Optimal. Leaf size=46 \[ \frac {2 \tanh ^{-1}\left (\frac {\sqrt {2 \sqrt {3}-3} (1-x)}{\sqrt {1-x^3}}\right )}{\sqrt {2 \sqrt {3}-3}} \]
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Rubi [A] time = 0.11, antiderivative size = 46, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 36, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used = {2140, 206} \[ \frac {2 \tanh ^{-1}\left (\frac {\sqrt {2 \sqrt {3}-3} (1-x)}{\sqrt {1-x^3}}\right )}{\sqrt {2 \sqrt {3}-3}} \]
Antiderivative was successfully verified.
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Rule 206
Rule 2140
Rubi steps
\begin {align*} \int \frac {1+\sqrt {3}-x}{\left (1-\sqrt {3}-x\right ) \sqrt {1-x^3}} \, dx &=2 \operatorname {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 \tanh ^{-1}\left (\frac {\sqrt {-3+2 \sqrt {3}} (1-x)}{\sqrt {1-x^3}}\right )}{\sqrt {-3+2 \sqrt {3}}}\\ \end {align*}
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Mathematica [C] time = 0.49, size = 269, normalized size = 5.85 \[ \frac {2 \sqrt {6} \sqrt {\frac {i (x-1)}{\sqrt {3}-3 i}} \left (4 \sqrt {-2 i x+\sqrt {3}-i} \sqrt {x^2+x+1} \Pi \left (\frac {2 \sqrt {3}}{-3 i+(1+2 i) \sqrt {3}};\sin ^{-1}\left (\frac {\sqrt {-2 i x+\sqrt {3}-i}}{\sqrt {2} \sqrt [4]{3}}\right )|\frac {2 \sqrt {3}}{-3 i+\sqrt {3}}\right )+\sqrt {2 i x+\sqrt {3}+i} \left (\left ((1+2 i)-i \sqrt {3}\right ) x-\sqrt {3}+(2+i)\right ) F\left (\sin ^{-1}\left (\frac {\sqrt {-2 i x+\sqrt {3}-i}}{\sqrt {2} \sqrt [4]{3}}\right )|\frac {2 \sqrt {3}}{-3 i+\sqrt {3}}\right )\right )}{\left ((1+2 i) \sqrt {3}-3 i\right ) \sqrt {-2 i x+\sqrt {3}-i} \sqrt {1-x^3}} \]
Warning: Unable to verify antiderivative.
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fricas [B] time = 0.51, size = 207, normalized size = 4.50 \[ \frac {1}{6} \, \sqrt {3} \sqrt {2 \, \sqrt {3} + 3} \log \left (\frac {x^{8} + 16 \, x^{7} + 112 \, x^{6} + 16 \, x^{5} + 112 \, x^{4} - 224 \, x^{3} + 64 \, x^{2} + 4 \, {\left (2 \, x^{6} + 18 \, x^{5} + 42 \, x^{4} + 8 \, x^{3} - \sqrt {3} {\left (x^{6} + 12 \, x^{5} + 18 \, x^{4} + 16 \, x^{3} - 12 \, x^{2} - 8\right )} - 24 \, x + 8\right )} \sqrt {-x^{3} + 1} \sqrt {2 \, \sqrt {3} + 3} - 16 \, \sqrt {3} {\left (x^{7} + 2 \, x^{6} + 6 \, x^{5} - 5 \, x^{4} + 2 \, x^{3} - 6 \, x^{2} + 4 \, x - 4\right )} - 128 \, x + 112}{x^{8} - 8 \, x^{7} + 16 \, x^{6} + 16 \, x^{5} - 56 \, x^{4} - 32 \, x^{3} + 64 \, x^{2} + 64 \, x + 16}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.08, size = 243, normalized size = 5.28 \[ -\frac {2 i \sqrt {3}\, \sqrt {i \left (x +\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, \sqrt {\frac {x -1}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \sqrt {-i \left (x +\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, \EllipticF \left (\frac {\sqrt {3}\, \sqrt {i \left (x +\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}}{3}, \sqrt {\frac {i \sqrt {3}}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\right )}{3 \sqrt {-x^{3}+1}}+\frac {4 i \sqrt {i \left (x +\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, \sqrt {\frac {x -1}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \sqrt {-i \left (x +\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, \EllipticPi \left (\frac {\sqrt {3}\, \sqrt {i \left (x +\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}}{3}, \frac {i \sqrt {3}}{-\frac {3}{2}+\sqrt {3}+\frac {i \sqrt {3}}{2}}, \sqrt {\frac {i \sqrt {3}}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\right )}{\sqrt {-x^{3}+1}\, \left (-\frac {3}{2}+\sqrt {3}+\frac {i \sqrt {3}}{2}\right )} \]
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 \[ \int \frac {x - \sqrt {3} - 1}{\sqrt {-x^{3} + 1} {\left (x + \sqrt {3} - 1\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F(-1)] time = 0.00, size = -1, normalized size = -0.02 \[ \text {Hanged} \]
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 \[ \int \frac {x - \sqrt {3} - 1}{\sqrt {- \left (x - 1\right ) \left (x^{2} + x + 1\right )} \left (x - 1 + \sqrt {3}\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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