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

Optimal. Leaf size=46 \[ \frac{\tan ^{-1}\left (\frac{4 x+\sqrt{5}}{\sqrt{3}}\right )}{\sqrt{3}}-\frac{\tan ^{-1}\left (\frac{\sqrt{5}-4 x}{\sqrt{3}}\right )}{\sqrt{3}} \]

[Out]

-(ArcTan[(Sqrt[5] - 4*x)/Sqrt[3]]/Sqrt[3]) + ArcTan[(Sqrt[5] + 4*x)/Sqrt[3]]/Sqr
t[3]

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Rubi [A]  time = 0.0779575, antiderivative size = 46, 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 \[ \frac{\tan ^{-1}\left (\frac{4 x+\sqrt{5}}{\sqrt{3}}\right )}{\sqrt{3}}-\frac{\tan ^{-1}\left (\frac{\sqrt{5}-4 x}{\sqrt{3}}\right )}{\sqrt{3}} \]

Antiderivative was successfully verified.

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

[Out]

-(ArcTan[(Sqrt[5] - 4*x)/Sqrt[3]]/Sqrt[3]) + ArcTan[(Sqrt[5] + 4*x)/Sqrt[3]]/Sqr
t[3]

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Rubi in Sympy [A]  time = 8.74153, size = 49, normalized size = 1.07 \[ \frac{\sqrt{3} \operatorname{atan}{\left (\sqrt{3} \left (\frac{4 x}{3} - \frac{\sqrt{5}}{3}\right ) \right )}}{3} + \frac{\sqrt{3} \operatorname{atan}{\left (\sqrt{3} \left (\frac{4 x}{3} + \frac{\sqrt{5}}{3}\right ) \right )}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sqrt(3)*atan(sqrt(3)*(4*x/3 - sqrt(5)/3))/3 + sqrt(3)*atan(sqrt(3)*(4*x/3 + sqrt
(5)/3))/3

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Mathematica [C]  time = 0.476232, size = 101, normalized size = 2.2 \[ \frac{\left (\sqrt{15}-5 i\right ) \tan ^{-1}\left (\frac{2 x}{\sqrt{\frac{1}{2} \left (-1-i \sqrt{15}\right )}}\right )}{\sqrt{30 \left (-1-i \sqrt{15}\right )}}+\frac{\left (\sqrt{15}+5 i\right ) \tan ^{-1}\left (\frac{2 x}{\sqrt{\frac{1}{2} \left (-1+i \sqrt{15}\right )}}\right )}{\sqrt{30 \left (-1+i \sqrt{15}\right )}} \]

Antiderivative was successfully verified.

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

[Out]

((-5*I + Sqrt[15])*ArcTan[(2*x)/Sqrt[(-1 - I*Sqrt[15])/2]])/Sqrt[30*(-1 - I*Sqrt
[15])] + ((5*I + Sqrt[15])*ArcTan[(2*x)/Sqrt[(-1 + I*Sqrt[15])/2]])/Sqrt[30*(-1
+ I*Sqrt[15])]

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Maple [A]  time = 0.036, size = 40, normalized size = 0.9 \[{\frac{\sqrt{3}}{3}\arctan \left ({\frac{ \left ( 4\,x+\sqrt{5} \right ) \sqrt{3}}{3}} \right ) }+{\frac{\sqrt{3}}{3}\arctan \left ({\frac{ \left ( 4\,x-\sqrt{5} \right ) \sqrt{3}}{3}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((2*x^2+1)/(4*x^4-x^2+1),x)

[Out]

1/3*arctan(1/3*(4*x+5^(1/2))*3^(1/2))*3^(1/2)+1/3*3^(1/2)*arctan(1/3*(4*x-5^(1/2
))*3^(1/2))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.272119, size = 35, normalized size = 0.76 \[ \frac{1}{3} \, \sqrt{3}{\left (\arctan \left (\frac{1}{3} \, \sqrt{3}{\left (4 \, x^{3} + x\right )}\right ) + \arctan \left (\frac{2}{3} \, \sqrt{3} x\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3*sqrt(3)*(arctan(1/3*sqrt(3)*(4*x^3 + x)) + arctan(2/3*sqrt(3)*x))

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Sympy [A]  time = 0.250869, size = 42, normalized size = 0.91 \[ \frac{\sqrt{3} \left (2 \operatorname{atan}{\left (\frac{2 \sqrt{3} x}{3} \right )} + 2 \operatorname{atan}{\left (\frac{4 \sqrt{3} x^{3}}{3} + \frac{\sqrt{3} x}{3} \right )}\right )}{6} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sqrt(3)*(2*atan(2*sqrt(3)*x/3) + 2*atan(4*sqrt(3)*x**3/3 + sqrt(3)*x/3))/6

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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