Optimal. Leaf size=49 \[ \frac{\tan ^{-1}\left (\sqrt{\frac{2}{3+\sqrt{5}}} x\right )}{\sqrt{5}}+\frac{\tan ^{-1}\left (\sqrt{\frac{1}{2} \left (3+\sqrt{5}\right )} x\right )}{\sqrt{5}} \]
[Out]
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Rubi [A] time = 0.122879, antiderivative size = 49, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111 \[ \frac{\tan ^{-1}\left (\sqrt{\frac{2}{3+\sqrt{5}}} x\right )}{\sqrt{5}}+\frac{\tan ^{-1}\left (\sqrt{\frac{1}{2} \left (3+\sqrt{5}\right )} x\right )}{\sqrt{5}} \]
Antiderivative was successfully verified.
[In] Int[(1 + x^2)/(1 + 3*x^2 + x^4),x]
[Out]
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Rubi in Sympy [A] time = 8.94797, size = 87, normalized size = 1.78 \[ \frac{\sqrt{2} \left (- \frac{\sqrt{5}}{10} + \frac{1}{2}\right ) \operatorname{atan}{\left (\frac{\sqrt{2} x}{\sqrt{- \sqrt{5} + 3}} \right )}}{\sqrt{- \sqrt{5} + 3}} + \frac{\sqrt{2} \left (\frac{\sqrt{5}}{10} + \frac{1}{2}\right ) \operatorname{atan}{\left (\frac{\sqrt{2} x}{\sqrt{\sqrt{5} + 3}} \right )}}{\sqrt{\sqrt{5} + 3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((x**2+1)/(x**4+3*x**2+1),x)
[Out]
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Mathematica [A] time = 0.164721, size = 83, normalized size = 1.69 \[ \frac{\left (\sqrt{5}-1\right ) \tan ^{-1}\left (\sqrt{\frac{2}{3-\sqrt{5}}} x\right )}{\sqrt{10 \left (3-\sqrt{5}\right )}}+\frac{\left (1+\sqrt{5}\right ) \tan ^{-1}\left (\sqrt{\frac{2}{3+\sqrt{5}}} x\right )}{\sqrt{10 \left (3+\sqrt{5}\right )}} \]
Antiderivative was successfully verified.
[In] Integrate[(1 + x^2)/(1 + 3*x^2 + x^4),x]
[Out]
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Maple [B] time = 0.042, size = 104, normalized size = 2.1 \[{\frac{2\,\sqrt{5}}{10\,\sqrt{5}+10}\arctan \left ( 4\,{\frac{x}{2\,\sqrt{5}+2}} \right ) }+2\,{\frac{1}{2\,\sqrt{5}+2}\arctan \left ( 4\,{\frac{x}{2\,\sqrt{5}+2}} \right ) }-{\frac{2\,\sqrt{5}}{-10+10\,\sqrt{5}}\arctan \left ( 4\,{\frac{x}{-2+2\,\sqrt{5}}} \right ) }+2\,{\frac{1}{-2+2\,\sqrt{5}}\arctan \left ( 4\,{\frac{x}{-2+2\,\sqrt{5}}} \right ) } \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((x^2+1)/(x^4+3*x^2+1),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{2} + 1}{x^{4} + 3 \, x^{2} + 1}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((x^2 + 1)/(x^4 + 3*x^2 + 1),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.280968, size = 35, normalized size = 0.71 \[ \frac{1}{5} \, \sqrt{5}{\left (\arctan \left (\frac{1}{5} \, \sqrt{5}{\left (x^{3} + 4 \, x\right )}\right ) + \arctan \left (\frac{1}{5} \, \sqrt{5} x\right )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((x^2 + 1)/(x^4 + 3*x^2 + 1),x, algorithm="fricas")
[Out]
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Sympy [A] time = 0.237262, size = 41, normalized size = 0.84 \[ \frac{\sqrt{5} \left (2 \operatorname{atan}{\left (\frac{\sqrt{5} x}{5} \right )} + 2 \operatorname{atan}{\left (\frac{\sqrt{5} x^{3}}{5} + \frac{4 \sqrt{5} x}{5} \right )}\right )}{10} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((x**2+1)/(x**4+3*x**2+1),x)
[Out]
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GIAC/XCAS [A] time = 0.2728, size = 35, normalized size = 0.71 \[ \frac{1}{10} \, \sqrt{5}{\left (\pi{\rm sign}\left (x\right ) + 2 \, \arctan \left (\frac{\sqrt{5}{\left (x^{2} - 1\right )}}{5 \, x}\right )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((x^2 + 1)/(x^4 + 3*x^2 + 1),x, algorithm="giac")
[Out]