3.136 \(\int \frac{x^2}{\left (a^2-x^2\right )^{3/2}} \, dx\)

Optimal. Leaf size=34 \[ \frac{x}{\sqrt{a^2-x^2}}-\tan ^{-1}\left (\frac{x}{\sqrt{a^2-x^2}}\right ) \]

[Out]

x/Sqrt[a^2 - x^2] - ArcTan[x/Sqrt[a^2 - x^2]]

_______________________________________________________________________________________

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]

x/Sqrt[a^2 - x^2] - ArcTan[x/Sqrt[a^2 - x^2]]

_______________________________________________________________________________________

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]

x/sqrt(a**2 - x**2) - atan(x/sqrt(a**2 - x**2))

_______________________________________________________________________________________

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]

x/Sqrt[a^2 - x^2] - ArcTan[x/Sqrt[a^2 - x^2]]

_______________________________________________________________________________________

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]

-arctan(x/(a^2-x^2)^(1/2))+x/(a^2-x^2)^(1/2)

_______________________________________________________________________________________

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]

x/sqrt(a^2 - x^2) - arcsin(x/sqrt(a^2))

_______________________________________________________________________________________

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]

-(a*x - 2*(a^2 - x^2 - sqrt(a^2 - x^2)*a)*arctan(-(a - sqrt(a^2 - x^2))/x) - sqr
t(a^2 - x^2)*x)/(a^2 - x^2 - sqrt(a^2 - x^2)*a)

_______________________________________________________________________________________

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]

Piecewise((I*acosh(x/a) - I*x/(a*sqrt(-1 + x**2/a**2)), Abs(x**2/a**2) > 1), (-a
sin(x/a) + x/(a*sqrt(1 - x**2/a**2)), True))

_______________________________________________________________________________________

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]

-arcsin(x/a)*sign(a) + x/sqrt(a^2 - x^2)