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

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

[Out]

x^2/(2*Sqrt[1 + x^4])

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Rubi [A]  time = 0.0110394, antiderivative size = 16, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091 \[ \frac{x^2}{2 \sqrt{x^4+1}} \]

Antiderivative was successfully verified.

[In]  Int[x/(1 + x^4)^(3/2),x]

[Out]

x^2/(2*Sqrt[1 + x^4])

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Rubi in Sympy [A]  time = 1.97324, size = 12, normalized size = 0.75 \[ \frac{x^{2}}{2 \sqrt{x^{4} + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x**2/(2*sqrt(x**4 + 1))

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Mathematica [A]  time = 0.0086853, size = 16, normalized size = 1. \[ \frac{x^2}{2 \sqrt{x^4+1}} \]

Antiderivative was successfully verified.

[In]  Integrate[x/(1 + x^4)^(3/2),x]

[Out]

x^2/(2*Sqrt[1 + x^4])

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Maple [A]  time = 0.003, size = 13, normalized size = 0.8 \[{\frac{{x}^{2}}{2}{\frac{1}{\sqrt{{x}^{4}+1}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x/(x^4+1)^(3/2),x)

[Out]

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

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Maxima [A]  time = 1.41794, size = 16, normalized size = 1. \[ \frac{x^{2}}{2 \, \sqrt{x^{4} + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*x^2/sqrt(x^4 + 1)

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Fricas [A]  time = 0.262134, size = 28, normalized size = 1.75 \[ \frac{1}{2 \,{\left (x^{4} - \sqrt{x^{4} + 1} x^{2} + 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2/(x^4 - sqrt(x^4 + 1)*x^2 + 1)

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Sympy [A]  time = 1.57566, size = 12, normalized size = 0.75 \[ \frac{x^{2}}{2 \sqrt{x^{4} + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x/(x**4+1)**(3/2),x)

[Out]

x**2/(2*sqrt(x**4 + 1))

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GIAC/XCAS [A]  time = 0.228867, size = 16, normalized size = 1. \[ \frac{x^{2}}{2 \, \sqrt{x^{4} + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*x^2/sqrt(x^4 + 1)