3.138 \(\int \frac{-7 x+4 x^3}{4-5 x^2+x^4} \, dx\)

Optimal. Leaf size=25 \[ \frac{1}{2} \log \left (1-x^2\right )+\frac{3}{2} \log \left (4-x^2\right ) \]

[Out]

Log[1 - x^2]/2 + (3*Log[4 - x^2])/2

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Rubi [A]  time = 0.0519601, antiderivative size = 25, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182 \[ \frac{1}{2} \log \left (1-x^2\right )+\frac{3}{2} \log \left (4-x^2\right ) \]

Antiderivative was successfully verified.

[In]  Int[(-7*x + 4*x^3)/(4 - 5*x^2 + x^4),x]

[Out]

Log[1 - x^2]/2 + (3*Log[4 - x^2])/2

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Rubi in Sympy [A]  time = 13.5832, size = 17, normalized size = 0.68 \[ \frac{\log{\left (- x^{2} + 1 \right )}}{2} + \frac{3 \log{\left (- x^{2} + 4 \right )}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((4*x**3-7*x)/(x**4-5*x**2+4),x)

[Out]

log(-x**2 + 1)/2 + 3*log(-x**2 + 4)/2

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Mathematica [A]  time = 0.00813973, size = 25, normalized size = 1. \[ \frac{1}{2} \log \left (1-x^2\right )+\frac{3}{2} \log \left (4-x^2\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[(-7*x + 4*x^3)/(4 - 5*x^2 + x^4),x]

[Out]

Log[1 - x^2]/2 + (3*Log[4 - x^2])/2

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Maple [A]  time = 0.008, size = 18, normalized size = 0.7 \[{\frac{\ln \left ({x}^{2}-1 \right ) }{2}}+{\frac{3\,\ln \left ({x}^{2}-4 \right ) }{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((4*x^3-7*x)/(x^4-5*x^2+4),x)

[Out]

1/2*ln(x^2-1)+3/2*ln(x^2-4)

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Maxima [A]  time = 0.697192, size = 34, normalized size = 1.36 \[ \frac{3}{2} \, \log \left (x + 2\right ) + \frac{1}{2} \, \log \left (x + 1\right ) + \frac{1}{2} \, \log \left (x - 1\right ) + \frac{3}{2} \, \log \left (x - 2\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((4*x^3 - 7*x)/(x^4 - 5*x^2 + 4),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.251158, size = 23, normalized size = 0.92 \[ \frac{1}{2} \, \log \left (x^{2} - 1\right ) + \frac{3}{2} \, \log \left (x^{2} - 4\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((4*x^3 - 7*x)/(x^4 - 5*x^2 + 4),x, algorithm="fricas")

[Out]

1/2*log(x^2 - 1) + 3/2*log(x^2 - 4)

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Sympy [A]  time = 0.217721, size = 17, normalized size = 0.68 \[ \frac{3 \log{\left (x^{2} - 4 \right )}}{2} + \frac{\log{\left (x^{2} - 1 \right )}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((4*x**3-7*x)/(x**4-5*x**2+4),x)

[Out]

3*log(x**2 - 4)/2 + log(x**2 - 1)/2

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GIAC/XCAS [A]  time = 0.272121, size = 26, normalized size = 1.04 \[ \frac{1}{2} \,{\rm ln}\left ({\left | x^{2} - 1 \right |}\right ) + \frac{3}{2} \,{\rm ln}\left ({\left | x^{2} - 4 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((4*x^3 - 7*x)/(x^4 - 5*x^2 + 4),x, algorithm="giac")

[Out]

1/2*ln(abs(x^2 - 1)) + 3/2*ln(abs(x^2 - 4))