Integrand size = 17, antiderivative size = 36 \[ \int \frac {x^3 \arcsin (x)}{\left (1-x^2\right )^{3/2}} \, dx=-x+\frac {\arcsin (x)}{\sqrt {1-x^2}}+\sqrt {1-x^2} \arcsin (x)-\text {arctanh}(x) \]
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Time = 0.06 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.294, Rules used = {272, 45, 4779, 396, 212} \[ \int \frac {x^3 \arcsin (x)}{\left (1-x^2\right )^{3/2}} \, dx=\sqrt {1-x^2} \arcsin (x)+\frac {\arcsin (x)}{\sqrt {1-x^2}}-\text {arctanh}(x)-x \]
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Rule 45
Rule 212
Rule 272
Rule 396
Rule 4779
Rubi steps \begin{align*} \text {integral}& = \frac {\arcsin (x)}{\sqrt {1-x^2}}+\sqrt {1-x^2} \arcsin (x)-\int \frac {2-x^2}{1-x^2} \, dx \\ & = -x+\frac {\arcsin (x)}{\sqrt {1-x^2}}+\sqrt {1-x^2} \arcsin (x)-\int \frac {1}{1-x^2} \, dx \\ & = -x+\frac {\arcsin (x)}{\sqrt {1-x^2}}+\sqrt {1-x^2} \arcsin (x)-\text {arctanh}(x) \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.11 \[ \int \frac {x^3 \arcsin (x)}{\left (1-x^2\right )^{3/2}} \, dx=\frac {1}{2} \left (-2 x-\frac {2 \left (-2+x^2\right ) \arcsin (x)}{\sqrt {1-x^2}}+\log (1-x)-\log (1+x)\right ) \]
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Result contains complex when optimal does not.
Time = 0.63 (sec) , antiderivative size = 102, normalized size of antiderivative = 2.83
method | result | size |
default | \(\frac {\left (\arcsin \left (x \right )+i\right ) \left (i x +\sqrt {-x^{2}+1}\right )}{2}-\frac {\left (-\sqrt {-x^{2}+1}+i x \right ) \left (\arcsin \left (x \right )-i\right )}{2}-\frac {\sqrt {-x^{2}+1}\, \arcsin \left (x \right )}{x^{2}-1}-\ln \left (i x +\sqrt {-x^{2}+1}+i\right )+\ln \left (i x +\sqrt {-x^{2}+1}-i\right )\) | \(102\) |
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none
Time = 0.26 (sec) , antiderivative size = 57, normalized size of antiderivative = 1.58 \[ \int \frac {x^3 \arcsin (x)}{\left (1-x^2\right )^{3/2}} \, dx=-\frac {2 \, x^{3} - 2 \, {\left (x^{2} - 2\right )} \sqrt {-x^{2} + 1} \arcsin \left (x\right ) + {\left (x^{2} - 1\right )} \log \left (x + 1\right ) - {\left (x^{2} - 1\right )} \log \left (x - 1\right ) - 2 \, x}{2 \, {\left (x^{2} - 1\right )}} \]
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Time = 6.02 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.03 \[ \int \frac {x^3 \arcsin (x)}{\left (1-x^2\right )^{3/2}} \, dx=- x - \left (- \sqrt {1 - x^{2}} - \frac {1}{\sqrt {1 - x^{2}}}\right ) \operatorname {asin}{\left (x \right )} + \frac {\log {\left (x - 1 \right )}}{2} - \frac {\log {\left (x + 1 \right )}}{2} \]
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none
Time = 0.28 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.25 \[ \int \frac {x^3 \arcsin (x)}{\left (1-x^2\right )^{3/2}} \, dx=-{\left (\frac {x^{2}}{\sqrt {-x^{2} + 1}} - \frac {2}{\sqrt {-x^{2} + 1}}\right )} \arcsin \left (x\right ) - x - \frac {1}{2} \, \log \left (x + 1\right ) + \frac {1}{2} \, \log \left (x - 1\right ) \]
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none
Time = 0.29 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.11 \[ \int \frac {x^3 \arcsin (x)}{\left (1-x^2\right )^{3/2}} \, dx={\left (\sqrt {-x^{2} + 1} + \frac {1}{\sqrt {-x^{2} + 1}}\right )} \arcsin \left (x\right ) - x - \frac {1}{2} \, \log \left ({\left | x + 1 \right |}\right ) + \frac {1}{2} \, \log \left ({\left | x - 1 \right |}\right ) \]
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Timed out. \[ \int \frac {x^3 \arcsin (x)}{\left (1-x^2\right )^{3/2}} \, dx=\int \frac {x^3\,\mathrm {asin}\left (x\right )}{{\left (1-x^2\right )}^{3/2}} \,d x \]
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