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

Optimal. Leaf size=19 \[ \frac{x}{2 \left (x^2+1\right )}+\frac{1}{2} \tan ^{-1}(x) \]

[Out]

x/(2*(1 + x^2)) + ArcTan[x]/2

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Rubi [A]  time = 0.0109537, antiderivative size = 19, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ \frac{x}{2 \left (x^2+1\right )}+\frac{1}{2} \tan ^{-1}(x) \]

Antiderivative was successfully verified.

[In]  Int[(1 + 2*x^2 + x^4)^(-1),x]

[Out]

x/(2*(1 + x^2)) + ArcTan[x]/2

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Rubi in Sympy [A]  time = 2.54353, size = 12, normalized size = 0.63 \[ \frac{x}{2 \left (x^{2} + 1\right )} + \frac{\operatorname{atan}{\left (x \right )}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(x**4+2*x**2+1),x)

[Out]

x/(2*(x**2 + 1)) + atan(x)/2

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Mathematica [A]  time = 0.0100036, size = 16, normalized size = 0.84 \[ \frac{1}{2} \left (\frac{x}{x^2+1}+\tan ^{-1}(x)\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[(1 + 2*x^2 + x^4)^(-1),x]

[Out]

(x/(1 + x^2) + ArcTan[x])/2

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Maple [A]  time = 0.005, size = 16, normalized size = 0.8 \[{\frac{x}{2\,{x}^{2}+2}}+{\frac{\arctan \left ( x \right ) }{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(x^4+2*x^2+1),x)

[Out]

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

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Maxima [A]  time = 0.76385, size = 20, normalized size = 1.05 \[ \frac{x}{2 \,{\left (x^{2} + 1\right )}} + \frac{1}{2} \, \arctan \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(x^4 + 2*x^2 + 1),x, algorithm="maxima")

[Out]

1/2*x/(x^2 + 1) + 1/2*arctan(x)

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Fricas [A]  time = 0.256915, size = 26, normalized size = 1.37 \[ \frac{{\left (x^{2} + 1\right )} \arctan \left (x\right ) + x}{2 \,{\left (x^{2} + 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(x^4 + 2*x^2 + 1),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.191109, size = 12, normalized size = 0.63 \[ \frac{x}{2 x^{2} + 2} + \frac{\operatorname{atan}{\left (x \right )}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(x**4+2*x**2+1),x)

[Out]

x/(2*x**2 + 2) + atan(x)/2

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GIAC/XCAS [A]  time = 0.268799, size = 20, normalized size = 1.05 \[ \frac{x}{2 \,{\left (x^{2} + 1\right )}} + \frac{1}{2} \, \arctan \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(x^4 + 2*x^2 + 1),x, algorithm="giac")

[Out]

1/2*x/(x^2 + 1) + 1/2*arctan(x)