Optimal. Leaf size=144 \[ \frac{2 \sqrt{x^3-1}}{-x-\sqrt{3}+1}-\frac{\sqrt [4]{3} \sqrt{2+\sqrt{3}} (1-x) \sqrt{\frac{x^2+x+1}{\left (-x-\sqrt{3}+1\right )^2}} E\left (\sin ^{-1}\left (\frac{-x+\sqrt{3}+1}{-x-\sqrt{3}+1}\right )|-7+4 \sqrt{3}\right )}{\sqrt{-\frac{1-x}{\left (-x-\sqrt{3}+1\right )^2}} \sqrt{x^3-1}} \]
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Rubi [A] time = 0.0777194, antiderivative size = 144, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.05 \[ \frac{2 \sqrt{x^3-1}}{-x-\sqrt{3}+1}-\frac{\sqrt [4]{3} \sqrt{2+\sqrt{3}} (1-x) \sqrt{\frac{x^2+x+1}{\left (-x-\sqrt{3}+1\right )^2}} E\left (\sin ^{-1}\left (\frac{-x+\sqrt{3}+1}{-x-\sqrt{3}+1}\right )|-7+4 \sqrt{3}\right )}{\sqrt{-\frac{1-x}{\left (-x-\sqrt{3}+1\right )^2}} \sqrt{x^3-1}} \]
Antiderivative was successfully verified.
[In] Int[(1 + Sqrt[3] - x)/Sqrt[-1 + x^3],x]
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Rubi in Sympy [A] time = 6.76168, size = 109, normalized size = 0.76 \[ \frac{2 \sqrt{x^{3} - 1}}{- x - \sqrt{3} + 1} - \frac{\sqrt [4]{3} \sqrt{\frac{x^{2} + x + 1}{\left (- x - \sqrt{3} + 1\right )^{2}}} \sqrt{\sqrt{3} + 2} \left (- x + 1\right ) E\left (\operatorname{asin}{\left (\frac{- x + 1 + \sqrt{3}}{- x - \sqrt{3} + 1} \right )}\middle | -7 + 4 \sqrt{3}\right )}{\sqrt{\frac{x - 1}{\left (- x - \sqrt{3} + 1\right )^{2}}} \sqrt{x^{3} - 1}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((1-x+3**(1/2))/(x**3-1)**(1/2),x)
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Mathematica [C] time = 0.11397, size = 110, normalized size = 0.76 \[ \frac{2 \sqrt [4]{3} \sqrt{(-1)^{5/6} (x-1)} \sqrt{x^2+x+1} \left (i F\left (\sin ^{-1}\left (\frac{\sqrt{-i x-(-1)^{5/6}}}{\sqrt [4]{3}}\right )|\sqrt [3]{-1}\right )+(-1)^{2/3} E\left (\sin ^{-1}\left (\frac{\sqrt{-i x-(-1)^{5/6}}}{\sqrt [4]{3}}\right )|\sqrt [3]{-1}\right )\right )}{\sqrt{x^3-1}} \]
Warning: Unable to verify antiderivative.
[In] Integrate[(1 + Sqrt[3] - x)/Sqrt[-1 + x^3],x]
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Maple [B] time = 0.015, size = 407, normalized size = 2.8 \[ 2\,{\frac{-3/2-i/2\sqrt{3}}{\sqrt{{x}^{3}-1}}\sqrt{{\frac{-1+x}{-3/2-i/2\sqrt{3}}}}\sqrt{{\frac{x+1/2-i/2\sqrt{3}}{3/2-i/2\sqrt{3}}}}\sqrt{{\frac{x+1/2+i/2\sqrt{3}}{3/2+i/2\sqrt{3}}}}{\it EllipticF} \left ( \sqrt{{\frac{-1+x}{-3/2-i/2\sqrt{3}}}},\sqrt{{\frac{3/2+i/2\sqrt{3}}{3/2-i/2\sqrt{3}}}} \right ) }+2\,{\frac{\sqrt{3} \left ( -3/2-i/2\sqrt{3} \right ) }{\sqrt{{x}^{3}-1}}\sqrt{{\frac{-1+x}{-3/2-i/2\sqrt{3}}}}\sqrt{{\frac{x+1/2-i/2\sqrt{3}}{3/2-i/2\sqrt{3}}}}\sqrt{{\frac{x+1/2+i/2\sqrt{3}}{3/2+i/2\sqrt{3}}}}{\it EllipticF} \left ( \sqrt{{\frac{-1+x}{-3/2-i/2\sqrt{3}}}},\sqrt{{\frac{3/2+i/2\sqrt{3}}{3/2-i/2\sqrt{3}}}} \right ) }-2\,{\frac{-3/2-i/2\sqrt{3}}{\sqrt{{x}^{3}-1}}\sqrt{{\frac{-1+x}{-3/2-i/2\sqrt{3}}}}\sqrt{{\frac{x+1/2-i/2\sqrt{3}}{3/2-i/2\sqrt{3}}}}\sqrt{{\frac{x+1/2+i/2\sqrt{3}}{3/2+i/2\sqrt{3}}}} \left ( \left ( 3/2-i/2\sqrt{3} \right ){\it EllipticE} \left ( \sqrt{{\frac{-1+x}{-3/2-i/2\sqrt{3}}}},\sqrt{{\frac{3/2+i/2\sqrt{3}}{3/2-i/2\sqrt{3}}}} \right ) + \left ( -1/2+i/2\sqrt{3} \right ){\it EllipticF} \left ( \sqrt{{\frac{-1+x}{-3/2-i/2\sqrt{3}}}},\sqrt{{\frac{3/2+i/2\sqrt{3}}{3/2-i/2\sqrt{3}}}} \right ) \right ) } \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((1-x+3^(1/2))/(x^3-1)^(1/2),x)
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ -\int \frac{x - \sqrt{3} - 1}{\sqrt{x^{3} - 1}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-(x - sqrt(3) - 1)/sqrt(x^3 - 1),x, algorithm="maxima")
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (-\frac{x - \sqrt{3} - 1}{\sqrt{x^{3} - 1}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-(x - sqrt(3) - 1)/sqrt(x^3 - 1),x, algorithm="fricas")
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Sympy [A] time = 2.40001, size = 82, normalized size = 0.57 \[ \frac{i x^{2} \Gamma \left (\frac{2}{3}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{2}, \frac{2}{3} \\ \frac{5}{3} \end{matrix}\middle |{x^{3}} \right )}}{3 \Gamma \left (\frac{5}{3}\right )} - \frac{\sqrt{3} i x \Gamma \left (\frac{1}{3}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{3}, \frac{1}{2} \\ \frac{4}{3} \end{matrix}\middle |{x^{3}} \right )}}{3 \Gamma \left (\frac{4}{3}\right )} - \frac{i x \Gamma \left (\frac{1}{3}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{3}, \frac{1}{2} \\ \frac{4}{3} \end{matrix}\middle |{x^{3}} \right )}}{3 \Gamma \left (\frac{4}{3}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((1-x+3**(1/2))/(x**3-1)**(1/2),x)
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int -\frac{x - \sqrt{3} - 1}{\sqrt{x^{3} - 1}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-(x - sqrt(3) - 1)/sqrt(x^3 - 1),x, algorithm="giac")
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