Optimal. Leaf size=46 \[ \frac{\tan ^{-1}\left (\frac{4 x+\sqrt{3}}{\sqrt{5}}\right )}{\sqrt{5}}-\frac{\tan ^{-1}\left (\frac{\sqrt{3}-4 x}{\sqrt{5}}\right )}{\sqrt{5}} \]
[Out]
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Rubi [A] time = 0.0788841, antiderivative size = 46, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15 \[ \frac{\tan ^{-1}\left (\frac{4 x+\sqrt{3}}{\sqrt{5}}\right )}{\sqrt{5}}-\frac{\tan ^{-1}\left (\frac{\sqrt{3}-4 x}{\sqrt{5}}\right )}{\sqrt{5}} \]
Antiderivative was successfully verified.
[In] Int[(1 + 2*x^2)/(1 + x^2 + 4*x^4),x]
[Out]
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Rubi in Sympy [A] time = 8.5618, size = 49, normalized size = 1.07 \[ \frac{\sqrt{5} \operatorname{atan}{\left (\sqrt{5} \left (\frac{4 x}{5} - \frac{\sqrt{3}}{5}\right ) \right )}}{5} + \frac{\sqrt{5} \operatorname{atan}{\left (\sqrt{5} \left (\frac{4 x}{5} + \frac{\sqrt{3}}{5}\right ) \right )}}{5} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((2*x**2+1)/(4*x**4+x**2+1),x)
[Out]
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Mathematica [C] time = 0.383883, size = 97, normalized size = 2.11 \[ \frac{\left (\sqrt{15}-3 i\right ) \tan ^{-1}\left (\frac{2 x}{\sqrt{\frac{1}{2} \left (1-i \sqrt{15}\right )}}\right )}{\sqrt{30-30 i \sqrt{15}}}+\frac{\left (\sqrt{15}+3 i\right ) \tan ^{-1}\left (\frac{2 x}{\sqrt{\frac{1}{2} \left (1+i \sqrt{15}\right )}}\right )}{\sqrt{30+30 i \sqrt{15}}} \]
Antiderivative was successfully verified.
[In] Integrate[(1 + 2*x^2)/(1 + x^2 + 4*x^4),x]
[Out]
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Maple [A] time = 0.034, size = 40, normalized size = 0.9 \[{\frac{\sqrt{5}}{5}\arctan \left ({\frac{ \left ( 4\,x+\sqrt{3} \right ) \sqrt{5}}{5}} \right ) }+{\frac{\sqrt{5}}{5}\arctan \left ({\frac{ \left ( 4\,x-\sqrt{3} \right ) \sqrt{5}}{5}} \right ) } \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((2*x^2+1)/(4*x^4+x^2+1),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{2 \, x^{2} + 1}{4 \, x^{4} + x^{2} + 1}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((2*x^2 + 1)/(4*x^4 + x^2 + 1),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.270759, size = 38, normalized size = 0.83 \[ \frac{1}{5} \, \sqrt{5}{\left (\arctan \left (\frac{1}{5} \, \sqrt{5}{\left (4 \, x^{3} + 3 \, x\right )}\right ) + \arctan \left (\frac{2}{5} \, \sqrt{5} x\right )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((2*x^2 + 1)/(4*x^4 + x^2 + 1),x, algorithm="fricas")
[Out]
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Sympy [A] time = 0.247556, size = 44, normalized size = 0.96 \[ \frac{\sqrt{5} \left (2 \operatorname{atan}{\left (\frac{2 \sqrt{5} x}{5} \right )} + 2 \operatorname{atan}{\left (\frac{4 \sqrt{5} x^{3}}{5} + \frac{3 \sqrt{5} x}{5} \right )}\right )}{10} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((2*x**2+1)/(4*x**4+x**2+1),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{2 \, x^{2} + 1}{4 \, x^{4} + x^{2} + 1}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((2*x^2 + 1)/(4*x^4 + x^2 + 1),x, algorithm="giac")
[Out]