3.112 \(\int \frac{-x^2+2 x^4}{1+2 x^2} \, dx\)

Optimal. Leaf size=25 \[ \frac{x^3}{3}-x+\frac{\tan ^{-1}\left (\sqrt{2} x\right )}{\sqrt{2}} \]

[Out]

-x + x^3/3 + ArcTan[Sqrt[2]*x]/Sqrt[2]

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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]

-x + x^3/3 + ArcTan[Sqrt[2]*x]/Sqrt[2]

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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]

x**3/3 - x + sqrt(2)*atan(sqrt(2)*x)/2

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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]

-x + x^3/3 + ArcTan[Sqrt[2]*x]/Sqrt[2]

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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]

-x+1/3*x^3+1/2*arctan(x*2^(1/2))*2^(1/2)

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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]

1/3*x^3 + 1/2*sqrt(2)*arctan(sqrt(2)*x) - x

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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]

1/6*sqrt(2)*(sqrt(2)*(x^3 - 3*x) + 3*arctan(sqrt(2)*x))

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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]

x**3/3 - x + sqrt(2)*atan(sqrt(2)*x)/2

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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]

1/3*x^3 + 1/2*sqrt(2)*arctan(sqrt(2)*x) - x