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 = 0.201202, 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 \[ \frac{2 \tanh ^{-1}\left (\frac{\sqrt{3+2 \sqrt{3}} (1-x)}{\sqrt{x^3-1}}\right )}{\sqrt{3+2 \sqrt{3}}} \]
Antiderivative was successfully verified.
[In] Int[(1 - Sqrt[3] - x)/((1 + Sqrt[3] - x)*Sqrt[-1 + x^3]),x]
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Rubi in Sympy [A] time = 33.0019, size = 133, normalized size = 3.02 \[ \frac{2 \cdot 3^{\frac{3}{4}} \sqrt{\frac{x^{2} + x + 1}{\left (- x - \sqrt{3} + 1\right )^{2}}} \left (- x + 1\right ) \operatorname{atanh}{\left (\frac{\left (\sqrt{3} + 2\right ) \sqrt{- \frac{\left (- x + 1 + \sqrt{3}\right )^{2}}{\left (x - 1 + \sqrt{3}\right )^{2}} + 1}}{\sqrt{\frac{\left (- x + 1 + \sqrt{3}\right )^{2}}{\left (x - 1 + \sqrt{3}\right )^{2}} + 4 \sqrt{3} + 7}} \right )}}{3 \sqrt{\frac{x - 1}{\left (- x - \sqrt{3} + 1\right )^{2}}} \sqrt{\sqrt{3} + 2} \sqrt{x^{3} - 1}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((1-x-3**(1/2))/(1-x+3**(1/2))/(x**3-1)**(1/2),x)
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Mathematica [C] time = 0.477289, 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.
[In] Integrate[(1 - Sqrt[3] - x)/((1 + Sqrt[3] - x)*Sqrt[-1 + x^3]),x]
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Maple [C] time = 0.031, size = 245, normalized size = 5.6 \[ 2\,{\frac{-3/2-i/2\sqrt{3}}{\sqrt{{x}^{3}-1}}\sqrt{{\frac{-1+x}{-3/2-i/2\sqrt{3}}}}\sqrt{{\frac{x+1/2-i/2\sqrt{3}}{3/2-i/2\sqrt{3}}}}\sqrt{{\frac{x+1/2+i/2\sqrt{3}}{3/2+i/2\sqrt{3}}}}{\it EllipticF} \left ( \sqrt{{\frac{-1+x}{-3/2-i/2\sqrt{3}}}},\sqrt{{\frac{3/2+i/2\sqrt{3}}{3/2-i/2\sqrt{3}}}} \right ) }-4\,{\frac{-3/2-i/2\sqrt{3}}{\sqrt{{x}^{3}-1}}\sqrt{{\frac{-1+x}{-3/2-i/2\sqrt{3}}}}\sqrt{{\frac{x+1/2-i/2\sqrt{3}}{3/2-i/2\sqrt{3}}}}\sqrt{{\frac{x+1/2+i/2\sqrt{3}}{3/2+i/2\sqrt{3}}}}{\it EllipticPi} \left ( \sqrt{{\frac{-1+x}{-3/2-i/2\sqrt{3}}}},-1/3\, \left ( 3/2+i/2\sqrt{3} \right ) \sqrt{3},\sqrt{{\frac{3/2+i/2\sqrt{3}}{3/2-i/2\sqrt{3}}}} \right ) } \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((1-x-3^(1/2))/(1-x+3^(1/2))/(x^3-1)^(1/2),x)
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{x + \sqrt{3} - 1}{\sqrt{x^{3} - 1}{\left (x - \sqrt{3} - 1\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((x + sqrt(3) - 1)/(sqrt(x^3 - 1)*(x - sqrt(3) - 1)),x, algorithm="maxima")
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Fricas [A] time = 0.317479, size = 366, normalized size = 8.32 \[ \frac{1}{6} \, \sqrt{3} \sqrt{2 \, \sqrt{3} - 3} \log \left (\frac{6322680 \, x^{8} + 13553256 \, x^{7} + 26133432 \, x^{6} + 63422352 \, x^{5} + 113743056 \, x^{4} + 136435776 \, x^{3} + 102727296 \, x^{2} - 4 \,{\left (1694157 \, x^{6} + 5868732 \, x^{5} + 10586298 \, x^{4} + 12840912 \, x^{3} + 9886740 \, x^{2} - 2 \, \sqrt{3}{\left (489061 \, x^{6} + 1694157 \, x^{5} + 3056001 \, x^{4} + 3706852 \, x^{3} + 2854056 \, x^{2} + 1198884 \, x + 205636\right )} + 4153056 \, x + 712344\right )} \sqrt{x^{3} - 1} \sqrt{2 \, \sqrt{3} - 3} - \sqrt{3}{\left (3650401 \, x^{8} + 7824976 \, x^{7} + 15088144 \, x^{6} + 36616912 \, x^{5} + 65669584 \, x^{4} + 78771232 \, x^{3} + 59309632 \, x^{2} + 24558208 \, x + 4193392\right )} + 42536064 \, x + 7263168}{6322680 \, x^{8} + 37028184 \, x^{7} + 94872792 \, x^{6} + 138903408 \, x^{5} + 127105440 \, x^{4} + 74438112 \, x^{3} + 27246240 \, x^{2} - \sqrt{3}{\left (3650401 \, x^{8} + 21378232 \, x^{7} + 54774832 \, x^{6} + 80195920 \, x^{5} + 73384360 \, x^{4} + 42976864 \, x^{3} + 15730624 \, x^{2} + 3290176 \, x + 301072\right )} + 5698752 \, x + 521472}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((x + sqrt(3) - 1)/(sqrt(x^3 - 1)*(x - sqrt(3) - 1)),x, algorithm="fricas")
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{x - 1 + \sqrt{3}}{\sqrt{\left (x - 1\right ) \left (x^{2} + x + 1\right )} \left (x - \sqrt{3} - 1\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((1-x-3**(1/2))/(1-x+3**(1/2))/(x**3-1)**(1/2),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{x + \sqrt{3} - 1}{\sqrt{x^{3} - 1}{\left (x - \sqrt{3} - 1\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((x + sqrt(3) - 1)/(sqrt(x^3 - 1)*(x - sqrt(3) - 1)),x, algorithm="giac")
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