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 = 0.114319, antiderivative size = 46, normalized size of antiderivative = 1., 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 2140
Rule 206
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*}
Mathematica [C] time = 0.455526, 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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Maple [C] time = 0.049, size = 243, normalized size = 5.3 \begin{align*}{-{\frac{2\,i}{3}}\sqrt{3}\sqrt{i \left ( x+{\frac{1}{2}}-{\frac{i}{2}}\sqrt{3} \right ) \sqrt{3}}\sqrt{{\frac{x-1}{-{\frac{3}{2}}+{\frac{i}{2}}\sqrt{3}}}}\sqrt{-i \left ( x+{\frac{1}{2}}+{\frac{i}{2}}\sqrt{3} \right ) \sqrt{3}}{\it EllipticF} \left ({\frac{\sqrt{3}}{3}\sqrt{i \left ( x+{\frac{1}{2}}-{\frac{i}{2}}\sqrt{3} \right ) \sqrt{3}}},\sqrt{{\frac{i\sqrt{3}}{-{\frac{3}{2}}+{\frac{i}{2}}\sqrt{3}}}} \right ){\frac{1}{\sqrt{-{x}^{3}+1}}}}+{\frac{4\,i}{-{\frac{3}{2}}+{\frac{i}{2}}\sqrt{3}+\sqrt{3}}\sqrt{i \left ( x+{\frac{1}{2}}-{\frac{i}{2}}\sqrt{3} \right ) \sqrt{3}}\sqrt{{\frac{x-1}{-{\frac{3}{2}}+{\frac{i}{2}}\sqrt{3}}}}\sqrt{-i \left ( x+{\frac{1}{2}}+{\frac{i}{2}}\sqrt{3} \right ) \sqrt{3}}{\it EllipticPi} \left ({\frac{\sqrt{3}}{3}\sqrt{i \left ( x+{\frac{1}{2}}-{\frac{i}{2}}\sqrt{3} \right ) \sqrt{3}}},{\frac{i\sqrt{3}}{-{\frac{3}{2}}+{\frac{i}{2}}\sqrt{3}+\sqrt{3}}},\sqrt{{\frac{i\sqrt{3}}{-{\frac{3}{2}}+{\frac{i}{2}}\sqrt{3}}}} \right ){\frac{1}{\sqrt{-{x}^{3}+1}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x - \sqrt{3} - 1}{\sqrt{-x^{3} + 1}{\left (x + \sqrt{3} - 1\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.50751, size = 533, normalized size = 11.59 \begin{align*} \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 ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x - \sqrt{3} - 1}{\sqrt{- \left (x - 1\right ) \left (x^{2} + x + 1\right )} \left (x - 1 + \sqrt{3}\right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x - \sqrt{3} - 1}{\sqrt{-x^{3} + 1}{\left (x + \sqrt{3} - 1\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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