Integrand size = 4, antiderivative size = 32 \[ \int x \arcsin (x) \, dx=\frac {1}{4} x \sqrt {1-x^2}-\frac {\arcsin (x)}{4}+\frac {1}{2} x^2 \arcsin (x) \]
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Time = 0.01 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.750, Rules used = {4723, 327, 222} \[ \int x \arcsin (x) \, dx=\frac {1}{2} x^2 \arcsin (x)-\frac {\arcsin (x)}{4}+\frac {1}{4} \sqrt {1-x^2} x \]
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Rule 222
Rule 327
Rule 4723
Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} x^2 \arcsin (x)-\frac {1}{2} \int \frac {x^2}{\sqrt {1-x^2}} \, dx \\ & = \frac {1}{4} x \sqrt {1-x^2}+\frac {1}{2} x^2 \arcsin (x)-\frac {1}{4} \int \frac {1}{\sqrt {1-x^2}} \, dx \\ & = \frac {1}{4} x \sqrt {1-x^2}-\frac {\arcsin (x)}{4}+\frac {1}{2} x^2 \arcsin (x) \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.88 \[ \int x \arcsin (x) \, dx=\frac {1}{4} \left (x \sqrt {1-x^2}+\left (-1+2 x^2\right ) \arcsin (x)\right ) \]
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Time = 0.02 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.78
method | result | size |
default | \(-\frac {\arcsin \left (x \right )}{4}+\frac {x^{2} \arcsin \left (x \right )}{2}+\frac {x \sqrt {-x^{2}+1}}{4}\) | \(25\) |
parts | \(-\frac {\arcsin \left (x \right )}{4}+\frac {x^{2} \arcsin \left (x \right )}{2}+\frac {x \sqrt {-x^{2}+1}}{4}\) | \(25\) |
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Time = 0.27 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.75 \[ \int x \arcsin (x) \, dx=\frac {1}{4} \, {\left (2 \, x^{2} - 1\right )} \arcsin \left (x\right ) + \frac {1}{4} \, \sqrt {-x^{2} + 1} x \]
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Time = 0.09 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.75 \[ \int x \arcsin (x) \, dx=\frac {x^{2} \operatorname {asin}{\left (x \right )}}{2} + \frac {x \sqrt {1 - x^{2}}}{4} - \frac {\operatorname {asin}{\left (x \right )}}{4} \]
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Time = 0.27 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.75 \[ \int x \arcsin (x) \, dx=\frac {1}{2} \, x^{2} \arcsin \left (x\right ) + \frac {1}{4} \, \sqrt {-x^{2} + 1} x - \frac {1}{4} \, \arcsin \left (x\right ) \]
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Time = 0.33 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.81 \[ \int x \arcsin (x) \, dx=\frac {1}{2} \, {\left (x^{2} - 1\right )} \arcsin \left (x\right ) + \frac {1}{4} \, \sqrt {-x^{2} + 1} x + \frac {1}{4} \, \arcsin \left (x\right ) \]
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Time = 0.03 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.75 \[ \int x \arcsin (x) \, dx=\frac {x\,\sqrt {1-x^2}}{4}+\frac {\mathrm {asin}\left (x\right )\,\left (2\,x^2-1\right )}{4} \]
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