Optimal. Leaf size=46 \[ \frac{\tan ^{-1}\left (\frac{4 x+\sqrt{5}}{\sqrt{3}}\right )}{\sqrt{3}}-\frac{\tan ^{-1}\left (\frac{\sqrt{5}-4 x}{\sqrt{3}}\right )}{\sqrt{3}} \]
[Out]
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Rubi [A] time = 0.0779575, antiderivative size = 46, 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 \[ \frac{\tan ^{-1}\left (\frac{4 x+\sqrt{5}}{\sqrt{3}}\right )}{\sqrt{3}}-\frac{\tan ^{-1}\left (\frac{\sqrt{5}-4 x}{\sqrt{3}}\right )}{\sqrt{3}} \]
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 = 8.74153, size = 49, normalized size = 1.07 \[ \frac{\sqrt{3} \operatorname{atan}{\left (\sqrt{3} \left (\frac{4 x}{3} - \frac{\sqrt{5}}{3}\right ) \right )}}{3} + \frac{\sqrt{3} \operatorname{atan}{\left (\sqrt{3} \left (\frac{4 x}{3} + \frac{\sqrt{5}}{3}\right ) \right )}}{3} \]
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 = 0.476232, size = 101, normalized size = 2.2 \[ \frac{\left (\sqrt{15}-5 i\right ) \tan ^{-1}\left (\frac{2 x}{\sqrt{\frac{1}{2} \left (-1-i \sqrt{15}\right )}}\right )}{\sqrt{30 \left (-1-i \sqrt{15}\right )}}+\frac{\left (\sqrt{15}+5 i\right ) \tan ^{-1}\left (\frac{2 x}{\sqrt{\frac{1}{2} \left (-1+i \sqrt{15}\right )}}\right )}{\sqrt{30 \left (-1+i \sqrt{15}\right )}} \]
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 = 0.036, size = 40, normalized size = 0.9 \[{\frac{\sqrt{3}}{3}\arctan \left ({\frac{ \left ( 4\,x+\sqrt{5} \right ) \sqrt{3}}{3}} \right ) }+{\frac{\sqrt{3}}{3}\arctan \left ({\frac{ \left ( 4\,x-\sqrt{5} \right ) \sqrt{3}}{3}} \right ) } \]
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 = 0.272119, size = 35, normalized size = 0.76 \[ \frac{1}{3} \, \sqrt{3}{\left (\arctan \left (\frac{1}{3} \, \sqrt{3}{\left (4 \, x^{3} + x\right )}\right ) + \arctan \left (\frac{2}{3} \, \sqrt{3} x\right )\right )} \]
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 = 0.250869, size = 42, normalized size = 0.91 \[ \frac{\sqrt{3} \left (2 \operatorname{atan}{\left (\frac{2 \sqrt{3} x}{3} \right )} + 2 \operatorname{atan}{\left (\frac{4 \sqrt{3} x^{3}}{3} + \frac{\sqrt{3} x}{3} \right )}\right )}{6} \]
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]