Optimal. Leaf size=136 \[ -\frac{2 \left (1-a^2 x^2\right )^{3/2}}{27 a^3}+\frac{14 \sqrt{1-a^2 x^2}}{9 a^3}-\frac{x^2 \sqrt{1-a^2 x^2} \cos ^{-1}(a x)^2}{3 a}-\frac{2 \sqrt{1-a^2 x^2} \cos ^{-1}(a x)^2}{3 a^3}-\frac{4 x \cos ^{-1}(a x)}{3 a^2}+\frac{1}{3} x^3 \cos ^{-1}(a x)^3-\frac{2}{9} x^3 \cos ^{-1}(a x) \]
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Rubi [A] time = 0.240902, antiderivative size = 136, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 7, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.7, Rules used = {4628, 4708, 4678, 4620, 261, 266, 43} \[ -\frac{2 \left (1-a^2 x^2\right )^{3/2}}{27 a^3}+\frac{14 \sqrt{1-a^2 x^2}}{9 a^3}-\frac{x^2 \sqrt{1-a^2 x^2} \cos ^{-1}(a x)^2}{3 a}-\frac{2 \sqrt{1-a^2 x^2} \cos ^{-1}(a x)^2}{3 a^3}-\frac{4 x \cos ^{-1}(a x)}{3 a^2}+\frac{1}{3} x^3 \cos ^{-1}(a x)^3-\frac{2}{9} x^3 \cos ^{-1}(a x) \]
Antiderivative was successfully verified.
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Rule 4628
Rule 4708
Rule 4678
Rule 4620
Rule 261
Rule 266
Rule 43
Rubi steps
\begin{align*} \int x^2 \cos ^{-1}(a x)^3 \, dx &=\frac{1}{3} x^3 \cos ^{-1}(a x)^3+a \int \frac{x^3 \cos ^{-1}(a x)^2}{\sqrt{1-a^2 x^2}} \, dx\\ &=-\frac{x^2 \sqrt{1-a^2 x^2} \cos ^{-1}(a x)^2}{3 a}+\frac{1}{3} x^3 \cos ^{-1}(a x)^3-\frac{2}{3} \int x^2 \cos ^{-1}(a x) \, dx+\frac{2 \int \frac{x \cos ^{-1}(a x)^2}{\sqrt{1-a^2 x^2}} \, dx}{3 a}\\ &=-\frac{2}{9} x^3 \cos ^{-1}(a x)-\frac{2 \sqrt{1-a^2 x^2} \cos ^{-1}(a x)^2}{3 a^3}-\frac{x^2 \sqrt{1-a^2 x^2} \cos ^{-1}(a x)^2}{3 a}+\frac{1}{3} x^3 \cos ^{-1}(a x)^3-\frac{4 \int \cos ^{-1}(a x) \, dx}{3 a^2}-\frac{1}{9} (2 a) \int \frac{x^3}{\sqrt{1-a^2 x^2}} \, dx\\ &=-\frac{4 x \cos ^{-1}(a x)}{3 a^2}-\frac{2}{9} x^3 \cos ^{-1}(a x)-\frac{2 \sqrt{1-a^2 x^2} \cos ^{-1}(a x)^2}{3 a^3}-\frac{x^2 \sqrt{1-a^2 x^2} \cos ^{-1}(a x)^2}{3 a}+\frac{1}{3} x^3 \cos ^{-1}(a x)^3-\frac{4 \int \frac{x}{\sqrt{1-a^2 x^2}} \, dx}{3 a}-\frac{1}{9} a \operatorname{Subst}\left (\int \frac{x}{\sqrt{1-a^2 x}} \, dx,x,x^2\right )\\ &=\frac{4 \sqrt{1-a^2 x^2}}{3 a^3}-\frac{4 x \cos ^{-1}(a x)}{3 a^2}-\frac{2}{9} x^3 \cos ^{-1}(a x)-\frac{2 \sqrt{1-a^2 x^2} \cos ^{-1}(a x)^2}{3 a^3}-\frac{x^2 \sqrt{1-a^2 x^2} \cos ^{-1}(a x)^2}{3 a}+\frac{1}{3} x^3 \cos ^{-1}(a x)^3-\frac{1}{9} 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{14 \sqrt{1-a^2 x^2}}{9 a^3}-\frac{2 \left (1-a^2 x^2\right )^{3/2}}{27 a^3}-\frac{4 x \cos ^{-1}(a x)}{3 a^2}-\frac{2}{9} x^3 \cos ^{-1}(a x)-\frac{2 \sqrt{1-a^2 x^2} \cos ^{-1}(a x)^2}{3 a^3}-\frac{x^2 \sqrt{1-a^2 x^2} \cos ^{-1}(a x)^2}{3 a}+\frac{1}{3} x^3 \cos ^{-1}(a x)^3\\ \end{align*}
