Optimal. Leaf size=45 \[ -\frac{5 \sqrt{1-x^4}}{6 x^3}+\frac{1}{2 x^3 \sqrt{1-x^4}}+\frac{5}{6} F\left (\left .\sin ^{-1}(x)\right |-1\right ) \]
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Rubi [A] time = 0.034282, antiderivative size = 45, 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 \[ -\frac{5 \sqrt{1-x^4}}{6 x^3}+\frac{1}{2 x^3 \sqrt{1-x^4}}+\frac{5}{6} F\left (\left .\sin ^{-1}(x)\right |-1\right ) \]
Antiderivative was successfully verified.
[In] Int[1/(x^4*(1 - x^4)^(3/2)),x]
[Out]
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Rubi in Sympy [A] time = 4.41573, size = 39, normalized size = 0.87 \[ \frac{5 F\left (\operatorname{asin}{\left (x \right )}\middle | -1\right )}{6} - \frac{5 \sqrt{- x^{4} + 1}}{6 x^{3}} + \frac{1}{2 x^{3} \sqrt{- x^{4} + 1}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/x**4/(-x**4+1)**(3/2),x)
[Out]
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Mathematica [A] time = 0.0607152, size = 33, normalized size = 0.73 \[ \frac{1}{6} \left (\frac{5 x^4-2}{x^3 \sqrt{1-x^4}}+5 F\left (\left .\sin ^{-1}(x)\right |-1\right )\right ) \]
Antiderivative was successfully verified.
[In] Integrate[1/(x^4*(1 - x^4)^(3/2)),x]
[Out]
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Maple [A] time = 0.02, size = 59, normalized size = 1.3 \[{\frac{x}{2}{\frac{1}{\sqrt{-{x}^{4}+1}}}}-{\frac{1}{3\,{x}^{3}}\sqrt{-{x}^{4}+1}}+{\frac{5\,{\it EllipticF} \left ( x,i \right ) }{6}\sqrt{-{x}^{2}+1}\sqrt{{x}^{2}+1}{\frac{1}{\sqrt{-{x}^{4}+1}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/x^4/(-x^4+1)^(3/2),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (-x^{4} + 1\right )}^{\frac{3}{2}} x^{4}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((-x^4 + 1)^(3/2)*x^4),x, algorithm="maxima")
[Out]
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (-\frac{1}{{\left (x^{8} - x^{4}\right )} \sqrt{-x^{4} + 1}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((-x^4 + 1)^(3/2)*x^4),x, algorithm="fricas")
[Out]
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Sympy [A] time = 2.93578, size = 34, normalized size = 0.76 \[ \frac{\Gamma \left (- \frac{3}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{3}{4}, \frac{3}{2} \\ \frac{1}{4} \end{matrix}\middle |{x^{4} e^{2 i \pi }} \right )}}{4 x^{3} \Gamma \left (\frac{1}{4}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/x**4/(-x**4+1)**(3/2),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (-x^{4} + 1\right )}^{\frac{3}{2}} x^{4}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((-x^4 + 1)^(3/2)*x^4),x, algorithm="giac")
[Out]