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

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

[Out]

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

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Rubi [A]  time = 0.0788841, antiderivative size = 46, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15 \[ \frac{\tan ^{-1}\left (\frac{4 x+\sqrt{3}}{\sqrt{5}}\right )}{\sqrt{5}}-\frac{\tan ^{-1}\left (\frac{\sqrt{3}-4 x}{\sqrt{5}}\right )}{\sqrt{5}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 8.5618, size = 49, normalized size = 1.07 \[ \frac{\sqrt{5} \operatorname{atan}{\left (\sqrt{5} \left (\frac{4 x}{5} - \frac{\sqrt{3}}{5}\right ) \right )}}{5} + \frac{\sqrt{5} \operatorname{atan}{\left (\sqrt{5} \left (\frac{4 x}{5} + \frac{\sqrt{3}}{5}\right ) \right )}}{5} \]

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

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Mathematica [C]  time = 0.383883, size = 97, normalized size = 2.11 \[ \frac{\left (\sqrt{15}-3 i\right ) \tan ^{-1}\left (\frac{2 x}{\sqrt{\frac{1}{2} \left (1-i \sqrt{15}\right )}}\right )}{\sqrt{30-30 i \sqrt{15}}}+\frac{\left (\sqrt{15}+3 i\right ) \tan ^{-1}\left (\frac{2 x}{\sqrt{\frac{1}{2} \left (1+i \sqrt{15}\right )}}\right )}{\sqrt{30+30 i \sqrt{15}}} \]

Antiderivative was successfully verified.

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

[Out]

((-3*I + Sqrt[15])*ArcTan[(2*x)/Sqrt[(1 - I*Sqrt[15])/2]])/Sqrt[30 - (30*I)*Sqrt
[15]] + ((3*I + Sqrt[15])*ArcTan[(2*x)/Sqrt[(1 + I*Sqrt[15])/2]])/Sqrt[30 + (30*
I)*Sqrt[15]]

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Maple [A]  time = 0.034, size = 40, normalized size = 0.9 \[{\frac{\sqrt{5}}{5}\arctan \left ({\frac{ \left ( 4\,x+\sqrt{3} \right ) \sqrt{5}}{5}} \right ) }+{\frac{\sqrt{5}}{5}\arctan \left ({\frac{ \left ( 4\,x-\sqrt{3} \right ) \sqrt{5}}{5}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((2*x^2+1)/(4*x^4+x^2+1),x)

[Out]

1/5*arctan(1/5*(4*x+3^(1/2))*5^(1/2))*5^(1/2)+1/5*5^(1/2)*arctan(1/5*(4*x-3^(1/2
))*5^(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 = 0.270759, size = 38, normalized size = 0.83 \[ \frac{1}{5} \, \sqrt{5}{\left (\arctan \left (\frac{1}{5} \, \sqrt{5}{\left (4 \, x^{3} + 3 \, x\right )}\right ) + \arctan \left (\frac{2}{5} \, \sqrt{5} x\right )\right )} \]

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]

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

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Sympy [A]  time = 0.247556, size = 44, normalized size = 0.96 \[ \frac{\sqrt{5} \left (2 \operatorname{atan}{\left (\frac{2 \sqrt{5} x}{5} \right )} + 2 \operatorname{atan}{\left (\frac{4 \sqrt{5} x^{3}}{5} + \frac{3 \sqrt{5} x}{5} \right )}\right )}{10} \]

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

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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)