Optimal. Leaf size=66 \[ \frac{2}{27} \left (1-x^2\right )^{3/2}+\frac{4 \sqrt{1-x^2}}{9}+\frac{2}{9} x^3 \cos ^{-1}(x)-\frac{1}{3} \left (1-x^2\right )^{3/2} \cos ^{-1}(x)^2-\frac{2}{3} x \cos ^{-1}(x) \]
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Rubi [A] time = 0.0717082, antiderivative size = 66, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.235, Rules used = {4678, 4646, 444, 43} \[ \frac{2}{27} \left (1-x^2\right )^{3/2}+\frac{4 \sqrt{1-x^2}}{9}+\frac{2}{9} x^3 \cos ^{-1}(x)-\frac{1}{3} \left (1-x^2\right )^{3/2} \cos ^{-1}(x)^2-\frac{2}{3} x \cos ^{-1}(x) \]
Antiderivative was successfully verified.
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Rule 4678
Rule 4646
Rule 444
Rule 43
Rubi steps
\begin{align*} \int x \sqrt{1-x^2} \cos ^{-1}(x)^2 \, dx &=-\frac{1}{3} \left (1-x^2\right )^{3/2} \cos ^{-1}(x)^2-\frac{2}{3} \int \left (1-x^2\right ) \cos ^{-1}(x) \, dx\\ &=-\frac{2}{3} x \cos ^{-1}(x)+\frac{2}{9} x^3 \cos ^{-1}(x)-\frac{1}{3} \left (1-x^2\right )^{3/2} \cos ^{-1}(x)^2-\frac{2}{3} \int \frac{x \left (1-\frac{x^2}{3}\right )}{\sqrt{1-x^2}} \, dx\\ &=-\frac{2}{3} x \cos ^{-1}(x)+\frac{2}{9} x^3 \cos ^{-1}(x)-\frac{1}{3} \left (1-x^2\right )^{3/2} \cos ^{-1}(x)^2-\frac{1}{3} \operatorname{Subst}\left (\int \frac{1-\frac{x}{3}}{\sqrt{1-x}} \, dx,x,x^2\right )\\ &=-\frac{2}{3} x \cos ^{-1}(x)+\frac{2}{9} x^3 \cos ^{-1}(x)-\frac{1}{3} \left (1-x^2\right )^{3/2} \cos ^{-1}(x)^2-\frac{1}{3} \operatorname{Subst}\left (\int \left (\frac{2}{3 \sqrt{1-x}}+\frac{\sqrt{1-x}}{3}\right ) \, dx,x,x^2\right )\\ &=\frac{4 \sqrt{1-x^2}}{9}+\frac{2}{27} \left (1-x^2\right )^{3/2}-\frac{2}{3} x \cos ^{-1}(x)+\frac{2}{9} x^3 \cos ^{-1}(x)-\frac{1}{3} \left (1-x^2\right )^{3/2} \cos ^{-1}(x)^2\\ \end{align*}
Mathematica [A] time = 0.0531804, size = 50, normalized size = 0.76 \[ \frac{1}{27} \left (-2 \sqrt{1-x^2} \left (x^2-7\right )-9 \left (1-x^2\right )^{3/2} \cos ^{-1}(x)^2+6 x \left (x^2-3\right ) \cos ^{-1}(x)\right ) \]
Antiderivative was successfully verified.
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Maple [C] time = 0.14, size = 158, normalized size = 2.4 \begin{align*} -{\frac{6\,i\arccos \left ( x \right ) +9\, \left ( \arccos \left ( x \right ) \right ) ^{2}-2}{216} \left ( 4\,i{x}^{3}-4\,\sqrt{-{x}^{2}+1}{x}^{2}-3\,ix+\sqrt{-{x}^{2}+1} \right ) }+{\frac{ \left ( \arccos \left ( x \right ) \right ) ^{2}-2+2\,i\arccos \left ( x \right ) }{8} \left ( ix-\sqrt{-{x}^{2}+1} \right ) }-{\frac{ \left ( \arccos \left ( x \right ) \right ) ^{2}-2-2\,i\arccos \left ( x \right ) }{8} \left ( ix+\sqrt{-{x}^{2}+1} \right ) }+{\frac{-6\,i\arccos \left ( x \right ) +9\, \left ( \arccos \left ( x \right ) \right ) ^{2}-2}{216} \left ( 4\,i{x}^{3}+4\,\sqrt{-{x}^{2}+1}{x}^{2}-3\,ix-\sqrt{-{x}^{2}+1} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.45235, size = 70, normalized size = 1.06 \begin{align*} -\frac{1}{3} \,{\left (-x^{2} + 1\right )}^{\frac{3}{2}} \arccos \left (x\right )^{2} - \frac{2}{27} \, \sqrt{-x^{2} + 1} x^{2} + \frac{2}{9} \,{\left (x^{3} - 3 \, x\right )} \arccos \left (x\right ) + \frac{14}{27} \, \sqrt{-x^{2} + 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.52164, size = 119, normalized size = 1.8 \begin{align*} \frac{2}{9} \,{\left (x^{3} - 3 \, x\right )} \arccos \left (x\right ) + \frac{1}{27} \,{\left (9 \,{\left (x^{2} - 1\right )} \arccos \left (x\right )^{2} - 2 \, x^{2} + 14\right )} \sqrt{-x^{2} + 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 2.94496, size = 78, normalized size = 1.18 \begin{align*} \frac{2 x^{3} \operatorname{acos}{\left (x \right )}}{9} + \frac{x^{2} \sqrt{1 - x^{2}} \operatorname{acos}^{2}{\left (x \right )}}{3} - \frac{2 x^{2} \sqrt{1 - x^{2}}}{27} - \frac{2 x \operatorname{acos}{\left (x \right )}}{3} - \frac{\sqrt{1 - x^{2}} \operatorname{acos}^{2}{\left (x \right )}}{3} + \frac{14 \sqrt{1 - x^{2}}}{27} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.11153, size = 72, normalized size = 1.09 \begin{align*} \frac{2}{9} \, x^{3} \arccos \left (x\right ) - \frac{1}{3} \,{\left (-x^{2} + 1\right )}^{\frac{3}{2}} \arccos \left (x\right )^{2} - \frac{2}{27} \, \sqrt{-x^{2} + 1} x^{2} - \frac{2}{3} \, x \arccos \left (x\right ) + \frac{14}{27} \, \sqrt{-x^{2} + 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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