3.49 \(\int \frac{1+2 x^2}{1+6 x^2+4 x^4} \, dx\)

Optimal. Leaf size=45 \[ \frac{\tan ^{-1}\left (\frac{2 x}{\sqrt{3-\sqrt{5}}}\right )}{\sqrt{10}}+\frac{\tan ^{-1}\left (\frac{2 x}{\sqrt{3+\sqrt{5}}}\right )}{\sqrt{10}} \]

[Out]

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

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Rubi [A]  time = 0.114716, antiderivative size = 45, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091 \[ \frac{\tan ^{-1}\left (\frac{2 x}{\sqrt{3-\sqrt{5}}}\right )}{\sqrt{10}}+\frac{\tan ^{-1}\left (\frac{2 x}{\sqrt{3+\sqrt{5}}}\right )}{\sqrt{10}} \]

Antiderivative was successfully verified.

[In]  Int[(1 + 2*x^2)/(1 + 6*x^2 + 4*x^4),x]

[Out]

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

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Rubi in Sympy [A]  time = 9.45092, size = 70, normalized size = 1.56 \[ \frac{\left (- \frac{\sqrt{5}}{5} + 1\right ) \operatorname{atan}{\left (\frac{2 x}{\sqrt{- \sqrt{5} + 3}} \right )}}{2 \sqrt{- \sqrt{5} + 3}} + \frac{\left (\frac{\sqrt{5}}{5} + 1\right ) \operatorname{atan}{\left (\frac{2 x}{\sqrt{\sqrt{5} + 3}} \right )}}{2 \sqrt{\sqrt{5} + 3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((2*x**2+1)/(4*x**4+6*x**2+1),x)

[Out]

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

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Mathematica [A]  time = 0.122975, size = 83, normalized size = 1.84 \[ \frac{\left (\sqrt{5}-1\right ) \tan ^{-1}\left (\frac{2 x}{\sqrt{3-\sqrt{5}}}\right )}{2 \sqrt{5 \left (3-\sqrt{5}\right )}}+\frac{\left (1+\sqrt{5}\right ) \tan ^{-1}\left (\frac{2 x}{\sqrt{3+\sqrt{5}}}\right )}{2 \sqrt{5 \left (3+\sqrt{5}\right )}} \]

Antiderivative was successfully verified.

[In]  Integrate[(1 + 2*x^2)/(1 + 6*x^2 + 4*x^4),x]

[Out]

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

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Maple [B]  time = 0.06, size = 136, normalized size = 3. \[ -{\frac{2\,\sqrt{5}}{10\,\sqrt{10}-10\,\sqrt{2}}\arctan \left ( 8\,{\frac{x}{2\,\sqrt{10}-2\,\sqrt{2}}} \right ) }+2\,{\frac{1}{2\,\sqrt{10}-2\,\sqrt{2}}\arctan \left ( 8\,{\frac{x}{2\,\sqrt{10}-2\,\sqrt{2}}} \right ) }+{\frac{2\,\sqrt{5}}{10\,\sqrt{10}+10\,\sqrt{2}}\arctan \left ( 8\,{\frac{x}{2\,\sqrt{10}+2\,\sqrt{2}}} \right ) }+2\,{\frac{1}{2\,\sqrt{10}+2\,\sqrt{2}}\arctan \left ( 8\,{\frac{x}{2\,\sqrt{10}+2\,\sqrt{2}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((2*x^2+1)/(4*x^4+6*x^2+1),x)

[Out]

-2/5*5^(1/2)/(2*10^(1/2)-2*2^(1/2))*arctan(8*x/(2*10^(1/2)-2*2^(1/2)))+2/(2*10^(
1/2)-2*2^(1/2))*arctan(8*x/(2*10^(1/2)-2*2^(1/2)))+2/5*5^(1/2)/(2*10^(1/2)+2*2^(
1/2))*arctan(8*x/(2*10^(1/2)+2*2^(1/2)))+2/(2*10^(1/2)+2*2^(1/2))*arctan(8*x/(2*
10^(1/2)+2*2^(1/2)))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{2 \, x^{2} + 1}{4 \, x^{4} + 6 \, x^{2} + 1}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((2*x^2 + 1)/(4*x^4 + 6*x^2 + 1),x, algorithm="maxima")

[Out]

integrate((2*x^2 + 1)/(4*x^4 + 6*x^2 + 1), x)

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Fricas [A]  time = 0.282201, size = 35, normalized size = 0.78 \[ \frac{1}{10} \, \sqrt{10}{\left (\arctan \left (\frac{2}{5} \, \sqrt{10}{\left (x^{3} + 2 \, x\right )}\right ) + \arctan \left (\frac{1}{5} \, \sqrt{10} x\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((2*x^2 + 1)/(4*x^4 + 6*x^2 + 1),x, algorithm="fricas")

[Out]

1/10*sqrt(10)*(arctan(2/5*sqrt(10)*(x^3 + 2*x)) + arctan(1/5*sqrt(10)*x))

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Sympy [A]  time = 0.257224, size = 42, normalized size = 0.93 \[ \frac{\sqrt{10} \left (2 \operatorname{atan}{\left (\frac{\sqrt{10} x}{5} \right )} + 2 \operatorname{atan}{\left (\frac{2 \sqrt{10} x^{3}}{5} + \frac{4 \sqrt{10} x}{5} \right )}\right )}{20} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((2*x**2+1)/(4*x**4+6*x**2+1),x)

[Out]

sqrt(10)*(2*atan(sqrt(10)*x/5) + 2*atan(2*sqrt(10)*x**3/5 + 4*sqrt(10)*x/5))/20

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GIAC/XCAS [A]  time = 0.275919, size = 53, normalized size = 1.18 \[ \frac{1}{10} \, \sqrt{10} \arctan \left (\frac{4 \, x}{\sqrt{10} + \sqrt{2}}\right ) + \frac{1}{10} \, \sqrt{10} \arctan \left (\frac{4 \, x}{\sqrt{10} - \sqrt{2}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((2*x^2 + 1)/(4*x^4 + 6*x^2 + 1),x, algorithm="giac")

[Out]

1/10*sqrt(10)*arctan(4*x/(sqrt(10) + sqrt(2))) + 1/10*sqrt(10)*arctan(4*x/(sqrt(
10) - sqrt(2)))