Optimal. Leaf size=23 \[ \frac{1}{3} \left (x^2-2\right )^{3/2}+\sqrt{x^2-2} \]
[Out]
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Rubi [A] time = 0.0526903, antiderivative size = 23, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.176 \[ \frac{1}{3} \left (x^2-2\right )^{3/2}+\sqrt{x^2-2} \]
Antiderivative was successfully verified.
[In] Int[(-x + x^3)/Sqrt[-2 + x^2],x]
[Out]
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Rubi in Sympy [A] time = 7.20262, size = 17, normalized size = 0.74 \[ \frac{\left (x^{2} - 2\right )^{\frac{3}{2}}}{3} + \sqrt{x^{2} - 2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((x**3-x)/(x**2-2)**(1/2),x)
[Out]
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Mathematica [A] time = 0.0121626, size = 18, normalized size = 0.78 \[ \frac{1}{3} \sqrt{x^2-2} \left (x^2+1\right ) \]
Antiderivative was successfully verified.
[In] Integrate[(-x + x^3)/Sqrt[-2 + x^2],x]
[Out]
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Maple [A] time = 0.008, size = 15, normalized size = 0.7 \[{\frac{{x}^{2}+1}{3}\sqrt{{x}^{2}-2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((x^3-x)/(x^2-2)^(1/2),x)
[Out]
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Maxima [A] time = 1.32909, size = 30, normalized size = 1.3 \[ \frac{1}{3} \, \sqrt{x^{2} - 2} x^{2} + \frac{1}{3} \, \sqrt{x^{2} - 2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((x^3 - x)/sqrt(x^2 - 2),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.23735, size = 93, normalized size = 4.04 \[ -\frac{2 \, x^{6} - 3 \, x^{4} - 3 \, x^{2} -{\left (2 \, x^{5} - x^{3} - 3 \, x\right )} \sqrt{x^{2} - 2} + 2}{3 \,{\left (2 \, x^{3} -{\left (2 \, x^{2} - 1\right )} \sqrt{x^{2} - 2} - 3 \, x\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((x^3 - x)/sqrt(x^2 - 2),x, algorithm="fricas")
[Out]
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Sympy [A] time = 0.480152, size = 22, normalized size = 0.96 \[ \frac{x^{2} \sqrt{x^{2} - 2}}{3} + \frac{\sqrt{x^{2} - 2}}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((x**3-x)/(x**2-2)**(1/2),x)
[Out]
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GIAC/XCAS [A] time = 0.214884, size = 23, normalized size = 1. \[ \frac{1}{3} \,{\left (x^{2} - 2\right )}^{\frac{3}{2}} + \sqrt{x^{2} - 2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((x^3 - x)/sqrt(x^2 - 2),x, algorithm="giac")
[Out]