3.538 \(\int \frac{x^3}{1+2 x^2+x^4} \, dx\)

Optimal. Leaf size=22 \[ \frac{1}{2 \left (x^2+1\right )}+\frac{1}{2} \log \left (x^2+1\right ) \]

[Out]

1/(2*(1 + x^2)) + Log[1 + x^2]/2

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Rubi [A]  time = 0.0294682, antiderivative size = 22, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.188 \[ \frac{1}{2 \left (x^2+1\right )}+\frac{1}{2} \log \left (x^2+1\right ) \]

Antiderivative was successfully verified.

[In]  Int[x^3/(1 + 2*x^2 + x^4),x]

[Out]

1/(2*(1 + x^2)) + Log[1 + x^2]/2

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Rubi in Sympy [A]  time = 4.95612, size = 15, normalized size = 0.68 \[ \frac{\log{\left (x^{2} + 1 \right )}}{2} + \frac{1}{2 \left (x^{2} + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**3/(x**4+2*x**2+1),x)

[Out]

log(x**2 + 1)/2 + 1/(2*(x**2 + 1))

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Mathematica [A]  time = 0.00729017, size = 18, normalized size = 0.82 \[ \frac{1}{2} \left (\frac{1}{x^2+1}+\log \left (x^2+1\right )\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[x^3/(1 + 2*x^2 + x^4),x]

[Out]

((1 + x^2)^(-1) + Log[1 + x^2])/2

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Maple [A]  time = 0.008, size = 19, normalized size = 0.9 \[{\frac{1}{2\,{x}^{2}+2}}+{\frac{\ln \left ({x}^{2}+1 \right ) }{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^3/(x^4+2*x^2+1),x)

[Out]

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

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Maxima [A]  time = 0.690458, size = 24, normalized size = 1.09 \[ \frac{1}{2 \,{\left (x^{2} + 1\right )}} + \frac{1}{2} \, \log \left (x^{2} + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^3/(x^4 + 2*x^2 + 1),x, algorithm="maxima")

[Out]

1/2/(x^2 + 1) + 1/2*log(x^2 + 1)

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Fricas [A]  time = 0.251669, size = 31, normalized size = 1.41 \[ \frac{{\left (x^{2} + 1\right )} \log \left (x^{2} + 1\right ) + 1}{2 \,{\left (x^{2} + 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^3/(x^4 + 2*x^2 + 1),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.154311, size = 15, normalized size = 0.68 \[ \frac{\log{\left (x^{2} + 1 \right )}}{2} + \frac{1}{2 x^{2} + 2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**3/(x**4+2*x**2+1),x)

[Out]

log(x**2 + 1)/2 + 1/(2*x**2 + 2)

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GIAC/XCAS [A]  time = 0.270716, size = 24, normalized size = 1.09 \[ \frac{1}{2 \,{\left (x^{2} + 1\right )}} + \frac{1}{2} \,{\rm ln}\left (x^{2} + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^3/(x^4 + 2*x^2 + 1),x, algorithm="giac")

[Out]

1/2/(x^2 + 1) + 1/2*ln(x^2 + 1)