3.916 \(\int \frac{x}{10+2 x^2+x^4} \, dx\)

Optimal. Leaf size=14 \[ \frac{1}{6} \tan ^{-1}\left (\frac{1}{3} \left (x^2+1\right )\right ) \]

[Out]

ArcTan[(1 + x^2)/3]/6

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Rubi [A]  time = 0.0351037, antiderivative size = 14, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214 \[ \frac{1}{6} \tan ^{-1}\left (\frac{1}{3} \left (x^2+1\right )\right ) \]

Antiderivative was successfully verified.

[In]  Int[x/(10 + 2*x^2 + x^4),x]

[Out]

ArcTan[(1 + x^2)/3]/6

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Rubi in Sympy [A]  time = 4.7103, size = 10, normalized size = 0.71 \[ \frac{\operatorname{atan}{\left (\frac{x^{2}}{3} + \frac{1}{3} \right )}}{6} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x/(x**4+2*x**2+10),x)

[Out]

atan(x**2/3 + 1/3)/6

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Mathematica [A]  time = 0.00851059, size = 14, normalized size = 1. \[ \frac{1}{6} \tan ^{-1}\left (\frac{1}{3} \left (x^2+1\right )\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[x/(10 + 2*x^2 + x^4),x]

[Out]

ArcTan[(1 + x^2)/3]/6

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Maple [A]  time = 0.004, size = 11, normalized size = 0.8 \[{\frac{1}{6}\arctan \left ({\frac{{x}^{2}}{3}}+{\frac{1}{3}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x/(x^4+2*x^2+10),x)

[Out]

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

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Maxima [A]  time = 0.761683, size = 14, normalized size = 1. \[ \frac{1}{6} \, \arctan \left (\frac{1}{3} \, x^{2} + \frac{1}{3}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x/(x^4 + 2*x^2 + 10),x, algorithm="maxima")

[Out]

1/6*arctan(1/3*x^2 + 1/3)

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Fricas [A]  time = 0.269194, size = 14, normalized size = 1. \[ \frac{1}{6} \, \arctan \left (\frac{1}{3} \, x^{2} + \frac{1}{3}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x/(x^4 + 2*x^2 + 10),x, algorithm="fricas")

[Out]

1/6*arctan(1/3*x^2 + 1/3)

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Sympy [A]  time = 0.187189, size = 10, normalized size = 0.71 \[ \frac{\operatorname{atan}{\left (\frac{x^{2}}{3} + \frac{1}{3} \right )}}{6} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x/(x**4+2*x**2+10),x)

[Out]

atan(x**2/3 + 1/3)/6

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GIAC/XCAS [A]  time = 0.268994, size = 14, normalized size = 1. \[ \frac{1}{6} \, \arctan \left (\frac{1}{3} \, x^{2} + \frac{1}{3}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x/(x^4 + 2*x^2 + 10),x, algorithm="giac")

[Out]

1/6*arctan(1/3*x^2 + 1/3)