3.326 \(\int \frac{-1+x^4}{x^2 \sqrt{1+x^2+x^4}} \, dx\)

Optimal. Leaf size=16 \[ \frac{\sqrt{x^4+x^2+1}}{x} \]

[Out]

Sqrt[1 + x^2 + x^4]/x

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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]

Sqrt[1 + x^2 + x^4]/x

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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]

sqrt(x**4 + x**2 + 1)/x

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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]

Sqrt[1 + x^2 + x^4]/x

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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]

(x^2+x+1)*(x^2-x+1)/(x^4+x^2+1)^(1/2)/x

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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]

sqrt(x^2 + x + 1)*sqrt(x^2 - x + 1)/x

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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]

sqrt(x^4 + x^2 + 1)/x

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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]

Integral((x - 1)*(x + 1)*(x**2 + 1)/(x**2*sqrt((x**2 - x + 1)*(x**2 + x + 1))),
x)

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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]

integrate((x^4 - 1)/(sqrt(x^4 + x^2 + 1)*x^2), x)