3.14 \(\int \frac{6-3 x^2+x^4}{4+5 x^2-5 x^4+x^6} \, dx\)

Optimal. Leaf size=39 \[ \tan ^{-1}\left (\frac{1}{2} x \left (x^4-3 x^2+1\right )\right )-\tan ^{-1}\left (\sqrt{3}-2 x\right )+\tan ^{-1}\left (2 x+\sqrt{3}\right ) \]

[Out]

-ArcTan[Sqrt[3] - 2*x] + ArcTan[Sqrt[3] + 2*x] + ArcTan[(x*(1 - 3*x^2 + x^4))/2]

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Rubi [A]  time = 0.0475696, antiderivative size = 39, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.036 \[ \tan ^{-1}\left (\frac{1}{2} x \left (x^4-3 x^2+1\right )\right )-\tan ^{-1}\left (\sqrt{3}-2 x\right )+\tan ^{-1}\left (2 x+\sqrt{3}\right ) \]

Antiderivative was successfully verified.

[In]  Int[(6 - 3*x^2 + x^4)/(4 + 5*x^2 - 5*x^4 + x^6),x]

[Out]

-ArcTan[Sqrt[3] - 2*x] + ArcTan[Sqrt[3] + 2*x] + ArcTan[(x*(1 - 3*x^2 + x^4))/2]

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Rubi in Sympy [A]  time = 8.63654, size = 36, normalized size = 0.92 \[ \operatorname{atan}{\left (\frac{x \left (42 x^{4} - 126 x^{2} + 42\right )}{84} \right )} + \operatorname{atan}{\left (2 x - \sqrt{3} \right )} + \operatorname{atan}{\left (2 x + \sqrt{3} \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((x**4-3*x**2+6)/(x**6-5*x**4+5*x**2+4),x)

[Out]

atan(x*(42*x**4 - 126*x**2 + 42)/84) + atan(2*x - sqrt(3)) + atan(2*x + sqrt(3))

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Mathematica [A]  time = 0.0178544, size = 41, normalized size = 1.05 \[ \frac{1}{2} \tan ^{-1}\left (\frac{x \left (x^2-3\right )}{x^2-2}\right )-\frac{1}{2} \tan ^{-1}\left (\frac{x \left (x^2-3\right )}{2-x^2}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[(6 - 3*x^2 + x^4)/(4 + 5*x^2 - 5*x^4 + x^6),x]

[Out]

-ArcTan[(x*(-3 + x^2))/(2 - x^2)]/2 + ArcTan[(x*(-3 + x^2))/(-2 + x^2)]/2

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Maple [A]  time = 0.024, size = 23, normalized size = 0.6 \[ \arctan \left ({\frac{{x}^{5}}{2}}-{\frac{3\,{x}^{3}}{2}}+{\frac{x}{2}} \right ) +\arctan \left ({x}^{3} \right ) +\arctan \left ( x \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((x^4-3*x^2+6)/(x^6-5*x^4+5*x^2+4),x)

[Out]

arctan(1/2*x^5-3/2*x^3+1/2*x)+arctan(x^3)+arctan(x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{4} - 3 \, x^{2} + 6}{x^{6} - 5 \, x^{4} + 5 \, x^{2} + 4}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x^4 - 3*x^2 + 6)/(x^6 - 5*x^4 + 5*x^2 + 4),x, algorithm="maxima")

[Out]

integrate((x^4 - 3*x^2 + 6)/(x^6 - 5*x^4 + 5*x^2 + 4), x)

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Fricas [A]  time = 0.218482, size = 30, normalized size = 0.77 \[ \arctan \left (\frac{1}{2} \, x^{5} - \frac{3}{2} \, x^{3} + \frac{1}{2} \, x\right ) + \arctan \left (x^{3}\right ) + \arctan \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x^4 - 3*x^2 + 6)/(x^6 - 5*x^4 + 5*x^2 + 4),x, algorithm="fricas")

[Out]

arctan(1/2*x^5 - 3/2*x^3 + 1/2*x) + arctan(x^3) + arctan(x)

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Sympy [A]  time = 0.141792, size = 24, normalized size = 0.62 \[ \operatorname{atan}{\left (x \right )} + \operatorname{atan}{\left (x^{3} \right )} + \operatorname{atan}{\left (\frac{x^{5}}{2} - \frac{3 x^{3}}{2} + \frac{x}{2} \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x**4-3*x**2+6)/(x**6-5*x**4+5*x**2+4),x)

[Out]

atan(x) + atan(x**3) + atan(x**5/2 - 3*x**3/2 + x/2)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{4} - 3 \, x^{2} + 6}{x^{6} - 5 \, x^{4} + 5 \, x^{2} + 4}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x^4 - 3*x^2 + 6)/(x^6 - 5*x^4 + 5*x^2 + 4),x, algorithm="giac")

[Out]

integrate((x^4 - 3*x^2 + 6)/(x^6 - 5*x^4 + 5*x^2 + 4), x)