Integrand size = 17, antiderivative size = 66 \[ \int x \sqrt {1-x^2} \arccos (x)^2 \, dx=\frac {4 \sqrt {1-x^2}}{9}+\frac {2}{27} \left (1-x^2\right )^{3/2}-\frac {2}{3} x \arccos (x)+\frac {2}{9} x^3 \arccos (x)-\frac {1}{3} \left (1-x^2\right )^{3/2} \arccos (x)^2 \]
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Time = 0.06 (sec) , antiderivative size = 66, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.235, Rules used = {4768, 4740, 455, 45} \[ \int x \sqrt {1-x^2} \arccos (x)^2 \, dx=\frac {2}{9} x^3 \arccos (x)-\frac {1}{3} \left (1-x^2\right )^{3/2} \arccos (x)^2-\frac {2}{3} x \arccos (x)+\frac {2}{27} \left (1-x^2\right )^{3/2}+\frac {4 \sqrt {1-x^2}}{9} \]
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Rule 45
Rule 455
Rule 4740
Rule 4768
Rubi steps \begin{align*} \text {integral}& = -\frac {1}{3} \left (1-x^2\right )^{3/2} \arccos (x)^2-\frac {2}{3} \int \left (1-x^2\right ) \arccos (x) \, dx \\ & = -\frac {2}{3} x \arccos (x)+\frac {2}{9} x^3 \arccos (x)-\frac {1}{3} \left (1-x^2\right )^{3/2} \arccos (x)^2-\frac {2}{3} \int \frac {x \left (1-\frac {x^2}{3}\right )}{\sqrt {1-x^2}} \, dx \\ & = -\frac {2}{3} x \arccos (x)+\frac {2}{9} x^3 \arccos (x)-\frac {1}{3} \left (1-x^2\right )^{3/2} \arccos (x)^2-\frac {1}{3} \text {Subst}\left (\int \frac {1-\frac {x}{3}}{\sqrt {1-x}} \, dx,x,x^2\right ) \\ & = -\frac {2}{3} x \arccos (x)+\frac {2}{9} x^3 \arccos (x)-\frac {1}{3} \left (1-x^2\right )^{3/2} \arccos (x)^2-\frac {1}{3} \text {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 \arccos (x)+\frac {2}{9} x^3 \arccos (x)-\frac {1}{3} \left (1-x^2\right )^{3/2} \arccos (x)^2 \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.76 \[ \int x \sqrt {1-x^2} \arccos (x)^2 \, dx=\frac {1}{27} \left (-2 \sqrt {1-x^2} \left (-7+x^2\right )+6 x \left (-3+x^2\right ) \arccos (x)-9 \left (1-x^2\right )^{3/2} \arccos (x)^2\right ) \]
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Time = 0.47 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.89
method | result | size |
default | \(\frac {\left (x^{2}-1\right ) \sqrt {-x^{2}+1}\, \arccos \left (x \right )^{2}}{3}+\frac {2 \arccos \left (x \right ) \left (x^{2}-3\right ) x}{9}-\frac {2 \left (x^{2}-1\right ) \sqrt {-x^{2}+1}}{27}+\frac {4 \sqrt {-x^{2}+1}}{9}\) | \(59\) |
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Time = 0.25 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.62 \[ \int x \sqrt {1-x^2} \arccos (x)^2 \, dx=\frac {2}{9} \, {\left (x^{3} - 3 \, x\right )} \arccos \left (x\right ) + \frac {1}{27} \, {\left (9 \, {\left (x^{2} - 1\right )} \arccos \left (x\right )^{2} - 2 \, x^{2} + 14\right )} \sqrt {-x^{2} + 1} \]
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Time = 0.31 (sec) , antiderivative size = 78, normalized size of antiderivative = 1.18 \[ \int x \sqrt {1-x^2} \arccos (x)^2 \, dx=\frac {2 x^{3} \operatorname {acos}{\left (x \right )}}{9} + \frac {x^{2} \sqrt {1 - x^{2}} \operatorname {acos}^{2}{\left (x \right )}}{3} - \frac {2 x^{2} \sqrt {1 - x^{2}}}{27} - \frac {2 x \operatorname {acos}{\left (x \right )}}{3} - \frac {\sqrt {1 - x^{2}} \operatorname {acos}^{2}{\left (x \right )}}{3} + \frac {14 \sqrt {1 - x^{2}}}{27} \]
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Time = 0.28 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.79 \[ \int x \sqrt {1-x^2} \arccos (x)^2 \, dx=-\frac {1}{3} \, {\left (-x^{2} + 1\right )}^{\frac {3}{2}} \arccos \left (x\right )^{2} - \frac {2}{27} \, \sqrt {-x^{2} + 1} x^{2} + \frac {2}{9} \, {\left (x^{3} - 3 \, x\right )} \arccos \left (x\right ) + \frac {14}{27} \, \sqrt {-x^{2} + 1} \]
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Time = 0.30 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.80 \[ \int x \sqrt {1-x^2} \arccos (x)^2 \, dx=\frac {2}{9} \, x^{3} \arccos \left (x\right ) - \frac {1}{3} \, {\left (-x^{2} + 1\right )}^{\frac {3}{2}} \arccos \left (x\right )^{2} - \frac {2}{27} \, \sqrt {-x^{2} + 1} x^{2} - \frac {2}{3} \, x \arccos \left (x\right ) + \frac {14}{27} \, \sqrt {-x^{2} + 1} \]
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Timed out. \[ \int x \sqrt {1-x^2} \arccos (x)^2 \, dx=\int x\,{\mathrm {acos}\left (x\right )}^2\,\sqrt {1-x^2} \,d x \]
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