3.898 \(\int \frac{x^5}{\left (1-x^4\right )^{3/2}} \, dx\)

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

[Out]

x^2/(2*Sqrt[1 - x^4]) - ArcSin[x^2]/2

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Rubi [A]  time = 0.0363181, 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 \[ \frac{x^2}{2 \sqrt{1-x^4}}-\frac{1}{2} \sin ^{-1}\left (x^2\right ) \]

Antiderivative was successfully verified.

[In]  Int[x^5/(1 - x^4)^(3/2),x]

[Out]

x^2/(2*Sqrt[1 - x^4]) - ArcSin[x^2]/2

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Rubi in Sympy [A]  time = 4.93694, size = 19, normalized size = 0.7 \[ \frac{x^{2}}{2 \sqrt{- x^{4} + 1}} - \frac{\operatorname{asin}{\left (x^{2} \right )}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**5/(-x**4+1)**(3/2),x)

[Out]

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

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Mathematica [A]  time = 0.0374265, size = 26, normalized size = 0.96 \[ \frac{1}{2} \left (\frac{x^2}{\sqrt{1-x^4}}-\sin ^{-1}\left (x^2\right )\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[x^5/(1 - x^4)^(3/2),x]

[Out]

(x^2/Sqrt[1 - x^4] - ArcSin[x^2])/2

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Maple [B]  time = 0.021, size = 62, normalized size = 2.3 \[ -{\frac{\arcsin \left ({x}^{2} \right ) }{2}}-{\frac{1}{4\,{x}^{2}+4}\sqrt{- \left ({x}^{2}+1 \right ) ^{2}+2+2\,{x}^{2}}}-{\frac{1}{4\,{x}^{2}-4}\sqrt{- \left ({x}^{2}-1 \right ) ^{2}-2\,{x}^{2}+2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^5/(-x^4+1)^(3/2),x)

[Out]

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

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Maxima [A]  time = 1.63352, size = 42, normalized size = 1.56 \[ \frac{x^{2}}{2 \, \sqrt{-x^{4} + 1}} + \frac{1}{2} \, \arctan \left (\frac{\sqrt{-x^{4} + 1}}{x^{2}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^5/(-x^4 + 1)^(3/2),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.289074, size = 93, normalized size = 3.44 \[ -\frac{\sqrt{-x^{4} + 1} x^{2} - x^{2} - 2 \,{\left (x^{4} + \sqrt{-x^{4} + 1} - 1\right )} \arctan \left (\frac{\sqrt{-x^{4} + 1} - 1}{x^{2}}\right )}{2 \,{\left (x^{4} + \sqrt{-x^{4} + 1} - 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^5/(-x^4 + 1)^(3/2),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 4.86511, size = 46, normalized size = 1.7 \[ \begin{cases} - \frac{i x^{2}}{2 \sqrt{x^{4} - 1}} + \frac{i \operatorname{acosh}{\left (x^{2} \right )}}{2} & \text{for}\: \left |{x^{4}}\right | > 1 \\\frac{x^{2}}{2 \sqrt{- x^{4} + 1}} - \frac{\operatorname{asin}{\left (x^{2} \right )}}{2} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**5/(-x**4+1)**(3/2),x)

[Out]

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

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GIAC/XCAS [A]  time = 0.219655, size = 38, normalized size = 1.41 \[ -\frac{\sqrt{-x^{4} + 1} x^{2}}{2 \,{\left (x^{4} - 1\right )}} - \frac{1}{2} \, \arcsin \left (x^{2}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^5/(-x^4 + 1)^(3/2),x, algorithm="giac")

[Out]

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