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

Optimal. Leaf size=74 \[ \frac{\left (3+\sqrt{5}\right )^{3/2} \tan ^{-1}\left (\sqrt{\frac{1}{2} \left (3+\sqrt{5}\right )} x\right )}{2 \sqrt{10}}-\frac{1}{10} \sqrt{180-80 \sqrt{5}} \tan ^{-1}\left (\sqrt{\frac{2}{3+\sqrt{5}}} x\right ) \]

[Out]

-(Sqrt[180 - 80*Sqrt[5]]*ArcTan[Sqrt[2/(3 + Sqrt[5])]*x])/10 + ((3 + Sqrt[5])^(3
/2)*ArcTan[Sqrt[(3 + Sqrt[5])/2]*x])/(2*Sqrt[10])

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Rubi [A]  time = 0.132275, antiderivative size = 74, 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 \[ \frac{\left (3+\sqrt{5}\right )^{3/2} \tan ^{-1}\left (\sqrt{\frac{1}{2} \left (3+\sqrt{5}\right )} x\right )}{2 \sqrt{10}}-\sqrt{\frac{1}{5} \left (9-4 \sqrt{5}\right )} \tan ^{-1}\left (\sqrt{\frac{2}{3+\sqrt{5}}} x\right ) \]

Antiderivative was successfully verified.

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

[Out]

-(Sqrt[(9 - 4*Sqrt[5])/5]*ArcTan[Sqrt[2/(3 + Sqrt[5])]*x]) + ((3 + Sqrt[5])^(3/2
)*ArcTan[Sqrt[(3 + Sqrt[5])/2]*x])/(2*Sqrt[10])

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Rubi in Sympy [A]  time = 9.24372, size = 90, normalized size = 1.22 \[ \frac{\sqrt{2} \left (\frac{1}{2} + \frac{3 \sqrt{5}}{10}\right ) \operatorname{atan}{\left (\frac{\sqrt{2} x}{\sqrt{- \sqrt{5} + 3}} \right )}}{\sqrt{- \sqrt{5} + 3}} + \frac{\sqrt{2} \left (- \frac{3 \sqrt{5}}{10} + \frac{1}{2}\right ) \operatorname{atan}{\left (\frac{\sqrt{2} x}{\sqrt{\sqrt{5} + 3}} \right )}}{\sqrt{\sqrt{5} + 3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.158851, size = 73, normalized size = 0.99 \[ \frac{\left (3+\sqrt{5}\right )^{3/2} \tan ^{-1}\left (\sqrt{\frac{1}{2} \left (3+\sqrt{5}\right )} x\right )-\left (3-\sqrt{5}\right )^{3/2} \tan ^{-1}\left (\sqrt{\frac{2}{3+\sqrt{5}}} x\right )}{2 \sqrt{10}} \]

Antiderivative was successfully verified.

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

[Out]

(-((3 - Sqrt[5])^(3/2)*ArcTan[Sqrt[2/(3 + Sqrt[5])]*x]) + (3 + Sqrt[5])^(3/2)*Ar
cTan[Sqrt[(3 + Sqrt[5])/2]*x])/(2*Sqrt[10])

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Maple [B]  time = 0.03, size = 104, normalized size = 1.4 \[ 2\,{\frac{1}{2\,\sqrt{5}+2}\arctan \left ( 4\,{\frac{x}{2\,\sqrt{5}+2}} \right ) }-{\frac{6\,\sqrt{5}}{10\,\sqrt{5}+10}\arctan \left ( 4\,{\frac{x}{2\,\sqrt{5}+2}} \right ) }+2\,{\frac{1}{-2+2\,\sqrt{5}}\arctan \left ( 4\,{\frac{x}{-2+2\,\sqrt{5}}} \right ) }+{\frac{6\,\sqrt{5}}{-10+10\,\sqrt{5}}\arctan \left ( 4\,{\frac{x}{-2+2\,\sqrt{5}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((x^2+3)/(x^4+3*x^2+1),x)

[Out]

2/(2*5^(1/2)+2)*arctan(4*x/(2*5^(1/2)+2))-6/5*5^(1/2)/(2*5^(1/2)+2)*arctan(4*x/(
2*5^(1/2)+2))+2/(-2+2*5^(1/2))*arctan(4*x/(-2+2*5^(1/2)))+6/5*5^(1/2)/(-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. \[ \int \frac{x^{2} + 3}{x^{4} + 3 \, x^{2} + 1}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.290024, size = 196, normalized size = 2.65 \[ \frac{2}{5} \, \sqrt{\sqrt{5}{\left (9 \, \sqrt{5} - 20\right )}} \arctan \left (\frac{\sqrt{\sqrt{5}{\left (9 \, \sqrt{5} - 20\right )}}{\left (3 \, \sqrt{5} + 7\right )}}{2 \,{\left (\sqrt{5} \sqrt{\frac{1}{10}} \sqrt{\sqrt{5}{\left (\sqrt{5}{\left (2 \, x^{2} + 3\right )} + 5\right )}} + \sqrt{5} x\right )}}\right ) + \frac{2}{5} \, \sqrt{\sqrt{5}{\left (9 \, \sqrt{5} + 20\right )}} \arctan \left (\frac{\sqrt{\sqrt{5}{\left (9 \, \sqrt{5} + 20\right )}}{\left (3 \, \sqrt{5} - 7\right )}}{2 \,{\left (\sqrt{5} \sqrt{\frac{1}{10}} \sqrt{\sqrt{5}{\left (\sqrt{5}{\left (2 \, x^{2} + 3\right )} - 5\right )}} + \sqrt{5} x\right )}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/5*sqrt(sqrt(5)*(9*sqrt(5) - 20))*arctan(1/2*sqrt(sqrt(5)*(9*sqrt(5) - 20))*(3*
sqrt(5) + 7)/(sqrt(5)*sqrt(1/10)*sqrt(sqrt(5)*(sqrt(5)*(2*x^2 + 3) + 5)) + sqrt(
5)*x)) + 2/5*sqrt(sqrt(5)*(9*sqrt(5) + 20))*arctan(1/2*sqrt(sqrt(5)*(9*sqrt(5) +
20))*(3*sqrt(5) - 7)/(sqrt(5)*sqrt(1/10)*sqrt(sqrt(5)*(sqrt(5)*(2*x^2 + 3) - 5)
) + sqrt(5)*x))

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Sympy [A]  time = 0.500868, size = 46, normalized size = 0.62 \[ 2 \left (\frac{\sqrt{5}}{5} + \frac{1}{2}\right ) \operatorname{atan}{\left (\frac{2 x}{-1 + \sqrt{5}} \right )} - 2 \left (- \frac{\sqrt{5}}{5} + \frac{1}{2}\right ) \operatorname{atan}{\left (\frac{2 x}{1 + \sqrt{5}} \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.269401, size = 55, normalized size = 0.74 \[ \frac{1}{5} \,{\left (2 \, \sqrt{5} - 5\right )} \arctan \left (\frac{2 \, x}{\sqrt{5} + 1}\right ) + \frac{1}{5} \,{\left (2 \, \sqrt{5} + 5\right )} \arctan \left (\frac{2 \, x}{\sqrt{5} - 1}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/5*(2*sqrt(5) - 5)*arctan(2*x/(sqrt(5) + 1)) + 1/5*(2*sqrt(5) + 5)*arctan(2*x/(
sqrt(5) - 1))