3.897 \(\int \frac{x^9}{\left (1-x^4\right )^{3/2}} \, dx\)

Optimal. Leaf size=45 \[ -\frac{3}{4} \sin ^{-1}\left (x^2\right )+\frac{x^6}{2 \sqrt{1-x^4}}+\frac{3}{4} \sqrt{1-x^4} x^2 \]

[Out]

x^6/(2*Sqrt[1 - x^4]) + (3*x^2*Sqrt[1 - x^4])/4 - (3*ArcSin[x^2])/4

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Rubi [A]  time = 0.0593844, antiderivative size = 45, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267 \[ -\frac{3}{4} \sin ^{-1}\left (x^2\right )+\frac{x^6}{2 \sqrt{1-x^4}}+\frac{3}{4} \sqrt{1-x^4} x^2 \]

Antiderivative was successfully verified.

[In]  Int[x^9/(1 - x^4)^(3/2),x]

[Out]

x^6/(2*Sqrt[1 - x^4]) + (3*x^2*Sqrt[1 - x^4])/4 - (3*ArcSin[x^2])/4

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Rubi in Sympy [A]  time = 7.22949, size = 36, normalized size = 0.8 \[ \frac{x^{6}}{2 \sqrt{- x^{4} + 1}} + \frac{3 x^{2} \sqrt{- x^{4} + 1}}{4} - \frac{3 \operatorname{asin}{\left (x^{2} \right )}}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**9/(-x**4+1)**(3/2),x)

[Out]

x**6/(2*sqrt(-x**4 + 1)) + 3*x**2*sqrt(-x**4 + 1)/4 - 3*asin(x**2)/4

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Mathematica [A]  time = 0.0463003, size = 32, normalized size = 0.71 \[ -\frac{3}{4} \sin ^{-1}\left (x^2\right )-\frac{\left (x^4-3\right ) x^2}{4 \sqrt{1-x^4}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^9/(1 - x^4)^(3/2),x]

[Out]

-(x^2*(-3 + x^4))/(4*Sqrt[1 - x^4]) - (3*ArcSin[x^2])/4

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Maple [B]  time = 0.023, size = 76, normalized size = 1.7 \[ -{\frac{3\,\arcsin \left ({x}^{2} \right ) }{4}}+{\frac{{x}^{2}}{4}\sqrt{-{x}^{4}+1}}-{\frac{1}{4\,{x}^{2}+4}\sqrt{- \left ({x}^{2}+1 \right ) ^{2}+2+2\,{x}^{2}}}-{\frac{1}{4\,{x}^{2}-4}\sqrt{- \left ({x}^{2}-1 \right ) ^{2}-2\,{x}^{2}+2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^9/(-x^4+1)^(3/2),x)

[Out]

-3/4*arcsin(x^2)+1/4*x^2*(-x^4+1)^(1/2)-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 = 1.59205, size = 81, normalized size = 1.8 \[ -\frac{\frac{3 \,{\left (x^{4} - 1\right )}}{x^{4}} - 2}{4 \,{\left (\frac{\sqrt{-x^{4} + 1}}{x^{2}} + \frac{{\left (-x^{4} + 1\right )}^{\frac{3}{2}}}{x^{6}}\right )}} + \frac{3}{4} \, \arctan \left (\frac{\sqrt{-x^{4} + 1}}{x^{2}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^9/(-x^4 + 1)^(3/2),x, algorithm="maxima")

[Out]

-1/4*(3*(x^4 - 1)/x^4 - 2)/(sqrt(-x^4 + 1)/x^2 + (-x^4 + 1)^(3/2)/x^6) + 3/4*arc
tan(sqrt(-x^4 + 1)/x^2)

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Fricas [A]  time = 0.261884, size = 158, normalized size = 3.51 \[ -\frac{3 \, x^{10} - 13 \, x^{6} + 12 \, x^{2} - 6 \,{\left (x^{8} - 5 \, x^{4} +{\left (3 \, x^{4} - 4\right )} \sqrt{-x^{4} + 1} + 4\right )} \arctan \left (\frac{\sqrt{-x^{4} + 1} - 1}{x^{2}}\right ) -{\left (x^{10} - 7 \, x^{6} + 12 \, x^{2}\right )} \sqrt{-x^{4} + 1}}{4 \,{\left (x^{8} - 5 \, x^{4} +{\left (3 \, x^{4} - 4\right )} \sqrt{-x^{4} + 1} + 4\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^9/(-x^4 + 1)^(3/2),x, algorithm="fricas")

[Out]

-1/4*(3*x^10 - 13*x^6 + 12*x^2 - 6*(x^8 - 5*x^4 + (3*x^4 - 4)*sqrt(-x^4 + 1) + 4
)*arctan((sqrt(-x^4 + 1) - 1)/x^2) - (x^10 - 7*x^6 + 12*x^2)*sqrt(-x^4 + 1))/(x^
8 - 5*x^4 + (3*x^4 - 4)*sqrt(-x^4 + 1) + 4)

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Sympy [A]  time = 9.27667, size = 82, normalized size = 1.82 \[ \begin{cases} \frac{i x^{6}}{4 \sqrt{x^{4} - 1}} - \frac{3 i x^{2}}{4 \sqrt{x^{4} - 1}} + \frac{3 i \operatorname{acosh}{\left (x^{2} \right )}}{4} & \text{for}\: \left |{x^{4}}\right | > 1 \\- \frac{x^{6}}{4 \sqrt{- x^{4} + 1}} + \frac{3 x^{2}}{4 \sqrt{- x^{4} + 1}} - \frac{3 \operatorname{asin}{\left (x^{2} \right )}}{4} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**9/(-x**4+1)**(3/2),x)

[Out]

Piecewise((I*x**6/(4*sqrt(x**4 - 1)) - 3*I*x**2/(4*sqrt(x**4 - 1)) + 3*I*acosh(x
**2)/4, Abs(x**4) > 1), (-x**6/(4*sqrt(-x**4 + 1)) + 3*x**2/(4*sqrt(-x**4 + 1))
- 3*asin(x**2)/4, True))

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GIAC/XCAS [A]  time = 0.2178, size = 45, normalized size = 1. \[ \frac{{\left (x^{4} - 3\right )} \sqrt{-x^{4} + 1} x^{2}}{4 \,{\left (x^{4} - 1\right )}} - \frac{3}{4} \, \arcsin \left (x^{2}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^9/(-x^4 + 1)^(3/2),x, algorithm="giac")

[Out]

1/4*(x^4 - 3)*sqrt(-x^4 + 1)*x^2/(x^4 - 1) - 3/4*arcsin(x^2)