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

Optimal. Leaf size=29 \[ -\frac{1}{2} \log (1-x)+\frac{1}{3} \log (2-x)+\frac{1}{6} \log (x+1) \]

[Out]

-Log[1 - x]/2 + Log[2 - x]/3 + Log[1 + x]/6

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Rubi [A]  time = 0.0385765, antiderivative size = 29, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125 \[ -\frac{1}{2} \log (1-x)+\frac{1}{3} \log (2-x)+\frac{1}{6} \log (x+1) \]

Antiderivative was successfully verified.

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

[Out]

-Log[1 - x]/2 + Log[2 - x]/3 + Log[1 + x]/6

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Rubi in Sympy [A]  time = 26.8304, size = 22, normalized size = 0.76 \[ - 3 \log{\left (- x + \frac{8}{3} \right )} + 2 \log{\left (- x + \frac{26}{3} \right )} + \log{\left (x + \frac{28}{3} \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-3*log(-x + 8/3) + 2*log(-x + 26/3) + log(x + 28/3)

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Mathematica [A]  time = 0.00967629, size = 29, normalized size = 1. \[ -\frac{1}{2} \log (1-x)+\frac{1}{3} \log (2-x)+\frac{1}{6} \log (x+1) \]

Antiderivative was successfully verified.

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

[Out]

-Log[1 - x]/2 + Log[2 - x]/3 + Log[1 + x]/6

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Maple [A]  time = 0.01, size = 20, normalized size = 0.7 \[ -{\frac{\ln \left ( -1+x \right ) }{2}}+{\frac{\ln \left ( 1+x \right ) }{6}}+{\frac{\ln \left ( x-2 \right ) }{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((2+x)/(x^4-5*x^2+4),x)

[Out]

-1/2*ln(-1+x)+1/6*ln(1+x)+1/3*ln(x-2)

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Maxima [A]  time = 0.700268, size = 26, normalized size = 0.9 \[ \frac{1}{6} \, \log \left (x + 1\right ) - \frac{1}{2} \, \log \left (x - 1\right ) + \frac{1}{3} \, \log \left (x - 2\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/6*log(x + 1) - 1/2*log(x - 1) + 1/3*log(x - 2)

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Fricas [A]  time = 0.26206, size = 26, normalized size = 0.9 \[ \frac{1}{6} \, \log \left (x + 1\right ) - \frac{1}{2} \, \log \left (x - 1\right ) + \frac{1}{3} \, \log \left (x - 2\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/6*log(x + 1) - 1/2*log(x - 1) + 1/3*log(x - 2)

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Sympy [A]  time = 0.27699, size = 19, normalized size = 0.66 \[ \frac{\log{\left (x - 2 \right )}}{3} - \frac{\log{\left (x - 1 \right )}}{2} + \frac{\log{\left (x + 1 \right )}}{6} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

log(x - 2)/3 - log(x - 1)/2 + log(x + 1)/6

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GIAC/XCAS [A]  time = 0.28784, size = 30, normalized size = 1.03 \[ \frac{1}{6} \,{\rm ln}\left ({\left | x + 1 \right |}\right ) - \frac{1}{2} \,{\rm ln}\left ({\left | x - 1 \right |}\right ) + \frac{1}{3} \,{\rm ln}\left ({\left | x - 2 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/6*ln(abs(x + 1)) - 1/2*ln(abs(x - 1)) + 1/3*ln(abs(x - 2))