Integrand size = 17, antiderivative size = 34 \[ \int \frac {x^2 \arcsin (x)}{\sqrt {1-x^2}} \, dx=\frac {x^2}{4}-\frac {1}{2} x \sqrt {1-x^2} \arcsin (x)+\frac {\arcsin (x)^2}{4} \]
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Time = 0.04 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {4795, 4737, 30} \[ \int \frac {x^2 \arcsin (x)}{\sqrt {1-x^2}} \, dx=-\frac {1}{2} \sqrt {1-x^2} x \arcsin (x)+\frac {\arcsin (x)^2}{4}+\frac {x^2}{4} \]
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Rule 30
Rule 4737
Rule 4795
Rubi steps \begin{align*} \text {integral}& = -\frac {1}{2} x \sqrt {1-x^2} \arcsin (x)+\frac {\int x \, dx}{2}+\frac {1}{2} \int \frac {\arcsin (x)}{\sqrt {1-x^2}} \, dx \\ & = \frac {x^2}{4}-\frac {1}{2} x \sqrt {1-x^2} \arcsin (x)+\frac {\arcsin (x)^2}{4} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.82 \[ \int \frac {x^2 \arcsin (x)}{\sqrt {1-x^2}} \, dx=\frac {1}{4} \left (x^2-2 x \sqrt {1-x^2} \arcsin (x)+\arcsin (x)^2\right ) \]
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Time = 0.30 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.94
method | result | size |
default | \(\frac {\arcsin \left (x \right ) \left (-x \sqrt {-x^{2}+1}+\arcsin \left (x \right )\right )}{2}-\frac {\arcsin \left (x \right )^{2}}{4}+\frac {x^{2}}{4}\) | \(32\) |
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Time = 0.24 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.76 \[ \int \frac {x^2 \arcsin (x)}{\sqrt {1-x^2}} \, dx=-\frac {1}{2} \, \sqrt {-x^{2} + 1} x \arcsin \left (x\right ) + \frac {1}{4} \, x^{2} + \frac {1}{4} \, \arcsin \left (x\right )^{2} \]
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Time = 0.11 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.76 \[ \int \frac {x^2 \arcsin (x)}{\sqrt {1-x^2}} \, dx=\frac {x^{2}}{4} - \frac {x \sqrt {1 - x^{2}} \operatorname {asin}{\left (x \right )}}{2} + \frac {\operatorname {asin}^{2}{\left (x \right )}}{4} \]
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Time = 0.28 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.94 \[ \int \frac {x^2 \arcsin (x)}{\sqrt {1-x^2}} \, dx=\frac {1}{4} \, x^{2} - \frac {1}{2} \, {\left (\sqrt {-x^{2} + 1} x - \arcsin \left (x\right )\right )} \arcsin \left (x\right ) - \frac {1}{4} \, \arcsin \left (x\right )^{2} \]
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Time = 0.33 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.79 \[ \int \frac {x^2 \arcsin (x)}{\sqrt {1-x^2}} \, dx=-\frac {1}{2} \, \sqrt {-x^{2} + 1} x \arcsin \left (x\right ) + \frac {1}{4} \, x^{2} + \frac {1}{4} \, \arcsin \left (x\right )^{2} - \frac {1}{8} \]
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Timed out. \[ \int \frac {x^2 \arcsin (x)}{\sqrt {1-x^2}} \, dx=\int \frac {x^2\,\mathrm {asin}\left (x\right )}{\sqrt {1-x^2}} \,d x \]
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