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

Optimal. Leaf size=44 \[ \frac{\tan ^{-1}\left (2 \sqrt{2} x+\sqrt{3}\right )}{\sqrt{2}}-\frac{\tan ^{-1}\left (\sqrt{3}-2 \sqrt{2} x\right )}{\sqrt{2}} \]

[Out]

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

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Rubi [A]  time = 0.0680242, antiderivative size = 44, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136 \[ \frac{\tan ^{-1}\left (2 \sqrt{2} x+\sqrt{3}\right )}{\sqrt{2}}-\frac{\tan ^{-1}\left (\sqrt{3}-2 \sqrt{2} x\right )}{\sqrt{2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 8.72689, size = 46, normalized size = 1.05 \[ \frac{\sqrt{2} \operatorname{atan}{\left (\sqrt{2} \left (2 x - \frac{\sqrt{6}}{2}\right ) \right )}}{2} + \frac{\sqrt{2} \operatorname{atan}{\left (\sqrt{2} \left (2 x + \frac{\sqrt{6}}{2}\right ) \right )}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sqrt(2)*atan(sqrt(2)*(2*x - sqrt(6)/2))/2 + sqrt(2)*atan(sqrt(2)*(2*x + sqrt(6)/
2))/2

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Mathematica [C]  time = 0.171382, size = 99, normalized size = 2.25 \[ \frac{\left (\sqrt{3}-3 i\right ) \tan ^{-1}\left (\frac{2 x}{\sqrt{-1-i \sqrt{3}}}\right )}{2 \sqrt{3 \left (-1-i \sqrt{3}\right )}}+\frac{\left (\sqrt{3}+3 i\right ) \tan ^{-1}\left (\frac{2 x}{\sqrt{-1+i \sqrt{3}}}\right )}{2 \sqrt{3 \left (-1+i \sqrt{3}\right )}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.039, size = 40, normalized size = 0.9 \[{\frac{\sqrt{2}}{2}\arctan \left ({\frac{ \left ( 4\,x-\sqrt{6} \right ) \sqrt{2}}{2}} \right ) }+{\frac{\sqrt{2}}{2}\arctan \left ({\frac{ \left ( 4\,x+\sqrt{6} \right ) \sqrt{2}}{2}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*2^(1/2)*arctan(1/2*(4*x-6^(1/2))*2^(1/2))+1/2*2^(1/2)*arctan(1/2*(4*x+6^(1/2
))*2^(1/2))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.289893, size = 28, normalized size = 0.64 \[ \frac{1}{2} \, \sqrt{2}{\left (\arctan \left (2 \, \sqrt{2} x^{3}\right ) + \arctan \left (\sqrt{2} x\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 0.232445, size = 29, normalized size = 0.66 \[ \frac{\sqrt{2} \left (2 \operatorname{atan}{\left (\sqrt{2} x \right )} + 2 \operatorname{atan}{\left (2 \sqrt{2} x^{3} \right )}\right )}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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