Optimal. Leaf size=23 \[ \tan ^{-1}\left (4 x+\sqrt{7}\right )-\tan ^{-1}\left (\sqrt{7}-4 x\right ) \]
[Out]
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Rubi [A] time = 0.0495436, antiderivative size = 23, 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 \[ \tan ^{-1}\left (4 x+\sqrt{7}\right )-\tan ^{-1}\left (\sqrt{7}-4 x\right ) \]
Antiderivative was successfully verified.
[In] Int[(1 + 2*x^2)/(1 - 3*x^2 + 4*x^4),x]
[Out]
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Rubi in Sympy [A] time = 8.34717, size = 19, normalized size = 0.83 \[ \operatorname{atan}{\left (4 x - \sqrt{7} \right )} + \operatorname{atan}{\left (4 x + \sqrt{7} \right )} \]
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]
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Mathematica [A] time = 0.0112752, size = 14, normalized size = 0.61 \[ -\tan ^{-1}\left (\frac{x}{2 x^2-1}\right ) \]
Antiderivative was successfully verified.
[In] Integrate[(1 + 2*x^2)/(1 - 3*x^2 + 4*x^4),x]
[Out]
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Maple [A] time = 0.035, size = 20, normalized size = 0.9 \[ \arctan \left ( 4\,x-\sqrt{7} \right ) +\arctan \left ( 4\,x+\sqrt{7} \right ) \]
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]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{2 \, x^{2} + 1}{4 \, x^{4} - 3 \, x^{2} + 1}\,{d x} \]
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]
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Fricas [A] time = 0.303061, size = 20, normalized size = 0.87 \[ \arctan \left (4 \, x^{3} - x\right ) + \arctan \left (2 \, x\right ) \]
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]
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Sympy [A] time = 0.211849, size = 12, normalized size = 0.52 \[ \operatorname{atan}{\left (2 x \right )} + \operatorname{atan}{\left (4 x^{3} - x \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((2*x**2+1)/(4*x**4-3*x**2+1),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{2 \, x^{2} + 1}{4 \, x^{4} - 3 \, x^{2} + 1}\,{d x} \]
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]