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]
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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]
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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]
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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]
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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]
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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]
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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]
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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]
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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]