3.91 \(\int \frac{1-x^2}{1+3 x^2+x^4} \, dx\)

Optimal. Leaf size=39 \[ \tan ^{-1}\left (\sqrt{\frac{1}{2} \left (3+\sqrt{5}\right )} x\right )-\tan ^{-1}\left (\sqrt{\frac{2}{3+\sqrt{5}}} x\right ) \]

[Out]

-ArcTan[Sqrt[2/(3 + Sqrt[5])]*x] + ArcTan[Sqrt[(3 + Sqrt[5])/2]*x]

_______________________________________________________________________________________

Rubi [A]  time = 0.0922117, antiderivative size = 39, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1 \[ \tan ^{-1}\left (\sqrt{\frac{1}{2} \left (3+\sqrt{5}\right )} x\right )-\tan ^{-1}\left (\sqrt{\frac{2}{3+\sqrt{5}}} x\right ) \]

Antiderivative was successfully verified.

[In]  Int[(1 - x^2)/(1 + 3*x^2 + x^4),x]

[Out]

-ArcTan[Sqrt[2/(3 + Sqrt[5])]*x] + ArcTan[Sqrt[(3 + Sqrt[5])/2]*x]

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 9.2912, size = 88, normalized size = 2.26 \[ - \frac{\sqrt{2} \left (- \frac{\sqrt{5}}{2} + \frac{1}{2}\right ) \operatorname{atan}{\left (\frac{\sqrt{2} x}{\sqrt{- \sqrt{5} + 3}} \right )}}{\sqrt{- \sqrt{5} + 3}} - \frac{\sqrt{2} \left (\frac{1}{2} + \frac{\sqrt{5}}{2}\right ) \operatorname{atan}{\left (\frac{\sqrt{2} x}{\sqrt{\sqrt{5} + 3}} \right )}}{\sqrt{\sqrt{5} + 3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((-x**2+1)/(x**4+3*x**2+1),x)

[Out]

-sqrt(2)*(-sqrt(5)/2 + 1/2)*atan(sqrt(2)*x/sqrt(-sqrt(5) + 3))/sqrt(-sqrt(5) + 3
) - sqrt(2)*(1/2 + sqrt(5)/2)*atan(sqrt(2)*x/sqrt(sqrt(5) + 3))/sqrt(sqrt(5) + 3
)

_______________________________________________________________________________________

Mathematica [A]  time = 0.0100228, size = 10, normalized size = 0.26 \[ \tan ^{-1}\left (\frac{x}{x^2+1}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[(1 - x^2)/(1 + 3*x^2 + x^4),x]

[Out]

ArcTan[x/(1 + x^2)]

_______________________________________________________________________________________

Maple [B]  time = 0.017, size = 104, normalized size = 2.7 \[ -2\,{\frac{\sqrt{5}}{2\,\sqrt{5}+2}\arctan \left ( 4\,{\frac{x}{2\,\sqrt{5}+2}} \right ) }-2\,{\frac{1}{2\,\sqrt{5}+2}\arctan \left ( 4\,{\frac{x}{2\,\sqrt{5}+2}} \right ) }+2\,{\frac{\sqrt{5}}{-2+2\,\sqrt{5}}\arctan \left ( 4\,{\frac{x}{-2+2\,\sqrt{5}}} \right ) }-2\,{\frac{1}{-2+2\,\sqrt{5}}\arctan \left ( 4\,{\frac{x}{-2+2\,\sqrt{5}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((-x^2+1)/(x^4+3*x^2+1),x)

[Out]

-2*5^(1/2)/(2*5^(1/2)+2)*arctan(4*x/(2*5^(1/2)+2))-2/(2*5^(1/2)+2)*arctan(4*x/(2
*5^(1/2)+2))+2*5^(1/2)/(-2+2*5^(1/2))*arctan(4*x/(-2+2*5^(1/2)))-2/(-2+2*5^(1/2)
)*arctan(4*x/(-2+2*5^(1/2)))

_______________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \[ -\int \frac{x^{2} - 1}{x^{4} + 3 \, x^{2} + 1}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(x^2 - 1)/(x^4 + 3*x^2 + 1),x, algorithm="maxima")

[Out]

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

_______________________________________________________________________________________

Fricas [A]  time = 0.264358, size = 18, normalized size = 0.46 \[ \arctan \left (x^{3} + 2 \, x\right ) - \arctan \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(x^2 - 1)/(x^4 + 3*x^2 + 1),x, algorithm="fricas")

[Out]

arctan(x^3 + 2*x) - arctan(x)

_______________________________________________________________________________________

Sympy [A]  time = 0.201657, size = 10, normalized size = 0.26 \[ - \operatorname{atan}{\left (x \right )} + \operatorname{atan}{\left (x^{3} + 2 x \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-x**2+1)/(x**4+3*x**2+1),x)

[Out]

-atan(x) + atan(x**3 + 2*x)

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.274356, size = 35, normalized size = 0.9 \[ \frac{1}{4} \, \pi{\rm sign}\left (x\right ) - \frac{1}{2} \, \arctan \left (\frac{x^{4} + x^{2} + 1}{2 \,{\left (x^{3} + x\right )}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(x^2 - 1)/(x^4 + 3*x^2 + 1),x, algorithm="giac")

[Out]

1/4*pi*sign(x) - 1/2*arctan(1/2*(x^4 + x^2 + 1)/(x^3 + x))