3.67 \(\int \frac{2-x-2 x^2+x^3}{4-5 x^2+x^4} \, dx\)

Optimal. Leaf size=4 \[ \log (x+2) \]

[Out]

Log[2 + x]

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Rubi [A]  time = 0.0124115, antiderivative size = 4, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077 \[ \log (x+2) \]

Antiderivative was successfully verified.

[In]  Int[(2 - x - 2*x^2 + x^3)/(4 - 5*x^2 + x^4),x]

[Out]

Log[2 + x]

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Rubi in Sympy [A]  time = 4.91714, size = 3, normalized size = 0.75 \[ \log{\left (x + 2 \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((x**3-2*x**2-x+2)/(x**4-5*x**2+4),x)

[Out]

log(x + 2)

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Mathematica [A]  time = 0.00133337, size = 4, normalized size = 1. \[ \log (x+2) \]

Antiderivative was successfully verified.

[In]  Integrate[(2 - x - 2*x^2 + x^3)/(4 - 5*x^2 + x^4),x]

[Out]

Log[2 + x]

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Maple [A]  time = 0.002, size = 5, normalized size = 1.3 \[ \ln \left ( 2+x \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((x^3-2*x^2-x+2)/(x^4-5*x^2+4),x)

[Out]

ln(2+x)

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Maxima [A]  time = 0.701242, size = 5, normalized size = 1.25 \[ \log \left (x + 2\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x^3 - 2*x^2 - x + 2)/(x^4 - 5*x^2 + 4),x, algorithm="maxima")

[Out]

log(x + 2)

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Fricas [A]  time = 0.265592, size = 5, normalized size = 1.25 \[ \log \left (x + 2\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x^3 - 2*x^2 - x + 2)/(x^4 - 5*x^2 + 4),x, algorithm="fricas")

[Out]

log(x + 2)

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Sympy [A]  time = 0.088752, size = 3, normalized size = 0.75 \[ \log{\left (x + 2 \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x**3-2*x**2-x+2)/(x**4-5*x**2+4),x)

[Out]

log(x + 2)

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GIAC/XCAS [A]  time = 0.299861, size = 7, normalized size = 1.75 \[{\rm ln}\left ({\left | x + 2 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x^3 - 2*x^2 - x + 2)/(x^4 - 5*x^2 + 4),x, algorithm="giac")

[Out]

ln(abs(x + 2))