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]
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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]
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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]
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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]
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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]
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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]
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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]
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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]
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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]