Optimal. Leaf size=23 \[ \frac{\left (x^2-a^2\right )^{3/2}}{3 a^2 x^3} \]
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Rubi [A] time = 0.0040181, 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, Rules used = {264} \[ \frac{\left (x^2-a^2\right )^{3/2}}{3 a^2 x^3} \]
Antiderivative was successfully verified.
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Rule 264
Rubi steps
\begin{align*} \int \frac{\sqrt{-a^2+x^2}}{x^4} \, dx &=\frac{\left (-a^2+x^2\right )^{3/2}}{3 a^2 x^3}\\ \end{align*}
Mathematica [A] time = 0.0044987, size = 23, normalized size = 1. \[ \frac{\left (x^2-a^2\right )^{3/2}}{3 a^2 x^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 28, normalized size = 1.2 \begin{align*} -{\frac{ \left ( a+x \right ) \left ( a-x \right ) }{3\,{a}^{2}{x}^{3}}\sqrt{-{a}^{2}+{x}^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.0589, size = 57, normalized size = 2.48 \begin{align*} \frac{x^{3} +{\left (-a^{2} + x^{2}\right )}^{\frac{3}{2}}}{3 \, a^{2} x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.628714, size = 78, normalized size = 3.39 \begin{align*} \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}\: \frac{\left |{a^{2}}\right |}{\left |{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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.06475, size = 65, normalized size = 2.83 \begin{align*} \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}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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