Optimal. Leaf size=44 \[ \frac{2 \tanh ^{-1}\left (\frac{\sqrt{3+2 \sqrt{3}} (1-x)}{\sqrt{x^3-1}}\right )}{\sqrt{3+2 \sqrt{3}}} \]
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Rubi [A] time = 0.0898745, antiderivative size = 44, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 34, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.059, Rules used = {2140, 206} \[ \frac{2 \tanh ^{-1}\left (\frac{\sqrt{3+2 \sqrt{3}} (1-x)}{\sqrt{x^3-1}}\right )}{\sqrt{3+2 \sqrt{3}}} \]
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 = 0.289503, size = 265, normalized size = 6.02 \[ \frac{2 \sqrt{6} \sqrt{\frac{i (x-1)}{\sqrt{3}-3 i}} \left (\sqrt{2 i x+\sqrt{3}+i} \left (\left (\sqrt{3}+(2+i)\right ) x+i \sqrt{3}+(1+2 i)\right ) F\left (\sin ^{-1}\left (\frac{\sqrt{-2 i x+\sqrt{3}-i}}{\sqrt{2} \sqrt [4]{3}}\right )|\frac{2 \sqrt{3}}{-3 i+\sqrt{3}}\right )-4 i \sqrt{-2 i x+\sqrt{3}-i} \sqrt{x^2+x+1} \Pi \left (\frac{2 i \sqrt{3}}{3+(2+i) \sqrt{3}};\sin ^{-1}\left (\frac{\sqrt{-2 i x+\sqrt{3}-i}}{\sqrt{2} \sqrt [4]{3}}\right )|\frac{2 \sqrt{3}}{-3 i+\sqrt{3}}\right )\right )}{\left (3+(2+i) \sqrt{3}\right ) \sqrt{-2 i x+\sqrt{3}-i} \sqrt{x^3-1}} \]
Warning: Unable to verify antiderivative.
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Maple [C] time = 0.02, size = 245, normalized size = 5.6 \begin{align*} 2\,{\frac{-3/2-i/2\sqrt{3}}{\sqrt{{x}^{3}-1}}\sqrt{{\frac{x-1}{-3/2-i/2\sqrt{3}}}}\sqrt{{\frac{x+1/2-i/2\sqrt{3}}{3/2-i/2\sqrt{3}}}}\sqrt{{\frac{x+1/2+i/2\sqrt{3}}{3/2+i/2\sqrt{3}}}}{\it EllipticF} \left ( \sqrt{{\frac{x-1}{-3/2-i/2\sqrt{3}}}},\sqrt{{\frac{3/2+i/2\sqrt{3}}{3/2-i/2\sqrt{3}}}} \right ) }-4\,{\frac{-3/2-i/2\sqrt{3}}{\sqrt{{x}^{3}-1}}\sqrt{{\frac{x-1}{-3/2-i/2\sqrt{3}}}}\sqrt{{\frac{x+1/2-i/2\sqrt{3}}{3/2-i/2\sqrt{3}}}}\sqrt{{\frac{x+1/2+i/2\sqrt{3}}{3/2+i/2\sqrt{3}}}}{\it EllipticPi} \left ( \sqrt{{\frac{x-1}{-3/2-i/2\sqrt{3}}}},-1/3\, \left ( 3/2+i/2\sqrt{3} \right ) \sqrt{3},\sqrt{{\frac{3/2+i/2\sqrt{3}}{3/2-i/2\sqrt{3}}}} \right ) } \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 = 1.88947, size = 532, normalized size = 12.09 \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 - 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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