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 = 0.00, antiderivative size = 23, normalized size of antiderivative = 1.00, 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*}
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Mathematica [A] time = 0.01, size = 23, normalized size = 1.00 \[ \frac {\left (x^2-a^2\right )^{3/2}}{3 a^2 x^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.40, size = 23, normalized size = 1.00 \[ \frac {x^{3} + {\left (-a^{2} + x^{2}\right )}^{\frac {3}{2}}}{3 \, a^{2} x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.98, size = 48, normalized size = 2.09 \[ \frac {2 \, {\left (a^{4} + 3 \, {\left (x - \sqrt {-a^{2} + x^{2}}\right )}^{4}\right )}}{3 \, {\left (a^{2} + {\left (x - \sqrt {-a^{2} + x^{2}}\right )}^{2}\right )}^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 28, normalized size = 1.22 \[ -\frac {\left (a +x \right ) \left (a -x \right ) \sqrt {-a^{2}+x^{2}}}{3 a^{2} x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.97, size = 19, normalized size = 0.83 \[ \frac {{\left (-a^{2} + x^{2}\right )}^{\frac {3}{2}}}{3 \, a^{2} x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.34, size = 19, normalized size = 0.83 \[ \frac {{\left (x^2-a^2\right )}^{3/2}}{3\,a^2\,x^3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.76, size = 76, normalized size = 3.30 \[ \begin {cases} - \frac {i \sqrt {\frac {a^{2}}{x^{2}} - 1}}{3 x^{2}} + \frac {i \sqrt {\frac {a^{2}}{x^{2}} - 1}}{3 a^{2}} & \text {for}\: \left |{\frac {a^{2}}{x^{2}}}\right | > 1 \\- \frac {\sqrt {- \frac {a^{2}}{x^{2}} + 1}}{3 x^{2}} + \frac {\sqrt {- \frac {a^{2}}{x^{2}} + 1}}{3 a^{2}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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