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

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

[Out]

Sqrt[1 + x^4]/2

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Rubi [A]  time = 0.00693403, 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 \[ \frac{\sqrt{x^4+1}}{2} \]

Antiderivative was successfully verified.

[In]  Int[x^3/Sqrt[1 + x^4],x]

[Out]

Sqrt[1 + x^4]/2

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Rubi in Sympy [A]  time = 1.62751, size = 8, normalized size = 0.62 \[ \frac{\sqrt{x^{4} + 1}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sqrt(x**4 + 1)/2

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Mathematica [A]  time = 0.00453576, size = 13, normalized size = 1. \[ \frac{\sqrt{x^4+1}}{2} \]

Antiderivative was successfully verified.

[In]  Integrate[x^3/Sqrt[1 + x^4],x]

[Out]

Sqrt[1 + x^4]/2

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Maple [A]  time = 0.006, size = 10, normalized size = 0.8 \[{\frac{1}{2}\sqrt{{x}^{4}+1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^3/(x^4+1)^(1/2),x)

[Out]

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

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Maxima [A]  time = 1.43518, 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/sqrt(x^4 + 1),x, algorithm="maxima")

[Out]

1/2*sqrt(x^4 + 1)

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Fricas [A]  time = 0.261994, 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/sqrt(x^4 + 1),x, algorithm="fricas")

[Out]

1/2*sqrt(x^4 + 1)

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Sympy [A]  time = 0.378164, size = 8, normalized size = 0.62 \[ \frac{\sqrt{x^{4} + 1}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sqrt(x**4 + 1)/2

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GIAC/XCAS [A]  time = 0.210733, 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/sqrt(x^4 + 1),x, algorithm="giac")

[Out]

1/2*sqrt(x^4 + 1)