Optimal. Leaf size=20 \[ -2 \tan ^{-1}\left (\frac{1-x}{\sqrt{1-x^3}}\right ) \]
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Rubi [A] time = 0.132165, antiderivative size = 20, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.069 \[ -2 \tan ^{-1}\left (\frac{1-x}{\sqrt{1-x^3}}\right ) \]
Antiderivative was successfully verified.
[In] Int[(2 + 2*x - x^2)/((2 + x^2)*Sqrt[1 - x^3]),x]
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Rubi in Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((-x**2+2*x+2)/(x**2+2)/(-x**3+1)**(1/2),x)
[Out]
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Mathematica [C] time = 1.04607, size = 280, normalized size = 14. \[ \frac{2 \sqrt{\frac{1-x}{1+\sqrt [3]{-1}}} \sqrt{x^2+x+1} \left (\frac{\sqrt{3} \left (1+\sqrt [3]{-1}\right ) \left (x+\sqrt [3]{-1}\right ) F\left (\sin ^{-1}\left (\sqrt{\frac{1-(-1)^{2/3} x}{1+\sqrt [3]{-1}}}\right )|\sqrt [3]{-1}\right )}{(-1)^{2/3} x-1}+\frac{6 \left (1+i \sqrt{2}\right ) \Pi \left (\frac{2 \sqrt{3}}{-i-2 \sqrt{2}+\sqrt{3}};\sin ^{-1}\left (\sqrt{\frac{1-(-1)^{2/3} x}{1+\sqrt [3]{-1}}}\right )|\sqrt [3]{-1}\right )}{i+2 \sqrt{2}-\sqrt{3}}+\frac{3 \left (1-i \sqrt{2}\right ) \Pi \left (\frac{2 \sqrt{3}}{-i+2 \sqrt{2}+\sqrt{3}};\sin ^{-1}\left (\sqrt{\frac{1-(-1)^{2/3} x}{1+\sqrt [3]{-1}}}\right )|\sqrt [3]{-1}\right )}{(-1)^{5/6}-\sqrt{2}}\right )}{3 \sqrt{1-x^3}} \]
Antiderivative was successfully verified.
[In] Integrate[(2 + 2*x - x^2)/((2 + x^2)*Sqrt[1 - x^3]),x]
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Maple [C] time = 0.097, size = 732, normalized size = 36.6 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((-x^2+2*x+2)/(x^2+2)/(-x^3+1)^(1/2),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ -\int \frac{x^{2} - 2 \, x - 2}{\sqrt{-x^{3} + 1}{\left (x^{2} + 2\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-(x^2 - 2*x - 2)/(sqrt(-x^3 + 1)*(x^2 + 2)),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.306091, size = 26, normalized size = 1.3 \[ \arctan \left (\frac{x^{2} + 2 \, x}{2 \, \sqrt{-x^{3} + 1}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-(x^2 - 2*x - 2)/(sqrt(-x^3 + 1)*(x^2 + 2)),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ - \int \left (- \frac{2 x}{x^{2} \sqrt{- x^{3} + 1} + 2 \sqrt{- x^{3} + 1}}\right )\, dx - \int \frac{x^{2}}{x^{2} \sqrt{- x^{3} + 1} + 2 \sqrt{- x^{3} + 1}}\, dx - \int \left (- \frac{2}{x^{2} \sqrt{- x^{3} + 1} + 2 \sqrt{- x^{3} + 1}}\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((-x**2+2*x+2)/(x**2+2)/(-x**3+1)**(1/2),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int -\frac{x^{2} - 2 \, x - 2}{\sqrt{-x^{3} + 1}{\left (x^{2} + 2\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-(x^2 - 2*x - 2)/(sqrt(-x^3 + 1)*(x^2 + 2)),x, algorithm="giac")
[Out]