Optimal. Leaf size=25 \[ \frac{x^3}{3}-x+\frac{\tan ^{-1}\left (\sqrt{2} x\right )}{\sqrt{2}} \]
[Out]
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Rubi [A] time = 0.0591201, antiderivative size = 25, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.19 \[ \frac{x^3}{3}-x+\frac{\tan ^{-1}\left (\sqrt{2} x\right )}{\sqrt{2}} \]
Antiderivative was successfully verified.
[In] Int[(-x^2 + 2*x^4)/(1 + 2*x^2),x]
[Out]
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Rubi in Sympy [A] time = 11.6984, size = 20, normalized size = 0.8 \[ \frac{x^{3}}{3} - x + \frac{\sqrt{2} \operatorname{atan}{\left (\sqrt{2} x \right )}}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((2*x**4-x**2)/(2*x**2+1),x)
[Out]
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Mathematica [A] time = 0.0185529, size = 25, normalized size = 1. \[ \frac{x^3}{3}-x+\frac{\tan ^{-1}\left (\sqrt{2} x\right )}{\sqrt{2}} \]
Antiderivative was successfully verified.
[In] Integrate[(-x^2 + 2*x^4)/(1 + 2*x^2),x]
[Out]
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Maple [A] time = 0.005, size = 21, normalized size = 0.8 \[ -x+{\frac{{x}^{3}}{3}}+{\frac{\arctan \left ( x\sqrt{2} \right ) \sqrt{2}}{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((2*x^4-x^2)/(2*x^2+1),x)
[Out]
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Maxima [A] time = 1.50371, size = 27, normalized size = 1.08 \[ \frac{1}{3} \, x^{3} + \frac{1}{2} \, \sqrt{2} \arctan \left (\sqrt{2} x\right ) - x \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((2*x^4 - x^2)/(2*x^2 + 1),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.219631, size = 34, normalized size = 1.36 \[ \frac{1}{6} \, \sqrt{2}{\left (\sqrt{2}{\left (x^{3} - 3 \, x\right )} + 3 \, \arctan \left (\sqrt{2} x\right )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((2*x^4 - x^2)/(2*x^2 + 1),x, algorithm="fricas")
[Out]
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Sympy [A] time = 0.086394, size = 20, normalized size = 0.8 \[ \frac{x^{3}}{3} - x + \frac{\sqrt{2} \operatorname{atan}{\left (\sqrt{2} x \right )}}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((2*x**4-x**2)/(2*x**2+1),x)
[Out]
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GIAC/XCAS [A] time = 0.221173, size = 27, normalized size = 1.08 \[ \frac{1}{3} \, x^{3} + \frac{1}{2} \, \sqrt{2} \arctan \left (\sqrt{2} x\right ) - x \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((2*x^4 - x^2)/(2*x^2 + 1),x, algorithm="giac")
[Out]