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

Optimal. Leaf size=45 \[ \frac{1}{4} \left (2+\sqrt{2}\right ) \log \left (x-\sqrt{2}+1\right )+\frac{1}{4} \left (2-\sqrt{2}\right ) \log \left (x+\sqrt{2}+1\right ) \]

[Out]

((2 + Sqrt[2])*Log[1 - Sqrt[2] + x])/4 + ((2 - Sqrt[2])*Log[1 + Sqrt[2] + x])/4

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Rubi [A]  time = 0.0327935, antiderivative size = 45, 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 \[ \frac{1}{4} \left (2+\sqrt{2}\right ) \log \left (x-\sqrt{2}+1\right )+\frac{1}{4} \left (2-\sqrt{2}\right ) \log \left (x+\sqrt{2}+1\right ) \]

Antiderivative was successfully verified.

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

[Out]

((2 + Sqrt[2])*Log[1 - Sqrt[2] + x])/4 + ((2 - Sqrt[2])*Log[1 + Sqrt[2] + x])/4

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Rubi in Sympy [A]  time = 6.58991, size = 46, normalized size = 1.02 \[ - \frac{\sqrt{2} \left (- \sqrt{2} + 1\right ) \log{\left (x + 1 + \sqrt{2} \right )}}{4} + \frac{\sqrt{2} \left (1 + \sqrt{2}\right ) \log{\left (x - \sqrt{2} + 1 \right )}}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-sqrt(2)*(-sqrt(2) + 1)*log(x + 1 + sqrt(2))/4 + sqrt(2)*(1 + sqrt(2))*log(x - s
qrt(2) + 1)/4

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Mathematica [A]  time = 0.00756792, size = 42, normalized size = 0.93 \[ \frac{1}{4} \left (\left (2+\sqrt{2}\right ) \log \left (-x+\sqrt{2}-1\right )-\left (\sqrt{2}-2\right ) \log \left (x+\sqrt{2}+1\right )\right ) \]

Antiderivative was successfully verified.

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

[Out]

((2 + Sqrt[2])*Log[-1 + Sqrt[2] - x] - (-2 + Sqrt[2])*Log[1 + Sqrt[2] + x])/4

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Maple [A]  time = 0.003, size = 29, normalized size = 0.6 \[{\frac{\ln \left ({x}^{2}+2\,x-1 \right ) }{2}}-{\frac{\sqrt{2}}{2}{\it Artanh} \left ({\frac{ \left ( 2+2\,x \right ) \sqrt{2}}{4}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((x^2-4)/(x^3-5*x+2),x)

[Out]

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

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Maxima [A]  time = 0.918364, size = 51, normalized size = 1.13 \[ \frac{1}{4} \, \sqrt{2} \log \left (\frac{2 \,{\left (x - \sqrt{2} + 1\right )}}{2 \, x + 2 \, \sqrt{2} + 2}\right ) + \frac{1}{2} \, \log \left (x^{2} + 2 \, x - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/4*sqrt(2)*log(2*(x - sqrt(2) + 1)/((2*sqrt(2)) + 2*x + 2)) + 1/2*log(x^2 + 2*x
- 1)

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Fricas [A]  time = 0.276496, size = 65, normalized size = 1.44 \[ \frac{1}{4} \, \sqrt{2}{\left (\sqrt{2} \log \left (x^{2} + 2 \, x - 1\right ) + \log \left (\frac{\sqrt{2}{\left (x^{2} + 2 \, x + 3\right )} - 4 \, x - 4}{x^{2} + 2 \, x - 1}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 0.228284, size = 39, normalized size = 0.87 \[ \left (- \frac{\sqrt{2}}{4} + \frac{1}{2}\right ) \log{\left (x + 1 + \sqrt{2} \right )} + \left (\frac{\sqrt{2}}{4} + \frac{1}{2}\right ) \log{\left (x - \sqrt{2} + 1 \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(-sqrt(2)/4 + 1/2)*log(x + 1 + sqrt(2)) + (sqrt(2)/4 + 1/2)*log(x - sqrt(2) + 1)

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GIAC/XCAS [A]  time = 0.259936, size = 59, normalized size = 1.31 \[ \frac{1}{4} \, \sqrt{2}{\rm ln}\left (\frac{{\left | 2 \, x - 2 \, \sqrt{2} + 2 \right |}}{{\left | 2 \, x + 2 \, \sqrt{2} + 2 \right |}}\right ) + \frac{1}{2} \,{\rm ln}\left ({\left | x^{2} + 2 \, x - 1 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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