Integrand size = 15, antiderivative size = 37 \[ \int x \left (1-x^2\right )^{3/2} \arcsin (x) \, dx=\frac {x}{5}-\frac {2 x^3}{15}+\frac {x^5}{25}-\frac {1}{5} \left (1-x^2\right )^{5/2} \arcsin (x) \]
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Time = 0.03 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {4767, 200} \[ \int x \left (1-x^2\right )^{3/2} \arcsin (x) \, dx=-\frac {1}{5} \left (1-x^2\right )^{5/2} \arcsin (x)+\frac {x^5}{25}-\frac {2 x^3}{15}+\frac {x}{5} \]
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Rule 200
Rule 4767
Rubi steps \begin{align*} \text {integral}& = -\frac {1}{5} \left (1-x^2\right )^{5/2} \arcsin (x)+\frac {1}{5} \int \left (1-x^2\right )^2 \, dx \\ & = -\frac {1}{5} \left (1-x^2\right )^{5/2} \arcsin (x)+\frac {1}{5} \int \left (1-2 x^2+x^4\right ) \, dx \\ & = \frac {x}{5}-\frac {2 x^3}{15}+\frac {x^5}{25}-\frac {1}{5} \left (1-x^2\right )^{5/2} \arcsin (x) \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.95 \[ \int x \left (1-x^2\right )^{3/2} \arcsin (x) \, dx=\frac {1}{5} \left (x-\frac {2 x^3}{3}+\frac {x^5}{5}-\left (1-x^2\right )^{5/2} \arcsin (x)\right ) \]
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Time = 0.40 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.00
method | result | size |
default | \(-\frac {\left (x^{2}-1\right )^{2} \sqrt {-x^{2}+1}\, \arcsin \left (x \right )}{5}+\frac {\left (3 x^{4}-10 x^{2}+15\right ) x}{75}\) | \(37\) |
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Time = 0.26 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.00 \[ \int x \left (1-x^2\right )^{3/2} \arcsin (x) \, dx=\frac {1}{25} \, x^{5} - \frac {2}{15} \, x^{3} - \frac {1}{5} \, {\left (x^{4} - 2 \, x^{2} + 1\right )} \sqrt {-x^{2} + 1} \arcsin \left (x\right ) + \frac {1}{5} \, x \]
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Leaf count of result is larger than twice the leaf count of optimal. 63 vs. \(2 (27) = 54\).
Time = 0.38 (sec) , antiderivative size = 63, normalized size of antiderivative = 1.70 \[ \int x \left (1-x^2\right )^{3/2} \arcsin (x) \, dx=\frac {x^{5}}{25} - \frac {x^{4} \sqrt {1 - x^{2}} \operatorname {asin}{\left (x \right )}}{5} - \frac {2 x^{3}}{15} + \frac {2 x^{2} \sqrt {1 - x^{2}} \operatorname {asin}{\left (x \right )}}{5} + \frac {x}{5} - \frac {\sqrt {1 - x^{2}} \operatorname {asin}{\left (x \right )}}{5} \]
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Time = 0.29 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.73 \[ \int x \left (1-x^2\right )^{3/2} \arcsin (x) \, dx=\frac {1}{25} \, x^{5} - \frac {1}{5} \, {\left (-x^{2} + 1\right )}^{\frac {5}{2}} \arcsin \left (x\right ) - \frac {2}{15} \, x^{3} + \frac {1}{5} \, x \]
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Time = 0.29 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.92 \[ \int x \left (1-x^2\right )^{3/2} \arcsin (x) \, dx=\frac {1}{25} \, x^{5} - \frac {1}{5} \, {\left (x^{2} - 1\right )}^{2} \sqrt {-x^{2} + 1} \arcsin \left (x\right ) - \frac {2}{15} \, x^{3} + \frac {1}{5} \, x \]
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Timed out. \[ \int x \left (1-x^2\right )^{3/2} \arcsin (x) \, dx=\int x\,\mathrm {asin}\left (x\right )\,{\left (1-x^2\right )}^{3/2} \,d x \]
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