Integrand size = 36, antiderivative size = 46 \[ \int \frac {1-\sqrt {3}-x}{\left (1+\sqrt {3}-x\right ) \sqrt {1-x^3}} \, dx=\frac {2 \arctan \left (\frac {\sqrt {3+2 \sqrt {3}} (1-x)}{\sqrt {1-x^3}}\right )}{\sqrt {3+2 \sqrt {3}}} \]
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Time = 0.07 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used = {2165, 209} \[ \int \frac {1-\sqrt {3}-x}{\left (1+\sqrt {3}-x\right ) \sqrt {1-x^3}} \, dx=\frac {2 \arctan \left (\frac {\sqrt {3+2 \sqrt {3}} (1-x)}{\sqrt {1-x^3}}\right )}{\sqrt {3+2 \sqrt {3}}} \]
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Rule 209
Rule 2165
Rubi steps \begin{align*} \text {integral}& = 2 \text {Subst}\left (\int \frac {1}{1+\left (3+2 \sqrt {3}\right ) x^2} \, dx,x,\frac {1-x}{\sqrt {1-x^3}}\right ) \\ & = \frac {2 \tan ^{-1}\left (\frac {\sqrt {3+2 \sqrt {3}} (1-x)}{\sqrt {1-x^3}}\right )}{\sqrt {3+2 \sqrt {3}}} \\ \end{align*}
Time = 1.79 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.07 \[ \int \frac {1-\sqrt {3}-x}{\left (1+\sqrt {3}-x\right ) \sqrt {1-x^3}} \, dx=2 \sqrt {-1+\frac {2}{\sqrt {3}}} \arctan \left (\frac {\sqrt {3+2 \sqrt {3}} \sqrt {1-x^3}}{1+x+x^2}\right ) \]
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Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 2.47 (sec) , antiderivative size = 135, normalized size of antiderivative = 2.93
method | result | size |
trager | \(-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}-36+24 \sqrt {3}\right ) \ln \left (\frac {6 \operatorname {RootOf}\left (\textit {\_Z}^{2}-36+24 \sqrt {3}\right ) x^{2}-4 \operatorname {RootOf}\left (\textit {\_Z}^{2}-36+24 \sqrt {3}\right ) \sqrt {3}\, x^{2}-4 \sqrt {3}\, \operatorname {RootOf}\left (\textit {\_Z}^{2}-36+24 \sqrt {3}\right ) x +48 \sqrt {-x^{3}+1}\, \sqrt {3}-4 \operatorname {RootOf}\left (\textit {\_Z}^{2}-36+24 \sqrt {3}\right ) \sqrt {3}-72 \sqrt {-x^{3}+1}+12 \operatorname {RootOf}\left (\textit {\_Z}^{2}-36+24 \sqrt {3}\right )}{\left (x \sqrt {3}-x -2\right )^{2}}\right )}{6}\) | \(135\) |
default | \(-\frac {2 i \sqrt {3}\, \sqrt {i \left (x +\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, \sqrt {\frac {x -1}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \sqrt {-i \left (x +\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, F\left (\frac {\sqrt {3}\, \sqrt {i \left (x +\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}}{3}, \sqrt {\frac {i \sqrt {3}}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\right )}{3 \sqrt {-x^{3}+1}}-\frac {4 i \sqrt {i \left (x +\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, \sqrt {\frac {x -1}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \sqrt {-i \left (x +\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, \Pi \left (\frac {\sqrt {3}\, \sqrt {i \left (x +\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}}{3}, \frac {i \sqrt {3}}{-\frac {3}{2}-\sqrt {3}+\frac {i \sqrt {3}}{2}}, \sqrt {\frac {i \sqrt {3}}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\right )}{\sqrt {-x^{3}+1}\, \left (-\frac {3}{2}-\sqrt {3}+\frac {i \sqrt {3}}{2}\right )}\) | \(247\) |
elliptic | \(-\frac {2 i \sqrt {3}\, \sqrt {i \left (x +\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, \sqrt {\frac {x -1}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \sqrt {-i \left (x +\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, F\left (\frac {\sqrt {3}\, \sqrt {i \left (x +\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}}{3}, \sqrt {\frac {i \sqrt {3}}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\right )}{3 \sqrt {-x^{3}+1}}-\frac {4 i \sqrt {i \left (x +\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, \sqrt {\frac {x -1}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \sqrt {-i \left (x +\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, \Pi \left (\frac {\sqrt {3}\, \sqrt {i \left (x +\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}}{3}, \frac {i \sqrt {3}}{-\frac {3}{2}-\sqrt {3}+\frac {i \sqrt {3}}{2}}, \sqrt {\frac {i \sqrt {3}}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\right )}{\sqrt {-x^{3}+1}\, \left (-\frac {3}{2}-\sqrt {3}+\frac {i \sqrt {3}}{2}\right )}\) | \(247\) |
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none
Time = 0.30 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.28 \[ \int \frac {1-\sqrt {3}-x}{\left (1+\sqrt {3}-x\right ) \sqrt {1-x^3}} \, dx=\frac {1}{3} \, \sqrt {3} \sqrt {2 \, \sqrt {3} - 3} \arctan \left (\frac {\sqrt {-x^{3} + 1} {\left (\sqrt {3} {\left (x^{2} + 4 \, x - 2\right )} + 6 \, x - 6\right )} \sqrt {2 \, \sqrt {3} - 3}}{6 \, {\left (x^{3} - 1\right )}}\right ) \]
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\[ \int \frac {1-\sqrt {3}-x}{\left (1+\sqrt {3}-x\right ) \sqrt {1-x^3}} \, dx=\int \frac {x - 1 + \sqrt {3}}{\sqrt {- \left (x - 1\right ) \left (x^{2} + x + 1\right )} \left (x - \sqrt {3} - 1\right )}\, dx \]
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\[ \int \frac {1-\sqrt {3}-x}{\left (1+\sqrt {3}-x\right ) \sqrt {1-x^3}} \, dx=\int { \frac {x + \sqrt {3} - 1}{\sqrt {-x^{3} + 1} {\left (x - \sqrt {3} - 1\right )}} \,d x } \]
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Exception generated. \[ \int \frac {1-\sqrt {3}-x}{\left (1+\sqrt {3}-x\right ) \sqrt {1-x^3}} \, dx=\text {Exception raised: TypeError} \]
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Timed out. \[ \int \frac {1-\sqrt {3}-x}{\left (1+\sqrt {3}-x\right ) \sqrt {1-x^3}} \, dx=\text {Hanged} \]
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