Optimal. Leaf size=54 \[ \frac{\left (1-a^2 x^2\right )^{3/2}}{9 a^3}-\frac{\sqrt{1-a^2 x^2}}{3 a^3}+\frac{1}{3} x^3 \cos ^{-1}(a x) \]
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Rubi [A] time = 0.0346335, antiderivative size = 54, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.375, Rules used = {4628, 266, 43} \[ \frac{\left (1-a^2 x^2\right )^{3/2}}{9 a^3}-\frac{\sqrt{1-a^2 x^2}}{3 a^3}+\frac{1}{3} x^3 \cos ^{-1}(a x) \]
Antiderivative was successfully verified.
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Rule 4628
Rule 266
Rule 43
Rubi steps
\begin{align*} \int x^2 \cos ^{-1}(a x) \, dx &=\frac{1}{3} x^3 \cos ^{-1}(a x)+\frac{1}{3} a \int \frac{x^3}{\sqrt{1-a^2 x^2}} \, dx\\ &=\frac{1}{3} x^3 \cos ^{-1}(a x)+\frac{1}{6} a \operatorname{Subst}\left (\int \frac{x}{\sqrt{1-a^2 x}} \, dx,x,x^2\right )\\ &=\frac{1}{3} x^3 \cos ^{-1}(a x)+\frac{1}{6} a \operatorname{Subst}\left (\int \left (\frac{1}{a^2 \sqrt{1-a^2 x}}-\frac{\sqrt{1-a^2 x}}{a^2}\right ) \, dx,x,x^2\right )\\ &=-\frac{\sqrt{1-a^2 x^2}}{3 a^3}+\frac{\left (1-a^2 x^2\right )^{3/2}}{9 a^3}+\frac{1}{3} x^3 \cos ^{-1}(a x)\\ \end{align*}
Mathematica [A] time = 0.0246064, size = 42, normalized size = 0.78 \[ \frac{1}{3} x^3 \cos ^{-1}(a x)-\frac{\sqrt{1-a^2 x^2} \left (a^2 x^2+2\right )}{9 a^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 52, normalized size = 1. \begin{align*}{\frac{1}{{a}^{3}} \left ({\frac{{a}^{3}{x}^{3}\arccos \left ( ax \right ) }{3}}-{\frac{{a}^{2}{x}^{2}}{9}\sqrt{-{a}^{2}{x}^{2}+1}}-{\frac{2}{9}\sqrt{-{a}^{2}{x}^{2}+1}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.45727, size = 68, normalized size = 1.26 \begin{align*} \frac{1}{3} \, x^{3} \arccos \left (a x\right ) - \frac{1}{9} \, a{\left (\frac{\sqrt{-a^{2} x^{2} + 1} x^{2}}{a^{2}} + \frac{2 \, \sqrt{-a^{2} x^{2} + 1}}{a^{4}}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.89559, size = 92, normalized size = 1.7 \begin{align*} \frac{3 \, a^{3} x^{3} \arccos \left (a x\right ) -{\left (a^{2} x^{2} + 2\right )} \sqrt{-a^{2} x^{2} + 1}}{9 \, a^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.587228, size = 53, normalized size = 0.98 \begin{align*} \begin{cases} \frac{x^{3} \operatorname{acos}{\left (a x \right )}}{3} - \frac{x^{2} \sqrt{- a^{2} x^{2} + 1}}{9 a} - \frac{2 \sqrt{- a^{2} x^{2} + 1}}{9 a^{3}} & \text{for}\: a \neq 0 \\\frac{\pi x^{3}}{6} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.16271, size = 63, normalized size = 1.17 \begin{align*} \frac{1}{3} \, x^{3} \arccos \left (a x\right ) - \frac{\sqrt{-a^{2} x^{2} + 1} x^{2}}{9 \, a} - \frac{2 \, \sqrt{-a^{2} x^{2} + 1}}{9 \, a^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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