Optimal. Leaf size=35 \[ \frac{x^3}{2 \sqrt{1-x^4}}+\frac{1}{2} F\left (\left .\sin ^{-1}(x)\right |-1\right )-\frac{1}{2} E\left (\left .\sin ^{-1}(x)\right |-1\right ) \]
[Out]
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Rubi [A] time = 0.0600672, antiderivative size = 35, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333 \[ \frac{x^3}{2 \sqrt{1-x^4}}+\frac{1}{2} F\left (\left .\sin ^{-1}(x)\right |-1\right )-\frac{1}{2} E\left (\left .\sin ^{-1}(x)\right |-1\right ) \]
Antiderivative was successfully verified.
[In] Int[x^2/(1 - x^4)^(3/2),x]
[Out]
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Rubi in Sympy [A] time = 10.9784, size = 29, normalized size = 0.83 \[ \frac{x^{3}}{2 \sqrt{- x^{4} + 1}} - \frac{E\left (\operatorname{asin}{\left (x \right )}\middle | -1\right )}{2} + \frac{F\left (\operatorname{asin}{\left (x \right )}\middle | -1\right )}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**2/(-x**4+1)**(3/2),x)
[Out]
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Mathematica [A] time = 0.0506911, size = 30, normalized size = 0.86 \[ \frac{1}{2} \left (\frac{x^3}{\sqrt{1-x^4}}+F\left (\left .\sin ^{-1}(x)\right |-1\right )-E\left (\left .\sin ^{-1}(x)\right |-1\right )\right ) \]
Antiderivative was successfully verified.
[In] Integrate[x^2/(1 - x^4)^(3/2),x]
[Out]
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Maple [A] time = 0.014, size = 54, normalized size = 1.5 \[{\frac{{x}^{3}}{2}{\frac{1}{\sqrt{-{x}^{4}+1}}}}+{\frac{{\it EllipticF} \left ( x,i \right ) -{\it EllipticE} \left ( x,i \right ) }{2}\sqrt{-{x}^{2}+1}\sqrt{{x}^{2}+1}{\frac{1}{\sqrt{-{x}^{4}+1}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^2/(-x^4+1)^(3/2),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{2}}{{\left (-x^{4} + 1\right )}^{\frac{3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^2/(-x^4 + 1)^(3/2),x, algorithm="maxima")
[Out]
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (-\frac{x^{2}}{{\left (x^{4} - 1\right )} \sqrt{-x^{4} + 1}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^2/(-x^4 + 1)^(3/2),x, algorithm="fricas")
[Out]
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Sympy [A] time = 1.93959, size = 31, normalized size = 0.89 \[ \frac{x^{3} \Gamma \left (\frac{3}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{3}{4}, \frac{3}{2} \\ \frac{7}{4} \end{matrix}\middle |{x^{4} e^{2 i \pi }} \right )}}{4 \Gamma \left (\frac{7}{4}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**2/(-x**4+1)**(3/2),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{2}}{{\left (-x^{4} + 1\right )}^{\frac{3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^2/(-x^4 + 1)^(3/2),x, algorithm="giac")
[Out]