Optimal. Leaf size=46 \[ \frac{2 \tan ^{-1}\left (\frac{\sqrt{3+2 \sqrt{3}} (1-x)}{\sqrt{1-x^3}}\right )}{\sqrt{3+2 \sqrt{3}}} \]
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Rubi [A] time = 0.101986, 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, 203} \[ \frac{2 \tan ^{-1}\left (\frac{\sqrt{3+2 \sqrt{3}} (1-x)}{\sqrt{1-x^3}}\right )}{\sqrt{3+2 \sqrt{3}}} \]
Antiderivative was successfully verified.
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Rule 2140
Rule 203
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 \tan ^{-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 = 0.434934, size = 267, normalized size = 5.8 \[ \frac{2 \sqrt{6} \sqrt{\frac{i (x-1)}{\sqrt{3}-3 i}} \left (\sqrt{2 i x+\sqrt{3}+i} \left (\left (\sqrt{3}+(2+i)\right ) x+i \sqrt{3}+(1+2 i)\right ) F\left (\sin ^{-1}\left (\frac{\sqrt{-2 i x+\sqrt{3}-i}}{\sqrt{2} \sqrt [4]{3}}\right )|\frac{2 \sqrt{3}}{-3 i+\sqrt{3}}\right )-4 i \sqrt{-2 i x+\sqrt{3}-i} \sqrt{x^2+x+1} \Pi \left (\frac{2 i \sqrt{3}}{3+(2+i) \sqrt{3}};\sin ^{-1}\left (\frac{\sqrt{-2 i x+\sqrt{3}-i}}{\sqrt{2} \sqrt [4]{3}}\right )|\frac{2 \sqrt{3}}{-3 i+\sqrt{3}}\right )\right )}{\left (3+(2+i) \sqrt{3}\right ) \sqrt{-2 i x+\sqrt{3}-i} \sqrt{1-x^3}} \]
Warning: Unable to verify antiderivative.
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Maple [C] time = 0.025, size = 247, normalized size = 5.4 \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 [A] time = 1.67513, size = 170, normalized size = 3.7 \begin{align*} \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 ) \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 - 1 + \sqrt{3}}{\sqrt{- \left (x - 1\right ) \left (x^{2} + x + 1\right )} \left (x - \sqrt{3} - 1\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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