3.922 \(\int \frac{1}{x^3 \sqrt{1+x^4}} \, dx\)

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

[Out]

-Sqrt[1 + x^4]/(2*x^2)

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Rubi [A]  time = 0.0135103, 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 \[ -\frac{\sqrt{x^4+1}}{2 x^2} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^3*Sqrt[1 + x^4]),x]

[Out]

-Sqrt[1 + x^4]/(2*x^2)

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Rubi in Sympy [A]  time = 2.21692, size = 14, normalized size = 0.88 \[ - \frac{\sqrt{x^{4} + 1}}{2 x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**3/(x**4+1)**(1/2),x)

[Out]

-sqrt(x**4 + 1)/(2*x**2)

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Mathematica [A]  time = 0.00840659, size = 16, normalized size = 1. \[ -\frac{\sqrt{x^4+1}}{2 x^2} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x^3*Sqrt[1 + x^4]),x]

[Out]

-Sqrt[1 + x^4]/(2*x^2)

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Maple [A]  time = 0.004, size = 13, normalized size = 0.8 \[ -{\frac{1}{2\,{x}^{2}}\sqrt{{x}^{4}+1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^3/(x^4+1)^(1/2),x)

[Out]

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

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Maxima [A]  time = 1.43921, size = 16, normalized size = 1. \[ -\frac{\sqrt{x^{4} + 1}}{2 \, x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(x^4 + 1)*x^3),x, algorithm="maxima")

[Out]

-1/2*sqrt(x^4 + 1)/x^2

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Fricas [A]  time = 0.25324, size = 27, normalized size = 1.69 \[ \frac{1}{2 \,{\left (x^{4} - \sqrt{x^{4} + 1} x^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(x^4 + 1)*x^3),x, algorithm="fricas")

[Out]

1/2/(x^4 - sqrt(x^4 + 1)*x^2)

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Sympy [A]  time = 1.75109, size = 12, normalized size = 0.75 \[ - \frac{\sqrt{1 + \frac{1}{x^{4}}}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x**3/(x**4+1)**(1/2),x)

[Out]

-sqrt(1 + x**(-4))/2

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GIAC/XCAS [A]  time = 0.233927, size = 12, normalized size = 0.75 \[ -\frac{1}{2} \, \sqrt{\frac{1}{x^{4}} + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(x^4 + 1)*x^3),x, algorithm="giac")

[Out]

-1/2*sqrt(1/x^4 + 1)