Optimal. Leaf size=164 \[ \frac{2 \sqrt{\frac{7}{6}-\frac{2}{\sqrt{3}}} (1-x) \sqrt{\frac{x^2+x+1}{\left (-x-\sqrt{3}+1\right )^2}} F\left (\sin ^{-1}\left (\frac{-x+\sqrt{3}+1}{-x-\sqrt{3}+1}\right )|-7+4 \sqrt{3}\right )}{\sqrt [4]{3} \sqrt{-\frac{1-x}{\left (-x-\sqrt{3}+1\right )^2}} \sqrt{x^3-1}}-\frac{\sqrt{2} \tanh ^{-1}\left (\frac{\sqrt{3+2 \sqrt{3}} (1-x)}{\sqrt{x^3-1}}\right )}{3^{3/4}} \]
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Rubi [A] time = 0.218447, antiderivative size = 164, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.174, Rules used = {2141, 219, 2140, 206} \[ \frac{2 \sqrt{\frac{7}{6}-\frac{2}{\sqrt{3}}} (1-x) \sqrt{\frac{x^2+x+1}{\left (-x-\sqrt{3}+1\right )^2}} F\left (\sin ^{-1}\left (\frac{-x+\sqrt{3}+1}{-x-\sqrt{3}+1}\right )|-7+4 \sqrt{3}\right )}{\sqrt [4]{3} \sqrt{-\frac{1-x}{\left (-x-\sqrt{3}+1\right )^2}} \sqrt{x^3-1}}-\frac{\sqrt{2} \tanh ^{-1}\left (\frac{\sqrt{3+2 \sqrt{3}} (1-x)}{\sqrt{x^3-1}}\right )}{3^{3/4}} \]
Antiderivative was successfully verified.
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Rule 2141
Rule 219
Rule 2140
Rule 206
Rubi steps
\begin{align*} \int \frac{x}{\left (1+\sqrt{3}-x\right ) \sqrt{-1+x^3}} \, dx &=-\frac{\left (-1-\sqrt{3}\right ) \int \frac{\left (1+\sqrt{3}\right ) \left (-22+\left (1+\sqrt{3}\right )^3\right )-6 x}{\left (1+\sqrt{3}-x\right ) \sqrt{-1+x^3}} \, dx}{\left (1+\sqrt{3}\right ) \left (-28+\left (1+\sqrt{3}\right )^3\right )}-\frac{\left (-22+\left (1+\sqrt{3}\right )^3\right ) \int \frac{1}{\sqrt{-1+x^3}} \, dx}{-28+\left (1+\sqrt{3}\right )^3}\\ &=\frac{2 \sqrt{\frac{7}{6}-\frac{2}{\sqrt{3}}} (1-x) \sqrt{\frac{1+x+x^2}{\left (1-\sqrt{3}-x\right )^2}} F\left (\sin ^{-1}\left (\frac{1+\sqrt{3}-x}{1-\sqrt{3}-x}\right )|-7+4 \sqrt{3}\right )}{\sqrt [4]{3} \sqrt{-\frac{1-x}{\left (1-\sqrt{3}-x\right )^2}} \sqrt{-1+x^3}}-\frac{\left (12 \left (-1-\sqrt{3}\right )\right ) \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 )}{\left (1+\sqrt{3}\right ) \left (-28+\left (1+\sqrt{3}\right )^3\right )}\\ &=-\frac{\sqrt{2} \tanh ^{-1}\left (\frac{\sqrt{3+2 \sqrt{3}} (1-x)}{\sqrt{-1+x^3}}\right )}{3^{3/4}}+\frac{2 \sqrt{\frac{7}{6}-\frac{2}{\sqrt{3}}} (1-x) \sqrt{\frac{1+x+x^2}{\left (1-\sqrt{3}-x\right )^2}} F\left (\sin ^{-1}\left (\frac{1+\sqrt{3}-x}{1-\sqrt{3}-x}\right )|-7+4 \sqrt{3}\right )}{\sqrt [4]{3} \sqrt{-\frac{1-x}{\left (1-\sqrt{3}-x\right )^2}} \sqrt{-1+x^3}}\\ \end{align*}
Mathematica [C] time = 0.282499, size = 230, normalized size = 1.4 \[ \frac{2 i \sqrt{\frac{1-x}{1+\sqrt [3]{-1}}} \left (2 \left (1+\sqrt{3}\right ) \sqrt{x^2+x+1} \Pi \left (\frac{2 i \sqrt{3}}{3+(2+i) \sqrt{3}};\sin ^{-1}\left (\sqrt{\frac{1-(-1)^{2/3} x}{1+\sqrt [3]{-1}}}\right )|\sqrt [3]{-1}\right )+\frac{i \sqrt{\frac{(-1)^{2/3} x+\sqrt [3]{-1}}{1+\sqrt [3]{-1}}} \left (\left (3+(2+i) \sqrt{3}\right ) x+(1+2 i) \sqrt{3}+3 i\right ) F\left (\sin ^{-1}\left (\sqrt{\frac{1-(-1)^{2/3} x}{1+\sqrt [3]{-1}}}\right )|\sqrt [3]{-1}\right )}{\sqrt{\frac{1-(-1)^{2/3} x}{1+\sqrt [3]{-1}}}}\right )}{\left (3+(2+i) \sqrt{3}\right ) \sqrt{x^3-1}} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.017, size = 255, normalized size = 1.6 \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 ) }-{\frac{ \left ( -2\,\sqrt{3}-2 \right ) \left ( -{\frac{3}{2}}-{\frac{i}{2}}\sqrt{3} \right ) \sqrt{3}}{3}\sqrt{{\frac{x-1}{-{\frac{3}{2}}-{\frac{i}{2}}\sqrt{3}}}}\sqrt{{\frac{1}{{\frac{3}{2}}-{\frac{i}{2}}\sqrt{3}} \left ( x+{\frac{1}{2}}-{\frac{i}{2}}\sqrt{3} \right ) }}\sqrt{{\frac{1}{{\frac{3}{2}}+{\frac{i}{2}}\sqrt{3}} \left ( x+{\frac{1}{2}}+{\frac{i}{2}}\sqrt{3} \right ) }}{\it EllipticPi} \left ( \sqrt{{\frac{x-1}{-{\frac{3}{2}}-{\frac{i}{2}}\sqrt{3}}}},-{\frac{ \left ({\frac{3}{2}}+{\frac{i}{2}}\sqrt{3} \right ) \sqrt{3}}{3}},\sqrt{{\frac{{\frac{3}{2}}+{\frac{i}{2}}\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{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 [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-\frac{\sqrt{x^{3} - 1}{\left (x^{2} + \sqrt{3} x - x\right )}}{x^{5} - 2 \, x^{4} - 2 \, x^{3} - x^{2} + 2 \, x + 2}, x\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}{x \sqrt{x^{3} - 1} - \sqrt{3} \sqrt{x^{3} - 1} - \sqrt{x^{3} - 1}}\, 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{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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