\frac {2}{3}}}{6859 \, x^{2} + 240699 \, x - 220122}\right ) + \frac {1}{8} \, \log \left (\frac {x^{2} + 6 \, {\left (-x^{2} + 1\right )}^{\frac {1}{3}} {\left (x + 1\right )} - 6 \, x + 12 \, {\left (-x^{2} + 1\right )}^{\frac {2}{3}} + 9}{x^{2} - 6 \, x + 9}\right ) \]
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\[ \int \frac {1}{(3-x) \sqrt [3]{1-x^2}} \, dx=- \int \frac {1}{x \sqrt [3]{1 - x^{2}} - 3 \sqrt [3]{1 - x^{2}}}\, dx \]
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\[ \int \frac {1}{(3-x) \sqrt [3]{1-x^2}} \, dx=\int { -\frac {1}{{\left (-x^{2} + 1\right )}^{\frac {1}{3}} {\left (x - 3\right )}} \,d x } \]
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\[ \int \frac {1}{(3-x) \sqrt [3]{1-x^2}} \, dx=\int { -\frac {1}{{\left (-x^{2} + 1\right )}^{\frac {1}{3}} {\left (x - 3\right )}} \,d x } \]
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Timed out. \[ \int \frac {1}{(3-x) \sqrt [3]{1-x^2}} \, dx=-\int \frac {1}{{\left (1-x^2\right )}^{1/3}\,\left (x-3\right )} \,d x \]
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