3.1539 \(\int \frac{x^4}{\sqrt{1-x^{10}}} \, dx\)

Optimal. Leaf size=8 \[ \frac{1}{5} \sin ^{-1}\left (x^5\right ) \]

[Out]

ArcSin[x^5]/5

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Rubi [A]  time = 0.0182544, antiderivative size = 8, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133 \[ \frac{1}{5} \sin ^{-1}\left (x^5\right ) \]

Antiderivative was successfully verified.

[In]  Int[x^4/Sqrt[1 - x^10],x]

[Out]

ArcSin[x^5]/5

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Rubi in Sympy [A]  time = 2.87604, size = 5, normalized size = 0.62 \[ \frac{\operatorname{asin}{\left (x^{5} \right )}}{5} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**4/(-x**10+1)**(1/2),x)

[Out]

asin(x**5)/5

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Mathematica [A]  time = 0.00984012, size = 8, normalized size = 1. \[ \frac{1}{5} \sin ^{-1}\left (x^5\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[x^4/Sqrt[1 - x^10],x]

[Out]

ArcSin[x^5]/5

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Maple [A]  time = 0.051, size = 7, normalized size = 0.9 \[{\frac{\arcsin \left ({x}^{5} \right ) }{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^4/(-x^10+1)^(1/2),x)

[Out]

1/5*arcsin(x^5)

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Maxima [A]  time = 1.5795, size = 22, normalized size = 2.75 \[ -\frac{1}{5} \, \arctan \left (\frac{\sqrt{-x^{10} + 1}}{x^{5}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^4/sqrt(-x^10 + 1),x, algorithm="maxima")

[Out]

-1/5*arctan(sqrt(-x^10 + 1)/x^5)

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Fricas [A]  time = 0.232778, size = 24, normalized size = 3. \[ -\frac{2}{5} \, \arctan \left (\frac{\sqrt{-x^{10} + 1} - 1}{x^{5}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^4/sqrt(-x^10 + 1),x, algorithm="fricas")

[Out]

-2/5*arctan((sqrt(-x^10 + 1) - 1)/x^5)

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Sympy [A]  time = 3.52375, size = 19, normalized size = 2.38 \[ \begin{cases} - \frac{i \operatorname{acosh}{\left (x^{5} \right )}}{5} & \text{for}\: \left |{x^{10}}\right | > 1 \\\frac{\operatorname{asin}{\left (x^{5} \right )}}{5} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**4/(-x**10+1)**(1/2),x)

[Out]

Piecewise((-I*acosh(x**5)/5, Abs(x**10) > 1), (asin(x**5)/5, True))

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GIAC/XCAS [A]  time = 0.233258, size = 8, normalized size = 1. \[ \frac{1}{5} \, \arcsin \left (x^{5}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^4/sqrt(-x^10 + 1),x, algorithm="giac")

[Out]

1/5*arcsin(x^5)