Optimal. Leaf size=35 \[ -\frac{x}{\sqrt{1-x^2}}+\frac{x^3}{3 \left (1-x^2\right )^{3/2}}+\sin ^{-1}(x) \]
[Out]
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Rubi [A] time = 0.0289613, antiderivative size = 35, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133 \[ -\frac{x}{\sqrt{1-x^2}}+\frac{x^3}{3 \left (1-x^2\right )^{3/2}}+\sin ^{-1}(x) \]
Antiderivative was successfully verified.
[In] Int[x^4/(1 - x^2)^(5/2),x]
[Out]
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Rubi in Sympy [A] time = 2.56076, size = 26, normalized size = 0.74 \[ \frac{x^{3}}{3 \left (- x^{2} + 1\right )^{\frac{3}{2}}} - \frac{x}{\sqrt{- x^{2} + 1}} + \operatorname{asin}{\left (x \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**4/(-x**2+1)**(5/2),x)
[Out]
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Mathematica [A] time = 0.0440217, size = 26, normalized size = 0.74 \[ \frac{x \left (4 x^2-3\right )}{3 \left (1-x^2\right )^{3/2}}+\sin ^{-1}(x) \]
Antiderivative was successfully verified.
[In] Integrate[x^4/(1 - x^2)^(5/2),x]
[Out]
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Maple [A] time = 0., size = 30, normalized size = 0.9 \[{\frac{{x}^{3}}{3} \left ( -{x}^{2}+1 \right ) ^{-{\frac{3}{2}}}}+\arcsin \left ( x \right ) -{x{\frac{1}{\sqrt{-{x}^{2}+1}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^4/(-x^2+1)^(5/2),x)
[Out]
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Maxima [A] time = 1.47949, size = 59, normalized size = 1.69 \[ \frac{1}{3} \, x{\left (\frac{3 \, x^{2}}{{\left (-x^{2} + 1\right )}^{\frac{3}{2}}} - \frac{2}{{\left (-x^{2} + 1\right )}^{\frac{3}{2}}}\right )} - \frac{x}{3 \, \sqrt{-x^{2} + 1}} + \arcsin \left (x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^4/(-x^2 + 1)^(5/2),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.212308, size = 182, normalized size = 5.2 \[ -\frac{12 \, x^{5} - 25 \, x^{3} + 6 \,{\left (x^{6} - 6 \, x^{4} + 9 \, x^{2} +{\left (3 \, x^{4} - 7 \, x^{2} + 4\right )} \sqrt{-x^{2} + 1} - 4\right )} \arctan \left (\frac{\sqrt{-x^{2} + 1} - 1}{x}\right ) -{\left (4 \, x^{5} - 19 \, x^{3} + 12 \, x\right )} \sqrt{-x^{2} + 1} + 12 \, x}{3 \,{\left (x^{6} - 6 \, x^{4} + 9 \, x^{2} +{\left (3 \, x^{4} - 7 \, x^{2} + 4\right )} \sqrt{-x^{2} + 1} - 4\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^4/(-x^2 + 1)^(5/2),x, algorithm="fricas")
[Out]
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Sympy [A] time = 4.07598, size = 105, normalized size = 3. \[ \frac{3 x^{4} \operatorname{asin}{\left (x \right )}}{3 x^{4} - 6 x^{2} + 3} + \frac{4 x^{3} \sqrt{- x^{2} + 1}}{3 x^{4} - 6 x^{2} + 3} - \frac{6 x^{2} \operatorname{asin}{\left (x \right )}}{3 x^{4} - 6 x^{2} + 3} - \frac{3 x \sqrt{- x^{2} + 1}}{3 x^{4} - 6 x^{2} + 3} + \frac{3 \operatorname{asin}{\left (x \right )}}{3 x^{4} - 6 x^{2} + 3} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**4/(-x**2+1)**(5/2),x)
[Out]
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GIAC/XCAS [A] time = 0.207138, size = 39, normalized size = 1.11 \[ \frac{{\left (4 \, x^{2} - 3\right )} \sqrt{-x^{2} + 1} x}{3 \,{\left (x^{2} - 1\right )}^{2}} + \arcsin \left (x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^4/(-x^2 + 1)^(5/2),x, algorithm="giac")
[Out]