3.877 \(\int \frac{x^5}{\sqrt{1-x^4}} \, dx\)

Optimal. Leaf size=27 \[ \frac{1}{4} \sin ^{-1}\left (x^2\right )-\frac{1}{4} x^2 \sqrt{1-x^4} \]

[Out]

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

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Rubi [A]  time = 0.0377561, antiderivative size = 27, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ \frac{1}{4} \sin ^{-1}\left (x^2\right )-\frac{1}{4} x^2 \sqrt{1-x^4} \]

Antiderivative was successfully verified.

[In]  Int[x^5/Sqrt[1 - x^4],x]

[Out]

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

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Rubi in Sympy [A]  time = 5.17066, size = 19, normalized size = 0.7 \[ - \frac{x^{2} \sqrt{- x^{4} + 1}}{4} + \frac{\operatorname{asin}{\left (x^{2} \right )}}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**5/(-x**4+1)**(1/2),x)

[Out]

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

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Mathematica [A]  time = 0.016131, size = 27, normalized size = 1. \[ \frac{1}{4} \sin ^{-1}\left (x^2\right )-\frac{1}{4} x^2 \sqrt{1-x^4} \]

Antiderivative was successfully verified.

[In]  Integrate[x^5/Sqrt[1 - x^4],x]

[Out]

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

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Maple [A]  time = 0.015, size = 22, normalized size = 0.8 \[{\frac{\arcsin \left ({x}^{2} \right ) }{4}}-{\frac{{x}^{2}}{4}\sqrt{-{x}^{4}+1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^5/(-x^4+1)^(1/2),x)

[Out]

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

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Maxima [A]  time = 1.58951, size = 59, normalized size = 2.19 \[ \frac{\sqrt{-x^{4} + 1}}{4 \, x^{2}{\left (\frac{x^{4} - 1}{x^{4}} - 1\right )}} - \frac{1}{4} \, \arctan \left (\frac{\sqrt{-x^{4} + 1}}{x^{2}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^5/sqrt(-x^4 + 1),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.246919, size = 115, normalized size = 4.26 \[ \frac{2 \, x^{6} - 2 \, x^{2} - 2 \,{\left (x^{4} + 2 \, \sqrt{-x^{4} + 1} - 2\right )} \arctan \left (\frac{\sqrt{-x^{4} + 1} - 1}{x^{2}}\right ) -{\left (x^{6} - 2 \, x^{2}\right )} \sqrt{-x^{4} + 1}}{4 \,{\left (x^{4} + 2 \, \sqrt{-x^{4} + 1} - 2\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^5/sqrt(-x^4 + 1),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 5.91816, size = 61, normalized size = 2.26 \[ \begin{cases} - \frac{i x^{2} \sqrt{x^{4} - 1}}{4} - \frac{i \operatorname{acosh}{\left (x^{2} \right )}}{4} & \text{for}\: \left |{x^{4}}\right | > 1 \\\frac{x^{6}}{4 \sqrt{- x^{4} + 1}} - \frac{x^{2}}{4 \sqrt{- x^{4} + 1}} + \frac{\operatorname{asin}{\left (x^{2} \right )}}{4} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**5/(-x**4+1)**(1/2),x)

[Out]

Piecewise((-I*x**2*sqrt(x**4 - 1)/4 - I*acosh(x**2)/4, Abs(x**4) > 1), (x**6/(4*
sqrt(-x**4 + 1)) - x**2/(4*sqrt(-x**4 + 1)) + asin(x**2)/4, True))

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GIAC/XCAS [A]  time = 0.218245, size = 28, normalized size = 1.04 \[ -\frac{1}{4} \, \sqrt{-x^{4} + 1} x^{2} + \frac{1}{4} \, \arcsin \left (x^{2}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^5/sqrt(-x^4 + 1),x, algorithm="giac")

[Out]

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