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

Optimal. Leaf size=38 \[ \frac{\tan ^{-1}\left (\frac{4 x+1}{\sqrt{7}}\right )}{\sqrt{7}}-\frac{\tan ^{-1}\left (\frac{1-4 x}{\sqrt{7}}\right )}{\sqrt{7}} \]

[Out]

-(ArcTan[(1 - 4*x)/Sqrt[7]]/Sqrt[7]) + ArcTan[(1 + 4*x)/Sqrt[7]]/Sqrt[7]

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Rubi [A]  time = 0.0708948, antiderivative size = 38, 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 \[ \frac{\tan ^{-1}\left (\frac{4 x+1}{\sqrt{7}}\right )}{\sqrt{7}}-\frac{\tan ^{-1}\left (\frac{1-4 x}{\sqrt{7}}\right )}{\sqrt{7}} \]

Antiderivative was successfully verified.

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

[Out]

-(ArcTan[(1 - 4*x)/Sqrt[7]]/Sqrt[7]) + ArcTan[(1 + 4*x)/Sqrt[7]]/Sqrt[7]

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Rubi in Sympy [A]  time = 8.45441, size = 42, normalized size = 1.11 \[ \frac{\sqrt{7} \operatorname{atan}{\left (\sqrt{7} \left (\frac{4 x}{7} - \frac{1}{7}\right ) \right )}}{7} + \frac{\sqrt{7} \operatorname{atan}{\left (\sqrt{7} \left (\frac{4 x}{7} + \frac{1}{7}\right ) \right )}}{7} \]

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]

sqrt(7)*atan(sqrt(7)*(4*x/7 - 1/7))/7 + sqrt(7)*atan(sqrt(7)*(4*x/7 + 1/7))/7

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Mathematica [C]  time = 0.312761, size = 97, normalized size = 2.55 \[ \frac{\left (\sqrt{7}-i\right ) \tan ^{-1}\left (\frac{2 x}{\sqrt{\frac{1}{2} \left (3-i \sqrt{7}\right )}}\right )}{\sqrt{42-14 i \sqrt{7}}}+\frac{\left (\sqrt{7}+i\right ) \tan ^{-1}\left (\frac{2 x}{\sqrt{\frac{1}{2} \left (3+i \sqrt{7}\right )}}\right )}{\sqrt{42+14 i \sqrt{7}}} \]

Antiderivative was successfully verified.

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

[Out]

((-I + Sqrt[7])*ArcTan[(2*x)/Sqrt[(3 - I*Sqrt[7])/2]])/Sqrt[42 - (14*I)*Sqrt[7]]
+ ((I + Sqrt[7])*ArcTan[(2*x)/Sqrt[(3 + I*Sqrt[7])/2]])/Sqrt[42 + (14*I)*Sqrt[7
]]

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Maple [A]  time = 0.01, size = 34, normalized size = 0.9 \[{\frac{\sqrt{7}}{7}\arctan \left ({\frac{ \left ( 4\,x-1 \right ) \sqrt{7}}{7}} \right ) }+{\frac{\sqrt{7}}{7}\arctan \left ({\frac{ \left ( 1+4\,x \right ) \sqrt{7}}{7}} \right ) } \]

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]

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

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Maxima [A]  time = 0.827622, size = 45, normalized size = 1.18 \[ \frac{1}{7} \, \sqrt{7} \arctan \left (\frac{1}{7} \, \sqrt{7}{\left (4 \, x + 1\right )}\right ) + \frac{1}{7} \, \sqrt{7} \arctan \left (\frac{1}{7} \, \sqrt{7}{\left (4 \, x - 1\right )}\right ) \]

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="maxima")

[Out]

1/7*sqrt(7)*arctan(1/7*sqrt(7)*(4*x + 1)) + 1/7*sqrt(7)*arctan(1/7*sqrt(7)*(4*x
- 1))

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Fricas [A]  time = 0.281389, size = 38, normalized size = 1. \[ \frac{1}{7} \, \sqrt{7}{\left (\arctan \left (\frac{1}{7} \, \sqrt{7}{\left (4 \, x^{3} + 5 \, x\right )}\right ) + \arctan \left (\frac{2}{7} \, \sqrt{7} x\right )\right )} \]

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]

1/7*sqrt(7)*(arctan(1/7*sqrt(7)*(4*x^3 + 5*x)) + arctan(2/7*sqrt(7)*x))

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Sympy [A]  time = 0.250976, size = 44, normalized size = 1.16 \[ \frac{\sqrt{7} \left (2 \operatorname{atan}{\left (\frac{2 \sqrt{7} x}{7} \right )} + 2 \operatorname{atan}{\left (\frac{4 \sqrt{7} x^{3}}{7} + \frac{5 \sqrt{7} x}{7} \right )}\right )}{14} \]

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]

sqrt(7)*(2*atan(2*sqrt(7)*x/7) + 2*atan(4*sqrt(7)*x**3/7 + 5*sqrt(7)*x/7))/14

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GIAC/XCAS [A]  time = 0.269562, size = 45, normalized size = 1.18 \[ \frac{1}{7} \, \sqrt{7} \arctan \left (\frac{1}{7} \, \sqrt{7}{\left (4 \, x + 1\right )}\right ) + \frac{1}{7} \, \sqrt{7} \arctan \left (\frac{1}{7} \, \sqrt{7}{\left (4 \, x - 1\right )}\right ) \]

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="giac")

[Out]

1/7*sqrt(7)*arctan(1/7*sqrt(7)*(4*x + 1)) + 1/7*sqrt(7)*arctan(1/7*sqrt(7)*(4*x
- 1))