3.1391 \(\int \frac{1}{x^4 \sqrt{2+x^6}} \, dx\)

Optimal. Leaf size=16 \[ -\frac{\sqrt{x^6+2}}{6 x^3} \]

[Out]

-Sqrt[2 + x^6]/(6*x^3)

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Rubi [A]  time = 0.0127449, antiderivative size = 16, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077 \[ -\frac{\sqrt{x^6+2}}{6 x^3} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^4*Sqrt[2 + x^6]),x]

[Out]

-Sqrt[2 + x^6]/(6*x^3)

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Rubi in Sympy [A]  time = 2.31973, size = 14, normalized size = 0.88 \[ - \frac{\sqrt{x^{6} + 2}}{6 x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**4/(x**6+2)**(1/2),x)

[Out]

-sqrt(x**6 + 2)/(6*x**3)

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Mathematica [A]  time = 0.00881137, size = 16, normalized size = 1. \[ -\frac{\sqrt{x^6+2}}{6 x^3} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x^4*Sqrt[2 + x^6]),x]

[Out]

-Sqrt[2 + x^6]/(6*x^3)

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Maple [A]  time = 0.006, size = 13, normalized size = 0.8 \[ -{\frac{1}{6\,{x}^{3}}\sqrt{{x}^{6}+2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^4/(x^6+2)^(1/2),x)

[Out]

-1/6*(x^6+2)^(1/2)/x^3

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Maxima [A]  time = 1.42232, size = 16, normalized size = 1. \[ -\frac{\sqrt{x^{6} + 2}}{6 \, x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(x^6 + 2)*x^4),x, algorithm="maxima")

[Out]

-1/6*sqrt(x^6 + 2)/x^3

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Fricas [A]  time = 0.215941, size = 27, normalized size = 1.69 \[ \frac{1}{3 \,{\left (x^{6} - \sqrt{x^{6} + 2} x^{3}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(x^6 + 2)*x^4),x, algorithm="fricas")

[Out]

1/3/(x^6 - sqrt(x^6 + 2)*x^3)

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Sympy [A]  time = 2.02309, size = 12, normalized size = 0.75 \[ - \frac{\sqrt{1 + \frac{2}{x^{6}}}}{6} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x**4/(x**6+2)**(1/2),x)

[Out]

-sqrt(1 + 2/x**6)/6

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GIAC/XCAS [A]  time = 0.224655, size = 27, normalized size = 1.69 \[ -\frac{\sqrt{\frac{2}{x^{6}} + 1}}{6 \,{\rm sign}\left (x\right )} + \frac{1}{6} \,{\rm sign}\left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(x^6 + 2)*x^4),x, algorithm="giac")

[Out]

-1/6*sqrt(2/x^6 + 1)/sign(x) + 1/6*sign(x)