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]
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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]
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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]
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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]
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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]
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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]
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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]
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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]
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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]