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

Optimal. Leaf size=23 \[ \tan ^{-1}\left (4 x+\sqrt{7}\right )-\tan ^{-1}\left (\sqrt{7}-4 x\right ) \]

[Out]

-ArcTan[Sqrt[7] - 4*x] + ArcTan[Sqrt[7] + 4*x]

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Rubi [A]  time = 0.0495436, antiderivative size = 23, 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 \[ \tan ^{-1}\left (4 x+\sqrt{7}\right )-\tan ^{-1}\left (\sqrt{7}-4 x\right ) \]

Antiderivative was successfully verified.

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

[Out]

-ArcTan[Sqrt[7] - 4*x] + ArcTan[Sqrt[7] + 4*x]

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Rubi in Sympy [A]  time = 8.34717, size = 19, normalized size = 0.83 \[ \operatorname{atan}{\left (4 x - \sqrt{7} \right )} + \operatorname{atan}{\left (4 x + \sqrt{7} \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

atan(4*x - sqrt(7)) + atan(4*x + sqrt(7))

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Mathematica [A]  time = 0.0112752, size = 14, normalized size = 0.61 \[ -\tan ^{-1}\left (\frac{x}{2 x^2-1}\right ) \]

Antiderivative was successfully verified.

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

[Out]

-ArcTan[x/(-1 + 2*x^2)]

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Maple [A]  time = 0.035, size = 20, normalized size = 0.9 \[ \arctan \left ( 4\,x-\sqrt{7} \right ) +\arctan \left ( 4\,x+\sqrt{7} \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

arctan(4*x-7^(1/2))+arctan(4*x+7^(1/2))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.303061, size = 20, normalized size = 0.87 \[ \arctan \left (4 \, x^{3} - x\right ) + \arctan \left (2 \, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

arctan(4*x^3 - x) + arctan(2*x)

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Sympy [A]  time = 0.211849, size = 12, normalized size = 0.52 \[ \operatorname{atan}{\left (2 x \right )} + \operatorname{atan}{\left (4 x^{3} - x \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

atan(2*x) + atan(4*x**3 - x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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