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 = 0.03, antiderivative size = 54, normalized size of antiderivative = 1.00, 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 43
Rule 266
Rule 4628
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*}
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Mathematica [A] time = 0.03, 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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fricas [A] time = 1.31, size = 41, normalized size = 0.76 \[ \frac {3 \, a^{3} x^{3} \arccos \left (a x\right ) - {\left (a^{2} x^{2} + 2\right )} \sqrt {-a^{2} x^{2} + 1}}{9 \, a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.00, size = 47, normalized size = 0.87 \[ \frac {1}{3} \, x^{3} \arccos \left (a x\right ) - \frac {\sqrt {-a^{2} x^{2} + 1} x^{2}}{9 \, a} - \frac {2 \, \sqrt {-a^{2} x^{2} + 1}}{9 \, a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 52, normalized size = 0.96 \[ \frac {\frac {a^{3} x^{3} \arccos \left (a x \right )}{3}-\frac {a^{2} x^{2} \sqrt {-a^{2} x^{2}+1}}{9}-\frac {2 \sqrt {-a^{2} x^{2}+1}}{9}}{a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.42, size = 50, normalized size = 0.93 \[ \frac {1}{3} \, x^{3} \arccos \left (a x\right ) - \frac {1}{9} \, a {\left (\frac {\sqrt {-a^{2} x^{2} + 1} x^{2}}{a^{2}} + \frac {2 \, \sqrt {-a^{2} x^{2} + 1}}{a^{4}}\right )} \]
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 \[ \left \{\begin {array}{cl} \frac {x^3\,\mathrm {acos}\left (a\,x\right )}{3}-\frac {\sqrt {\frac {1}{a^2}-x^2}\,\left (\frac {2}{a^2}+x^2\right )}{9} & \text {\ if\ \ }0<a\\ \int x^2\,\mathrm {acos}\left (a\,x\right ) \,d x & \text {\ if\ \ }\neg 0<a \end {array}\right . \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.43, size = 53, normalized size = 0.98 \[ \begin {cases} \frac {x^{3} \operatorname {acos}{\left (a x \right )}}{3} - \frac {x^{2} \sqrt {- a^{2} x^{2} + 1}}{9 a} - \frac {2 \sqrt {- a^{2} x^{2} + 1}}{9 a^{3}} & \text {for}\: a \neq 0 \\\frac {\pi x^{3}}{6} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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