Optimal. Leaf size=16 \[ \frac{\sqrt{x^4+x^2+1}}{x} \]
[Out]
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Rubi [A] time = 0.0112432, antiderivative size = 16, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.048 \[ \frac{\sqrt{x^4+x^2+1}}{x} \]
Antiderivative was successfully verified.
[In] Int[(-1 + x^4)/(x^2*Sqrt[1 + x^2 + x^4]),x]
[Out]
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Rubi in Sympy [A] time = 5.12579, size = 12, normalized size = 0.75 \[ \frac{\sqrt{x^{4} + x^{2} + 1}}{x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((x**4-1)/x**2/(x**4+x**2+1)**(1/2),x)
[Out]
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Mathematica [A] time = 0.0158107, size = 16, normalized size = 1. \[ \frac{\sqrt{x^4+x^2+1}}{x} \]
Antiderivative was successfully verified.
[In] Integrate[(-1 + x^4)/(x^2*Sqrt[1 + x^2 + x^4]),x]
[Out]
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Maple [A] time = 0.009, size = 29, normalized size = 1.8 \[{\frac{ \left ({x}^{2}+x+1 \right ) \left ({x}^{2}-x+1 \right ) }{x}{\frac{1}{\sqrt{{x}^{4}+{x}^{2}+1}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((x^4-1)/x^2/(x^4+x^2+1)^(1/2),x)
[Out]
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Maxima [A] time = 1.63205, size = 30, normalized size = 1.88 \[ \frac{\sqrt{x^{2} + x + 1} \sqrt{x^{2} - x + 1}}{x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((x^4 - 1)/(sqrt(x^4 + x^2 + 1)*x^2),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.205142, size = 19, normalized size = 1.19 \[ \frac{\sqrt{x^{4} + x^{2} + 1}}{x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((x^4 - 1)/(sqrt(x^4 + x^2 + 1)*x^2),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (x - 1\right ) \left (x + 1\right ) \left (x^{2} + 1\right )}{x^{2} \sqrt{\left (x^{2} - x + 1\right ) \left (x^{2} + x + 1\right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((x**4-1)/x**2/(x**4+x**2+1)**(1/2),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{4} - 1}{\sqrt{x^{4} + x^{2} + 1} x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((x^4 - 1)/(sqrt(x^4 + x^2 + 1)*x^2),x, algorithm="giac")
[Out]