Integrand size = 19, antiderivative size = 73 \[ \int \frac {x^2 \arcsin (x)^3}{\sqrt {1-x^2}} \, dx=-\frac {3 x^2}{8}+\frac {3}{4} x \sqrt {1-x^2} \arcsin (x)-\frac {3 \arcsin (x)^2}{8}+\frac {3}{4} x^2 \arcsin (x)^2-\frac {1}{2} x \sqrt {1-x^2} \arcsin (x)^3+\frac {\arcsin (x)^4}{8} \]
[Out]
Time = 0.12 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.211, Rules used = {4795, 4737, 4723, 30} \[ \int \frac {x^2 \arcsin (x)^3}{\sqrt {1-x^2}} \, dx=-\frac {1}{2} x \sqrt {1-x^2} \arcsin (x)^3+\frac {3}{4} x^2 \arcsin (x)^2+\frac {3}{4} x \sqrt {1-x^2} \arcsin (x)+\frac {\arcsin (x)^4}{8}-\frac {3 \arcsin (x)^2}{8}-\frac {3 x^2}{8} \]
[In]
[Out]
Rule 30
Rule 4723
Rule 4737
Rule 4795
Rubi steps \begin{align*} \text {integral}& = -\frac {1}{2} x \sqrt {1-x^2} \arcsin (x)^3+\frac {1}{2} \int \frac {\arcsin (x)^3}{\sqrt {1-x^2}} \, dx+\frac {3}{2} \int x \arcsin (x)^2 \, dx \\ & = \frac {3}{4} x^2 \arcsin (x)^2-\frac {1}{2} x \sqrt {1-x^2} \arcsin (x)^3+\frac {\arcsin (x)^4}{8}-\frac {3}{2} \int \frac {x^2 \arcsin (x)}{\sqrt {1-x^2}} \, dx \\ & = \frac {3}{4} x \sqrt {1-x^2} \arcsin (x)+\frac {3}{4} x^2 \arcsin (x)^2-\frac {1}{2} x \sqrt {1-x^2} \arcsin (x)^3+\frac {\arcsin (x)^4}{8}-\frac {3 \int x \, dx}{4}-\frac {3}{4} \int \frac {\arcsin (x)}{\sqrt {1-x^2}} \, dx \\ & = -\frac {3 x^2}{8}+\frac {3}{4} x \sqrt {1-x^2} \arcsin (x)-\frac {3 \arcsin (x)^2}{8}+\frac {3}{4} x^2 \arcsin (x)^2-\frac {1}{2} x \sqrt {1-x^2} \arcsin (x)^3+\frac {\arcsin (x)^4}{8} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.82 \[ \int \frac {x^2 \arcsin (x)^3}{\sqrt {1-x^2}} \, dx=\frac {1}{8} \left (-3 x^2+6 x \sqrt {1-x^2} \arcsin (x)+\left (-3+6 x^2\right ) \arcsin (x)^2-4 x \sqrt {1-x^2} \arcsin (x)^3+\arcsin (x)^4\right ) \]
[In]
[Out]
Time = 0.34 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.95
method | result | size |
default | \(\frac {\arcsin \left (x \right )^{3} \left (-x \sqrt {-x^{2}+1}+\arcsin \left (x \right )\right )}{2}+\frac {3 \arcsin \left (x \right )^{2} \left (x^{2}-1\right )}{4}+\frac {3 \arcsin \left (x \right ) \left (x \sqrt {-x^{2}+1}+\arcsin \left (x \right )\right )}{4}-\frac {3 \arcsin \left (x \right )^{2}}{8}-\frac {3 x^{2}}{8}-\frac {3 \arcsin \left (x \right )^{4}}{8}\) | \(69\) |
[In]
[Out]
none
Time = 0.25 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.67 \[ \int \frac {x^2 \arcsin (x)^3}{\sqrt {1-x^2}} \, dx=\frac {1}{8} \, \arcsin \left (x\right )^{4} + \frac {3}{8} \, {\left (2 \, x^{2} - 1\right )} \arcsin \left (x\right )^{2} - \frac {3}{8} \, x^{2} - \frac {1}{4} \, {\left (2 \, x \arcsin \left (x\right )^{3} - 3 \, x \arcsin \left (x\right )\right )} \sqrt {-x^{2} + 1} \]
[In]
[Out]
Time = 0.21 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.90 \[ \int \frac {x^2 \arcsin (x)^3}{\sqrt {1-x^2}} \, dx=\frac {3 x^{2} \operatorname {asin}^{2}{\left (x \right )}}{4} - \frac {3 x^{2}}{8} - \frac {x \sqrt {1 - x^{2}} \operatorname {asin}^{3}{\left (x \right )}}{2} + \frac {3 x \sqrt {1 - x^{2}} \operatorname {asin}{\left (x \right )}}{4} + \frac {\operatorname {asin}^{4}{\left (x \right )}}{8} - \frac {3 \operatorname {asin}^{2}{\left (x \right )}}{8} \]
[In]
[Out]
\[ \int \frac {x^2 \arcsin (x)^3}{\sqrt {1-x^2}} \, dx=\int { \frac {x^{2} \arcsin \left (x\right )^{3}}{\sqrt {-x^{2} + 1}} \,d x } \]
[In]
[Out]
none
Time = 0.30 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.82 \[ \int \frac {x^2 \arcsin (x)^3}{\sqrt {1-x^2}} \, dx=-\frac {1}{2} \, \sqrt {-x^{2} + 1} x \arcsin \left (x\right )^{3} + \frac {1}{8} \, \arcsin \left (x\right )^{4} + \frac {3}{4} \, {\left (x^{2} - 1\right )} \arcsin \left (x\right )^{2} + \frac {3}{4} \, \sqrt {-x^{2} + 1} x \arcsin \left (x\right ) - \frac {3}{8} \, x^{2} + \frac {3}{8} \, \arcsin \left (x\right )^{2} + \frac {3}{16} \]
[In]
[Out]
Timed out. \[ \int \frac {x^2 \arcsin (x)^3}{\sqrt {1-x^2}} \, dx=\int \frac {x^2\,{\mathrm {asin}\left (x\right )}^3}{\sqrt {1-x^2}} \,d x \]
[In]
[Out]