3.133 \(\int \frac{\sqrt{-a^2+x^2}}{x^4} \, dx\)

Optimal. Leaf size=23 \[ \frac{\left (x^2-a^2\right )^{3/2}}{3 a^2 x^3} \]

[Out]

(-a^2 + x^2)^(3/2)/(3*a^2*x^3)

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Rubi [A]  time = 0.0160103, antiderivative size = 23, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.059 \[ \frac{\left (x^2-a^2\right )^{3/2}}{3 a^2 x^3} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[-a^2 + x^2]/x^4,x]

[Out]

(-a^2 + x^2)^(3/2)/(3*a^2*x^3)

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Rubi in Sympy [A]  time = 1.56211, size = 17, normalized size = 0.74 \[ \frac{\left (- a^{2} + x^{2}\right )^{\frac{3}{2}}}{3 a^{2} x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((-a**2+x**2)**(1/2)/x**4,x)

[Out]

(-a**2 + x**2)**(3/2)/(3*a**2*x**3)

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Mathematica [A]  time = 0.0118861, size = 23, normalized size = 1. \[ \frac{\left (x^2-a^2\right )^{3/2}}{3 a^2 x^3} \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[-a^2 + x^2]/x^4,x]

[Out]

(-a^2 + x^2)^(3/2)/(3*a^2*x^3)

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Maple [A]  time = 0.007, size = 28, normalized size = 1.2 \[ -{\frac{ \left ( a+x \right ) \left ( a-x \right ) }{3\,{a}^{2}{x}^{3}}\sqrt{-{a}^{2}+{x}^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((-a^2+x^2)^(1/2)/x^4,x)

[Out]

-1/3/x^3*(a+x)*(a-x)/a^2*(-a^2+x^2)^(1/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(-a^2 + x^2)/x^4,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.200202, size = 115, normalized size = 5. \[ -\frac{a^{4} - 6 \, a^{2} x^{2} + 6 \, x^{4} + 3 \,{\left (a^{2} x - 2 \, x^{3}\right )} \sqrt{-a^{2} + x^{2}}}{3 \,{\left (3 \, a^{2} x^{4} - 4 \, x^{6} -{\left (a^{2} x^{3} - 4 \, x^{5}\right )} \sqrt{-a^{2} + x^{2}}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(-a^2 + x^2)/x^4,x, algorithm="fricas")

[Out]

-1/3*(a^4 - 6*a^2*x^2 + 6*x^4 + 3*(a^2*x - 2*x^3)*sqrt(-a^2 + x^2))/(3*a^2*x^4 -
4*x^6 - (a^2*x^3 - 4*x^5)*sqrt(-a^2 + x^2))

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Sympy [A]  time = 1.03356, size = 76, normalized size = 3.3 \[ \begin{cases} - \frac{i \sqrt{\frac{a^{2}}{x^{2}} - 1}}{3 x^{2}} + \frac{i \sqrt{\frac{a^{2}}{x^{2}} - 1}}{3 a^{2}} & \text{for}\: \left |{\frac{a^{2}}{x^{2}}}\right | > 1 \\- \frac{\sqrt{- \frac{a^{2}}{x^{2}} + 1}}{3 x^{2}} + \frac{\sqrt{- \frac{a^{2}}{x^{2}} + 1}}{3 a^{2}} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-a**2+x**2)**(1/2)/x**4,x)

[Out]

Piecewise((-I*sqrt(a**2/x**2 - 1)/(3*x**2) + I*sqrt(a**2/x**2 - 1)/(3*a**2), Abs
(a**2/x**2) > 1), (-sqrt(-a**2/x**2 + 1)/(3*x**2) + sqrt(-a**2/x**2 + 1)/(3*a**2
), True))

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GIAC/XCAS [A]  time = 0.207051, size = 65, normalized size = 2.83 \[ \frac{2 \,{\left (a^{4} + 3 \,{\left (x - \sqrt{-a^{2} + x^{2}}\right )}^{4}\right )}}{3 \,{\left (a^{2} +{\left (x - \sqrt{-a^{2} + x^{2}}\right )}^{2}\right )}^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(-a^2 + x^2)/x^4,x, algorithm="giac")

[Out]

2/3*(a^4 + 3*(x - sqrt(-a^2 + x^2))^4)/(a^2 + (x - sqrt(-a^2 + x^2))^2)^3