Optimal. Leaf size=56 \[ \frac{1}{9} a^3 \left (1-\frac{x^2}{a^2}\right )^{3/2}-\frac{1}{3} a^3 \sqrt{1-\frac{x^2}{a^2}}+\frac{1}{3} x^3 \cos ^{-1}\left (\frac{x}{a}\right ) \]
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Rubi [A] time = 0.0424593, antiderivative size = 56, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4, Rules used = {5264, 4628, 266, 43} \[ \frac{1}{9} a^3 \left (1-\frac{x^2}{a^2}\right )^{3/2}-\frac{1}{3} a^3 \sqrt{1-\frac{x^2}{a^2}}+\frac{1}{3} x^3 \cos ^{-1}\left (\frac{x}{a}\right ) \]
Antiderivative was successfully verified.
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Rule 5264
Rule 4628
Rule 266
Rule 43
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{\operatorname{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{\operatorname{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*}
Mathematica [A] time = 0.0387233, size = 42, normalized size = 0.75 \[ \frac{1}{3} x^3 \sec ^{-1}\left (\frac{a}{x}\right )-\frac{1}{9} a \left (2 a^2+x^2\right ) \sqrt{1-\frac{x^2}{a^2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.175, size = 66, normalized size = 1.2 \begin{align*} -{a}^{3} \left ( -{\frac{{x}^{3}}{3\,{a}^{3}}{\rm arcsec} \left ({\frac{a}{x}}\right )}+{\frac{{x}^{4}}{9\,{a}^{4}} \left ( -1+{\frac{{a}^{2}}{{x}^{2}}} \right ) \left ( 2\,{\frac{{a}^{2}}{{x}^{2}}}+1 \right ){\frac{1}{\sqrt{{\frac{{x}^{2}}{{a}^{2}} \left ( -1+{\frac{{a}^{2}}{{x}^{2}}} \right ) }}}}} \right ) \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 = 2.46903, size = 88, normalized size = 1.57 \begin{align*} \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{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.599257, size = 51, normalized size = 0.91 \begin{align*} \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{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.08763, size = 63, normalized size = 1.12 \begin{align*} \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{align*}
Verification of antiderivative is not currently implemented for this CAS.
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