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

Optimal. Leaf size=13 \[ -\frac{1}{2 \sqrt{x^4+1}} \]

[Out]

-1/(2*Sqrt[1 + x^4])

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Rubi [A]  time = 0.00698683, antiderivative size = 13, 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 \[ -\frac{1}{2 \sqrt{x^4+1}} \]

Antiderivative was successfully verified.

[In]  Int[x^3/(1 + x^4)^(3/2),x]

[Out]

-1/(2*Sqrt[1 + x^4])

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Rubi in Sympy [A]  time = 1.61938, size = 12, normalized size = 0.92 \[ - \frac{1}{2 \sqrt{x^{4} + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/(2*sqrt(x**4 + 1))

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Mathematica [A]  time = 0.00428745, size = 13, normalized size = 1. \[ -\frac{1}{2 \sqrt{x^4+1}} \]

Antiderivative was successfully verified.

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

[Out]

-1/(2*Sqrt[1 + x^4])

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Maple [A]  time = 0.007, size = 10, normalized size = 0.8 \[ -{\frac{1}{2}{\frac{1}{\sqrt{{x}^{4}+1}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^3/(x^4+1)^(3/2),x)

[Out]

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

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Maxima [A]  time = 1.45428, size = 12, normalized size = 0.92 \[ -\frac{1}{2 \, \sqrt{x^{4} + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2/sqrt(x^4 + 1)

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Fricas [A]  time = 0.258313, size = 12, normalized size = 0.92 \[ -\frac{1}{2 \, \sqrt{x^{4} + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2/sqrt(x^4 + 1)

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Sympy [A]  time = 1.34648, size = 12, normalized size = 0.92 \[ - \frac{1}{2 \sqrt{x^{4} + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/(2*sqrt(x**4 + 1))

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GIAC/XCAS [A]  time = 0.227554, size = 12, normalized size = 0.92 \[ -\frac{1}{2 \, \sqrt{x^{4} + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2/sqrt(x^4 + 1)