Optimal. Leaf size=34 \[ \frac{x}{\sqrt{a^2-x^2}}-\tan ^{-1}\left (\frac{x}{\sqrt{a^2-x^2}}\right ) \]
[Out]
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Rubi [A] time = 0.0219736, antiderivative size = 34, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.176 \[ \frac{x}{\sqrt{a^2-x^2}}-\tan ^{-1}\left (\frac{x}{\sqrt{a^2-x^2}}\right ) \]
Antiderivative was successfully verified.
[In] Int[x^2/(a^2 - x^2)^(3/2),x]
[Out]
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Rubi in Sympy [A] time = 2.11096, size = 24, normalized size = 0.71 \[ \frac{x}{\sqrt{a^{2} - x^{2}}} - \operatorname{atan}{\left (\frac{x}{\sqrt{a^{2} - x^{2}}} \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**2/(a**2-x**2)**(3/2),x)
[Out]
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Mathematica [A] time = 0.0481261, size = 34, normalized size = 1. \[ \frac{x}{\sqrt{a^2-x^2}}-\tan ^{-1}\left (\frac{x}{\sqrt{a^2-x^2}}\right ) \]
Antiderivative was successfully verified.
[In] Integrate[x^2/(a^2 - x^2)^(3/2),x]
[Out]
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Maple [A] time = 0.011, size = 31, normalized size = 0.9 \[ -\arctan \left ({x{\frac{1}{\sqrt{{a}^{2}-{x}^{2}}}}} \right ) +{x{\frac{1}{\sqrt{{a}^{2}-{x}^{2}}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^2/(a^2-x^2)^(3/2),x)
[Out]
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Maxima [A] time = 1.50236, size = 32, normalized size = 0.94 \[ \frac{x}{\sqrt{a^{2} - x^{2}}} - \arcsin \left (\frac{x}{\sqrt{a^{2}}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^2/(a^2 - x^2)^(3/2),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.203253, size = 123, normalized size = 3.62 \[ -\frac{a x - 2 \,{\left (a^{2} - x^{2} - \sqrt{a^{2} - x^{2}} a\right )} \arctan \left (-\frac{a - \sqrt{a^{2} - x^{2}}}{x}\right ) - \sqrt{a^{2} - x^{2}} x}{a^{2} - x^{2} - \sqrt{a^{2} - x^{2}} a} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^2/(a^2 - x^2)^(3/2),x, algorithm="fricas")
[Out]
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Sympy [A] time = 2.6471, size = 49, normalized size = 1.44 \[ \begin{cases} i \operatorname{acosh}{\left (\frac{x}{a} \right )} - \frac{i x}{a \sqrt{-1 + \frac{x^{2}}{a^{2}}}} & \text{for}\: \left |{\frac{x^{2}}{a^{2}}}\right | > 1 \\- \operatorname{asin}{\left (\frac{x}{a} \right )} + \frac{x}{a \sqrt{1 - \frac{x^{2}}{a^{2}}}} & \text{otherwise} \end{cases} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**2/(a**2-x**2)**(3/2),x)
[Out]
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GIAC/XCAS [A] time = 0.221116, size = 32, normalized size = 0.94 \[ -\arcsin \left (\frac{x}{a}\right ){\rm sign}\left (a\right ) + \frac{x}{\sqrt{a^{2} - x^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^2/(a^2 - x^2)^(3/2),x, algorithm="giac")
[Out]