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

Optimal. Leaf size=49 \[ \frac{\tan ^{-1}\left (\sqrt{\frac{2}{3+\sqrt{5}}} x\right )}{\sqrt{5}}+\frac{\tan ^{-1}\left (\sqrt{\frac{1}{2} \left (3+\sqrt{5}\right )} x\right )}{\sqrt{5}} \]

[Out]

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

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Rubi [A]  time = 0.122879, antiderivative size = 49, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111 \[ \frac{\tan ^{-1}\left (\sqrt{\frac{2}{3+\sqrt{5}}} x\right )}{\sqrt{5}}+\frac{\tan ^{-1}\left (\sqrt{\frac{1}{2} \left (3+\sqrt{5}\right )} x\right )}{\sqrt{5}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 8.94797, size = 87, normalized size = 1.78 \[ \frac{\sqrt{2} \left (- \frac{\sqrt{5}}{10} + \frac{1}{2}\right ) \operatorname{atan}{\left (\frac{\sqrt{2} x}{\sqrt{- \sqrt{5} + 3}} \right )}}{\sqrt{- \sqrt{5} + 3}} + \frac{\sqrt{2} \left (\frac{\sqrt{5}}{10} + \frac{1}{2}\right ) \operatorname{atan}{\left (\frac{\sqrt{2} x}{\sqrt{\sqrt{5} + 3}} \right )}}{\sqrt{\sqrt{5} + 3}} \]

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)/10 + 1/2)*atan(sqrt(2)*x/sqrt(-sqrt(5) + 3))/sqrt(-sqrt(5) + 3
) + sqrt(2)*(sqrt(5)/10 + 1/2)*atan(sqrt(2)*x/sqrt(sqrt(5) + 3))/sqrt(sqrt(5) +
3)

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Mathematica [A]  time = 0.164721, size = 83, normalized size = 1.69 \[ \frac{\left (\sqrt{5}-1\right ) \tan ^{-1}\left (\sqrt{\frac{2}{3-\sqrt{5}}} x\right )}{\sqrt{10 \left (3-\sqrt{5}\right )}}+\frac{\left (1+\sqrt{5}\right ) \tan ^{-1}\left (\sqrt{\frac{2}{3+\sqrt{5}}} x\right )}{\sqrt{10 \left (3+\sqrt{5}\right )}} \]

Antiderivative was successfully verified.

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

[Out]

((-1 + Sqrt[5])*ArcTan[Sqrt[2/(3 - Sqrt[5])]*x])/Sqrt[10*(3 - Sqrt[5])] + ((1 +
Sqrt[5])*ArcTan[Sqrt[2/(3 + Sqrt[5])]*x])/Sqrt[10*(3 + Sqrt[5])]

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Maple [B]  time = 0.042, size = 104, normalized size = 2.1 \[{\frac{2\,\sqrt{5}}{10\,\sqrt{5}+10}\arctan \left ( 4\,{\frac{x}{2\,\sqrt{5}+2}} \right ) }+2\,{\frac{1}{2\,\sqrt{5}+2}\arctan \left ( 4\,{\frac{x}{2\,\sqrt{5}+2}} \right ) }-{\frac{2\,\sqrt{5}}{-10+10\,\sqrt{5}}\arctan \left ( 4\,{\frac{x}{-2+2\,\sqrt{5}}} \right ) }+2\,{\frac{1}{-2+2\,\sqrt{5}}\arctan \left ( 4\,{\frac{x}{-2+2\,\sqrt{5}}} \right ) } \]

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*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*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)))

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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)

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Fricas [A]  time = 0.280968, size = 35, normalized size = 0.71 \[ \frac{1}{5} \, \sqrt{5}{\left (\arctan \left (\frac{1}{5} \, \sqrt{5}{\left (x^{3} + 4 \, x\right )}\right ) + \arctan \left (\frac{1}{5} \, \sqrt{5} x\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/5*sqrt(5)*(arctan(1/5*sqrt(5)*(x^3 + 4*x)) + arctan(1/5*sqrt(5)*x))

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Sympy [A]  time = 0.237262, size = 41, normalized size = 0.84 \[ \frac{\sqrt{5} \left (2 \operatorname{atan}{\left (\frac{\sqrt{5} x}{5} \right )} + 2 \operatorname{atan}{\left (\frac{\sqrt{5} x^{3}}{5} + \frac{4 \sqrt{5} x}{5} \right )}\right )}{10} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sqrt(5)*(2*atan(sqrt(5)*x/5) + 2*atan(sqrt(5)*x**3/5 + 4*sqrt(5)*x/5))/10

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GIAC/XCAS [A]  time = 0.2728, size = 35, normalized size = 0.71 \[ \frac{1}{10} \, \sqrt{5}{\left (\pi{\rm sign}\left (x\right ) + 2 \, \arctan \left (\frac{\sqrt{5}{\left (x^{2} - 1\right )}}{5 \, x}\right )\right )} \]

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]

1/10*sqrt(5)*(pi*sign(x) + 2*arctan(1/5*sqrt(5)*(x^2 - 1)/x))