3.13 \(\int \frac{2+3 x^2}{4+9 x^4} \, dx\)

Optimal. Leaf size=40 \[ \frac{\tan ^{-1}\left (\sqrt{3} x+1\right )}{2 \sqrt{3}}-\frac{\tan ^{-1}\left (1-\sqrt{3} x\right )}{2 \sqrt{3}} \]

[Out]

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

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Rubi [A]  time = 0.0422042, antiderivative size = 40, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.176 \[ \frac{\tan ^{-1}\left (\sqrt{3} x+1\right )}{2 \sqrt{3}}-\frac{\tan ^{-1}\left (1-\sqrt{3} x\right )}{2 \sqrt{3}} \]

Antiderivative was successfully verified.

[In]  Int[(2 + 3*x^2)/(4 + 9*x^4),x]

[Out]

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

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Rubi in Sympy [A]  time = 5.34746, size = 32, normalized size = 0.8 \[ \frac{\sqrt{3} \operatorname{atan}{\left (\sqrt{3} x - 1 \right )}}{6} + \frac{\sqrt{3} \operatorname{atan}{\left (\sqrt{3} x + 1 \right )}}{6} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((3*x**2+2)/(9*x**4+4),x)

[Out]

sqrt(3)*atan(sqrt(3)*x - 1)/6 + sqrt(3)*atan(sqrt(3)*x + 1)/6

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Mathematica [A]  time = 0.0192236, size = 33, normalized size = 0.82 \[ \frac{\tan ^{-1}\left (\sqrt{3} x+1\right )-\tan ^{-1}\left (1-\sqrt{3} x\right )}{2 \sqrt{3}} \]

Antiderivative was successfully verified.

[In]  Integrate[(2 + 3*x^2)/(4 + 9*x^4),x]

[Out]

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

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Maple [B]  time = 0.009, size = 122, normalized size = 3.1 \[{\frac{\sqrt{6}\sqrt{2}}{12}\arctan \left ({\frac{\sqrt{6}x\sqrt{2}}{2}}-1 \right ) }+{\frac{\sqrt{6}\sqrt{2}}{48}\ln \left ({1 \left ({x}^{2}+{\frac{\sqrt{6}x\sqrt{2}}{3}}+{\frac{2}{3}} \right ) \left ({x}^{2}-{\frac{\sqrt{6}x\sqrt{2}}{3}}+{\frac{2}{3}} \right ) ^{-1}} \right ) }+{\frac{\sqrt{6}\sqrt{2}}{12}\arctan \left ({\frac{\sqrt{6}x\sqrt{2}}{2}}+1 \right ) }+{\frac{\sqrt{6}\sqrt{2}}{48}\ln \left ({1 \left ({x}^{2}-{\frac{\sqrt{6}x\sqrt{2}}{3}}+{\frac{2}{3}} \right ) \left ({x}^{2}+{\frac{\sqrt{6}x\sqrt{2}}{3}}+{\frac{2}{3}} \right ) ^{-1}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((3*x^2+2)/(9*x^4+4),x)

[Out]

1/12*6^(1/2)*2^(1/2)*arctan(1/2*6^(1/2)*x*2^(1/2)-1)+1/48*6^(1/2)*2^(1/2)*ln((x^
2+1/3*6^(1/2)*x*2^(1/2)+2/3)/(x^2-1/3*6^(1/2)*x*2^(1/2)+2/3))+1/12*6^(1/2)*2^(1/
2)*arctan(1/2*6^(1/2)*x*2^(1/2)+1)+1/48*6^(1/2)*2^(1/2)*ln((x^2-1/3*6^(1/2)*x*2^
(1/2)+2/3)/(x^2+1/3*6^(1/2)*x*2^(1/2)+2/3))

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Maxima [A]  time = 0.848098, size = 53, normalized size = 1.32 \[ \frac{1}{6} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (3 \, x + \sqrt{3}\right )}\right ) + \frac{1}{6} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (3 \, x - \sqrt{3}\right )}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((3*x^2 + 2)/(9*x^4 + 4),x, algorithm="maxima")

[Out]

1/6*sqrt(3)*arctan(1/3*sqrt(3)*(3*x + sqrt(3))) + 1/6*sqrt(3)*arctan(1/3*sqrt(3)
*(3*x - sqrt(3)))

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Fricas [A]  time = 0.287986, size = 38, normalized size = 0.95 \[ \frac{1}{6} \, \sqrt{3}{\left (\arctan \left (\frac{1}{4} \, \sqrt{3}{\left (3 \, x^{3} + 2 \, x\right )}\right ) + \arctan \left (\frac{1}{2} \, \sqrt{3} x\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((3*x^2 + 2)/(9*x^4 + 4),x, algorithm="fricas")

[Out]

1/6*sqrt(3)*(arctan(1/4*sqrt(3)*(3*x^3 + 2*x)) + arctan(1/2*sqrt(3)*x))

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Sympy [A]  time = 0.235763, size = 41, normalized size = 1.02 \[ \frac{\sqrt{3} \left (2 \operatorname{atan}{\left (\frac{\sqrt{3} x}{2} \right )} + 2 \operatorname{atan}{\left (\frac{3 \sqrt{3} x^{3}}{4} + \frac{\sqrt{3} x}{2} \right )}\right )}{12} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((3*x**2+2)/(9*x**4+4),x)

[Out]

sqrt(3)*(2*atan(sqrt(3)*x/2) + 2*atan(3*sqrt(3)*x**3/4 + sqrt(3)*x/2))/12

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GIAC/XCAS [A]  time = 0.274163, size = 70, normalized size = 1.75 \[ \frac{1}{6} \, \sqrt{3} \arctan \left (\frac{9}{8} \, \sqrt{2} \left (\frac{4}{9}\right )^{\frac{3}{4}}{\left (2 \, x + \sqrt{2} \left (\frac{4}{9}\right )^{\frac{1}{4}}\right )}\right ) + \frac{1}{6} \, \sqrt{3} \arctan \left (\frac{9}{8} \, \sqrt{2} \left (\frac{4}{9}\right )^{\frac{3}{4}}{\left (2 \, x - \sqrt{2} \left (\frac{4}{9}\right )^{\frac{1}{4}}\right )}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((3*x^2 + 2)/(9*x^4 + 4),x, algorithm="giac")

[Out]

1/6*sqrt(3)*arctan(9/8*sqrt(2)*(4/9)^(3/4)*(2*x + sqrt(2)*(4/9)^(1/4))) + 1/6*sq
rt(3)*arctan(9/8*sqrt(2)*(4/9)^(3/4)*(2*x - sqrt(2)*(4/9)^(1/4)))