Mathematica [A] time = 0.0496641, size = 95, normalized size = 0.7 \[ \frac{2 \sqrt{1-a^2 x^2} \left (a^2 x^2+20\right )+9 a^3 x^3 \cos ^{-1}(a x)^3-9 \sqrt{1-a^2 x^2} \left (a^2 x^2+2\right ) \cos ^{-1}(a x)^2-6 a x \left (a^2 x^2+6\right ) \cos ^{-1}(a x)}{27 a^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.051, size = 106, normalized size = 0.8 \begin{align*}{\frac{1}{{a}^{3}} \left ({\frac{{a}^{3}{x}^{3} \left ( \arccos \left ( ax \right ) \right ) ^{3}}{3}}-{\frac{ \left ( \arccos \left ( ax \right ) \right ) ^{2} \left ({a}^{2}{x}^{2}+2 \right ) }{3}\sqrt{-{a}^{2}{x}^{2}+1}}+{\frac{4}{3}\sqrt{-{a}^{2}{x}^{2}+1}}-{\frac{4\,ax\arccos \left ( ax \right ) }{3}}-{\frac{2\,{a}^{3}{x}^{3}\arccos \left ( ax \right ) }{9}}+{\frac{2\,{a}^{2}{x}^{2}+4}{27}\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 = 1.51405, size = 162, normalized size = 1.19 \begin{align*} \frac{1}{3} \, x^{3} \arccos \left (a x\right )^{3} - \frac{1}{3} \, a{\left (\frac{\sqrt{-a^{2} x^{2} + 1} x^{2}}{a^{2}} + \frac{2 \, \sqrt{-a^{2} x^{2} + 1}}{a^{4}}\right )} \arccos \left (a x\right )^{2} + \frac{2}{27} \, a{\left (\frac{\sqrt{-a^{2} x^{2} + 1} x^{2} + \frac{20 \, \sqrt{-a^{2} x^{2} + 1}}{a^{2}}}{a^{2}} - \frac{3 \,{\left (a^{2} x^{3} + 6 \, x\right )} \arccos \left (a x\right )}{a^{3}}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.2562, size = 189, normalized size = 1.39 \begin{align*} \frac{9 \, a^{3} x^{3} \arccos \left (a x\right )^{3} - 6 \,{\left (a^{3} x^{3} + 6 \, a x\right )} \arccos \left (a x\right ) +{\left (2 \, a^{2} x^{2} - 9 \,{\left (a^{2} x^{2} + 2\right )} \arccos \left (a x\right )^{2} + 40\right )} \sqrt{-a^{2} x^{2} + 1}}{27 \, a^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 2.45764, size = 134, normalized size = 0.99 \begin{align*} \begin{cases} \frac{x^{3} \operatorname{acos}^{3}{\left (a x \right )}}{3} - \frac{2 x^{3} \operatorname{acos}{\left (a x \right )}}{9} - \frac{x^{2} \sqrt{- a^{2} x^{2} + 1} \operatorname{acos}^{2}{\left (a x \right )}}{3 a} + \frac{2 x^{2} \sqrt{- a^{2} x^{2} + 1}}{27 a} - \frac{4 x \operatorname{acos}{\left (a x \right )}}{3 a^{2}} - \frac{2 \sqrt{- a^{2} x^{2} + 1} \operatorname{acos}^{2}{\left (a x \right )}}{3 a^{3}} + \frac{40 \sqrt{- a^{2} x^{2} + 1}}{27 a^{3}} & \text{for}\: a \neq 0 \\\frac{\pi ^{3} x^{3}}{24} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.15595, size = 158, normalized size = 1.16 \begin{align*} \frac{1}{3} \, x^{3} \arccos \left (a x\right )^{3} - \frac{2}{9} \, x^{3} \arccos \left (a x\right ) - \frac{\sqrt{-a^{2} x^{2} + 1} x^{2} \arccos \left (a x\right )^{2}}{3 \, a} + \frac{2 \, \sqrt{-a^{2} x^{2} + 1} x^{2}}{27 \, a} - \frac{4 \, x \arccos \left (a x\right )}{3 \, a^{2}} - \frac{2 \, \sqrt{-a^{2} x^{2} + 1} \arccos \left (a x\right )^{2}}{3 \, a^{3}} + \frac{40 \, \sqrt{-a^{2} x^{2} + 1}}{27 \, a^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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