3.943 \(\int \frac{1}{x^3 \left (1+x^4\right )^{3/2}} \, dx\)

Optimal. Leaf size=31 \[ -\frac{x^2}{\sqrt{x^4+1}}-\frac{1}{2 \sqrt{x^4+1} x^2} \]

[Out]

-1/(2*x^2*Sqrt[1 + x^4]) - x^2/Sqrt[1 + x^4]

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Rubi [A]  time = 0.0223854, antiderivative size = 31, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154 \[ -\frac{x^2}{\sqrt{x^4+1}}-\frac{1}{2 \sqrt{x^4+1} x^2} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^3*(1 + x^4)^(3/2)),x]

[Out]

-1/(2*x^2*Sqrt[1 + x^4]) - x^2/Sqrt[1 + x^4]

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Rubi in Sympy [A]  time = 2.85681, size = 27, normalized size = 0.87 \[ - \frac{x^{2}}{\sqrt{x^{4} + 1}} - \frac{1}{2 x^{2} \sqrt{x^{4} + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0130547, size = 23, normalized size = 0.74 \[ -\frac{2 x^4+1}{2 x^2 \sqrt{x^4+1}} \]

Antiderivative was successfully verified.

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

[Out]

-(1 + 2*x^4)/(2*x^2*Sqrt[1 + x^4])

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Maple [A]  time = 0.005, size = 20, normalized size = 0.7 \[ -{\frac{2\,{x}^{4}+1}{2\,{x}^{2}}{\frac{1}{\sqrt{{x}^{4}+1}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^3/(x^4+1)^(3/2),x)

[Out]

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

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Maxima [A]  time = 1.46181, size = 34, normalized size = 1.1 \[ -\frac{x^{2}}{2 \, \sqrt{x^{4} + 1}} - \frac{\sqrt{x^{4} + 1}}{2 \, x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((x^4 + 1)^(3/2)*x^3),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.265342, size = 45, normalized size = 1.45 \[ \frac{1}{2 \,{\left (2 \, x^{8} + 2 \, x^{4} -{\left (2 \, x^{6} + x^{2}\right )} \sqrt{x^{4} + 1}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((x^4 + 1)^(3/2)*x^3),x, algorithm="fricas")

[Out]

1/2/(2*x^8 + 2*x^4 - (2*x^6 + x^2)*sqrt(x^4 + 1))

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Sympy [A]  time = 2.3531, size = 42, normalized size = 1.35 \[ - \frac{2 x^{4} \sqrt{x^{4} + 1}}{2 x^{6} + 2 x^{2}} - \frac{\sqrt{x^{4} + 1}}{2 x^{6} + 2 x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*x**4*sqrt(x**4 + 1)/(2*x**6 + 2*x**2) - sqrt(x**4 + 1)/(2*x**6 + 2*x**2)

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GIAC/XCAS [A]  time = 0.233895, size = 30, normalized size = 0.97 \[ -\frac{x^{2}}{2 \, \sqrt{x^{4} + 1}} - \frac{1}{2} \, \sqrt{\frac{1}{x^{4}} + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((x^4 + 1)^(3/2)*x^3),x, algorithm="giac")

[Out]

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