3.961 \(\int \frac{x}{\sqrt{16-x^4}} \, dx\)

Optimal. Leaf size=12 \[ \frac{1}{2} \sin ^{-1}\left (\frac{x^2}{4}\right ) \]

[Out]

ArcSin[x^2/4]/2

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Rubi [A]  time = 0.014869, antiderivative size = 12, 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 \[ \frac{1}{2} \sin ^{-1}\left (\frac{x^2}{4}\right ) \]

Antiderivative was successfully verified.

[In]  Int[x/Sqrt[16 - x^4],x]

[Out]

ArcSin[x^2/4]/2

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Rubi in Sympy [A]  time = 2.54564, size = 7, normalized size = 0.58 \[ \frac{\operatorname{asin}{\left (\frac{x^{2}}{4} \right )}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x/(-x**4+16)**(1/2),x)

[Out]

asin(x**2/4)/2

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Mathematica [A]  time = 0.00853491, size = 12, normalized size = 1. \[ \frac{1}{2} \sin ^{-1}\left (\frac{x^2}{4}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[x/Sqrt[16 - x^4],x]

[Out]

ArcSin[x^2/4]/2

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Maple [A]  time = 0.01, size = 9, normalized size = 0.8 \[{\frac{1}{2}\arcsin \left ({\frac{{x}^{2}}{4}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x/(-x^4+16)^(1/2),x)

[Out]

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

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Maxima [A]  time = 1.60151, size = 22, normalized size = 1.83 \[ -\frac{1}{2} \, \arctan \left (\frac{\sqrt{-x^{4} + 16}}{x^{2}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x/sqrt(-x^4 + 16),x, algorithm="maxima")

[Out]

-1/2*arctan(sqrt(-x^4 + 16)/x^2)

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Fricas [A]  time = 0.254795, size = 24, normalized size = 2. \[ -\arctan \left (\frac{\sqrt{-x^{4} + 16} - 4}{x^{2}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x/sqrt(-x^4 + 16),x, algorithm="fricas")

[Out]

-arctan((sqrt(-x^4 + 16) - 4)/x^2)

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Sympy [A]  time = 3.3181, size = 24, normalized size = 2. \[ \begin{cases} - \frac{i \operatorname{acosh}{\left (\frac{x^{2}}{4} \right )}}{2} & \text{for}\: \frac{\left |{x^{4}}\right |}{16} > 1 \\\frac{\operatorname{asin}{\left (\frac{x^{2}}{4} \right )}}{2} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x/(-x**4+16)**(1/2),x)

[Out]

Piecewise((-I*acosh(x**2/4)/2, Abs(x**4)/16 > 1), (asin(x**2/4)/2, True))

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GIAC/XCAS [A]  time = 0.221435, size = 11, normalized size = 0.92 \[ \frac{1}{2} \, \arcsin \left (\frac{1}{4} \, x^{2}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x/sqrt(-x^4 + 16),x, algorithm="giac")

[Out]

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