Optimal. Leaf size=56 \[ -\frac {1}{3} a^3 \sqrt {1-\frac {x^2}{a^2}}+\frac {1}{9} a^3 \left (1-\frac {x^2}{a^2}\right )^{3/2}+\frac {1}{3} x^3 \text {ArcCos}\left (\frac {x}{a}\right ) \]
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Rubi [A]
time = 0.03, antiderivative size = 56, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 4, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {5372, 4724,
272, 45} \begin {gather*} \frac {1}{9} a^3 \left (1-\frac {x^2}{a^2}\right )^{3/2}-\frac {1}{3} a^3 \sqrt {1-\frac {x^2}{a^2}}+\frac {1}{3} x^3 \text {ArcCos}\left (\frac {x}{a}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 272
Rule 4724
Rule 5372
Rubi steps
\begin {align*} \int x^2 \sec ^{-1}\left (\frac {a}{x}\right ) \, dx &=\int x^2 \cos ^{-1}\left (\frac {x}{a}\right ) \, dx\\ &=\frac {1}{3} x^3 \cos ^{-1}\left (\frac {x}{a}\right )+\frac {\int \frac {x^3}{\sqrt {1-\frac {x^2}{a^2}}} \, dx}{3 a}\\ &=\frac {1}{3} x^3 \cos ^{-1}\left (\frac {x}{a}\right )+\frac {\text {Subst}\left (\int \frac {x}{\sqrt {1-\frac {x}{a^2}}} \, dx,x,x^2\right )}{6 a}\\ &=\frac {1}{3} x^3 \cos ^{-1}\left (\frac {x}{a}\right )+\frac {\text {Subst}\left (\int \left (\frac {a^2}{\sqrt {1-\frac {x}{a^2}}}-a^2 \sqrt {1-\frac {x}{a^2}}\right ) \, dx,x,x^2\right )}{6 a}\\ &=-\frac {1}{3} a^3 \sqrt {1-\frac {x^2}{a^2}}+\frac {1}{9} a^3 \left (1-\frac {x^2}{a^2}\right )^{3/2}+\frac {1}{3} x^3 \cos ^{-1}\left (\frac {x}{a}\right )\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 42, normalized size = 0.75 \begin {gather*} -\frac {1}{9} a \left (2 a^2+x^2\right ) \sqrt {1-\frac {x^2}{a^2}}+\frac {1}{3} x^3 \sec ^{-1}\left (\frac {a}{x}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.74, size = 66, normalized size = 1.18
method | result | size |
derivativedivides | \(-a^{3} \left (-\frac {x^{3} \mathrm {arcsec}\left (\frac {a}{x}\right )}{3 a^{3}}+\frac {\left (\frac {a^{2}}{x^{2}}-1\right ) \left (\frac {2 a^{2}}{x^{2}}+1\right ) x^{4}}{9 \sqrt {\frac {\left (\frac {a^{2}}{x^{2}}-1\right ) x^{2}}{a^{2}}}\, a^{4}}\right )\) | \(66\) |
default | \(-a^{3} \left (-\frac {x^{3} \mathrm {arcsec}\left (\frac {a}{x}\right )}{3 a^{3}}+\frac {\left (\frac {a^{2}}{x^{2}}-1\right ) \left (\frac {2 a^{2}}{x^{2}}+1\right ) x^{4}}{9 \sqrt {\frac {\left (\frac {a^{2}}{x^{2}}-1\right ) x^{2}}{a^{2}}}\, a^{4}}\right )\) | \(66\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.47, size = 54, normalized size = 0.96 \begin {gather*} \frac {1}{3} \, x^{3} \operatorname {arcsec}\left (\frac {a}{x}\right ) - \frac {2 \, a^{4} \sqrt {-\frac {x^{2}}{a^{2}} + 1} + a^{2} x^{2} \sqrt {-\frac {x^{2}}{a^{2}} + 1}}{9 \, a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.46, size = 39, normalized size = 0.70 \begin {gather*} \frac {1}{3} \, x^{3} \operatorname {arcsec}\left (\frac {a}{x}\right ) - \frac {1}{9} \, {\left (2 \, a^{2} x + x^{3}\right )} \sqrt {\frac {a^{2} - x^{2}}{x^{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.14, size = 51, normalized size = 0.91 \begin {gather*} \begin {cases} - \frac {2 a^{3} \sqrt {1 - \frac {x^{2}}{a^{2}}}}{9} - \frac {a x^{2} \sqrt {1 - \frac {x^{2}}{a^{2}}}}{9} + \frac {x^{3} \operatorname {asec}{\left (\frac {a}{x} \right )}}{3} & \text {for}\: a \neq 0 \\\tilde {\infty } x^{3} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.61, size = 47, normalized size = 0.84 \begin {gather*} \frac {1}{3} \, x^{3} \arccos \left (\frac {x}{a}\right ) - \frac {2}{9} \, a^{3} \sqrt {-\frac {x^{2}}{a^{2}} + 1} - \frac {1}{9} \, a x^{2} \sqrt {-\frac {x^{2}}{a^{2}} + 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \left \{\begin {array}{cl} \frac {x^3\,\mathrm {acos}\left (\frac {x}{a}\right )}{3}-\frac {\sqrt {a^2-x^2}\,\left (2\,a^2+x^2\right )}{9} & \text {\ if\ \ }0<a\\ \int x^2\,\mathrm {acos}\left (\frac {x}{a}\right ) \,d x & \text {\ if\ \ }\neg 0<a \end {array}\right . \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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